The complete mathematics formulary

Combinatorial analysis and enumerating formulas

All the formulas that follow will always have two cases:

• without repetition
• with repetition : in this case there will be a bar above to mean "with repetition"

For example, if we note the arrangements without repetition \(A_n\), we will note \(\overline{A_n}\) those with possible repetition.

Permutations

The number of permutations of the elements of a set (without repetition)

For any set \(E\) of \(n\) elements, the number of possible permutations without repetition is:

The number of permutations of the elements of a set (with repetition)

For any set \(E = \{e_1, e_2, e_3, \ ..., \ e_n \}\) with \(k_1, k_2, k_3, ...,k_{n}\) the number of occurrences of each element, the number of possible permutations is:

Arrangements (with a certain order)

The number of arrangements of the elements of a set (without repetition)

The number of arrangements without repetition of \(p\) elements taken from a set of \(n\) are worth:

The number of arrangements of the elements of a set (with repetition)

The number of arrangements with repetition of \(p\) elements taken from a set of \(n\) are worth:

Combinations (without any order)

The number of ways to take distinct elements from a set (without repetition)

The number of ways to take \(p\) elements (distinct and without repetition) in a set of \(n\) elements are worth:

(\(\Longrightarrow\) voir les propriétés du binôme )

The number of ways to take distinct elements from a set (with repetition)

The number of ways to take \(p\) elements (distinct and with repetition) in a set of \(n\) elements are worth:

The number of possible parts of a set

The number of possible parts of a set \(E = \{e_1, e_2, e_3, \ ..., e_n\}\), that's to say :

is worth :

Summary of combinatorics and enumeration formulas

The geometrical identity

We call geometrical identity, or Bernouilli's formula , the following expression:

The Newton's binomial \(: (a + b)^n \)

Newton's binomial tells us that :

We can use the Pascal's triangle to find binomials coefficients \(\binom{n}{p}\).

The properties of fractions

Elementary operations

Multiplication

Multiplying fractions together is the same as multiplying all the numerators (resp. all the denominators) together.

Division

Dividing by a fraction is the same as multiplying by its inverse.

Addition / substraction

Adding (resp. subtracting) fractions together requires putting them under a common denominator.

Properties

Cross product

Ratio between respectives numerators and denominators

The same ratios are possibles replacing all \( (+) \) by \( (-) \).

Ratio between respectives sums an differences

Addition of numerators and denominators

The same relation is possible replacing \( (+) \) by \( (-) \).

1. Generalization

On the whole, with a serie of \(n \) numerators and \(m \) denominators:

$$ \forall F \in \mathbb{Q}, \enspace \forall (a, c, e ...) \in \hspace{0.04em} \mathbb{N}^n, \enspace (b, d, f...) \in \hspace{0.04em} \mathbb{N}^m, \enspace \ \Bigl \{ (b \textcolor{rgb(85, 109, 229)}{\pm} d \textcolor{rgb(54 152 46)}{\pm} f \textcolor{rgb(192 52 52)}{\pm} ...) \Bigr \} \ \neq 0, $$
$$ F = \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \ ... \ \Longrightarrow F = \frac{a \textcolor{rgb(85, 109, 229)}{\pm} c \textcolor{rgb(54 152 46)}{\pm} e \textcolor{rgb(192 52 52)}{\pm} \ ...}{b \textcolor{rgb(85, 109, 229)}{\pm} d \textcolor{rgb(54 152 46)}{\pm} f \textcolor{rgb(192 52 52)}{\pm} \ ...}$$
$$ (\text{with the same-colour signs being the same}) $$

Recap table of the properties formulas of fractions

The properties of matrices

For what follows, it is important to establish the following definitions:

1. Operations on matrices
1. Matrices addition

Let \((A,B) \in \hspace{0.03em} \mathcal{M}_{n,p} (\mathbb{K})^2\) be two matrices of the same size.

$$ \forall (i, j) \in [\![1, n]\!] \times [\![1, p]\!],$$
$$(A + B)_{i,j} = a_{i,j} + b_{i,j} $$

In other words, we add each element of the left matrix with the element located at the same position of the right one:

$$ A + B = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} & \dots & a_{1, p} \\ a_{2,1} & a_{2,2} & a_{2,3} & \dots & a_{2, p} \\ \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \ddots & \hspace{0.5em} \vdots \\ a_{n,1} & a_{n,2} & a_{n,3} & \dots & a_{n, p} \end{pmatrix} + \begin{pmatrix} b_{1,1} & b_{1,2} & b_{1,3} & \dots & b_{1, p} \\ b_{2,1} & b_{2,2} & b_{2,3} & \dots & b_{2, p} \\ \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \ddots & \hspace{0.5em} \vdots \\ b_{n,1} & b_{n,2} & b_{n,3} & \dots & b_{n, p} \end{pmatrix} $$
$$ A + B = \begin{pmatrix} a_{1,1} + b_{1,1} & a_{1,2} + b_{1,2} & a_{1,3} + b_{1,3} & \dots & a_{1, p} + b_{1, p} \\ a_{2,1} + b_{2,1} & a_{2,2} + b_{2,2} & a_{2,3} + b_{2,3} & \dots & a_{2, p} + b_{2,p} \\ \hspace{2em} \vdots & \hspace{2em} \vdots & \hspace{2em} \vdots & \ddots & \hspace{2em} \vdots \\ a_{n,1} + b_{n,1} & a_{n,2} + b_{n,2} & a_{n,3} + b_{n,3} & \dots & a_{n, p} + b_{n,p} \end{pmatrix} $$
2. Matrices product

Let \(A \in \hspace{0.03em} \mathcal{M}_{n,p} (\mathbb{K})\) and \(B \in \hspace{0.03em} \mathcal{M}_{p,q} (\mathbb{K})\) be two matrices.

To multiply two matrices, we need the left matrix to have the same number of columns as the number of rows of the right one (here \(p\)). As a result, we obtain a matrix \(AB \in \hspace{0.03em} \mathcal{M}_{n,q} (\mathbb{K})\), so having \(n\) lines and \(q\) columns.

$$ \forall (i, j) \in [\![1, n]\!] \times [\![1, q]\!],$$
$$(A \times B)_{i,j} = \sum_{k = 1}^p a_{i,k} \times b_{k,j} $$

For example:

$$ A \times B = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} & \dots & a_{1, p} \\ a_{2,1} & a_{2,2} & a_{2,3} & \dots & a_{2, p} \\ \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \ddots & \hspace{0.5em} \vdots \\ a_{n,1} & a_{n,2} & a_{n,3} & \dots & a_{n, p} \end{pmatrix} \times \begin{pmatrix} b_{1,1} & b_{1,2} & b_{1,3} & \dots & b_{1, q} \\ b_{2,1} & b_{2,2} & b_{2,3} & \dots & b_{2, q} \\ \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \ddots & \hspace{0.5em} \vdots \\ b_{p,1} & b_{p,2} & b_{p,3} & \dots & b_{p, q} \end{pmatrix} $$
$$ A \times B = \begin{pmatrix} \Bigl[a_{1,1} b_{1,1} + a_{1,2} b_{2,1} \ + \ ... \ + \ a_{1,p} b_{p,1} \Bigr] & \Bigl[a_{1,1} b_{1,2} + a_{1,2} b_{2,2} \ + \ ... \ + \ a_{1,p} b_{p,2}\Bigr] & \hspace{1em} \dots \dots \dots \hspace{1em} & \Bigl[a_{1,1} b_{1,q} + a_{1,2} b_{2,q} \ + \ ... \ + \ a_{1,p} b_{p,q}\Bigr] \\ \Bigl[a_{2,1} b_{1,1} + a_{2,2} b_{2,1} \ + \ ... \ + \ a_{2,p} b_{p,1}\Bigr] & \Bigl[a_{2,1} b_{1,2} + a_{2,2} b_{2,2} \ + \ ... \ + \ a_{2,p} b_{p,2}\Bigr] & \hspace{1em} \dots \dots \dots \hspace{1em} & \Bigl[a_{2,1} b_{1,q} + a_{2,2} b_{2,q} \ + \ ... \ + \ a_{2,p} b_{p,q}\Bigr] \\ \hspace{8em} \vdots & \hspace{8em} \vdots & \hspace{1em} \ddots & \hspace{8em} \vdots \\ \hspace{8em} \vdots & \hspace{8em} \vdots & \hspace{1em} \ddots & \hspace{8em} \vdots \\ \Bigl[a_{n,1} b_{1,1} + a_{n,2} b_{2,1} \ + \ ... \ + \ a_{n,p} b_{p,1}\Bigr] & \Bigl[a_{n,1} b_{1,2} + a_{2,2} b_{2,2} \ + \ ... \ + \ a_{n,p} b_{p,2}\Bigr] & \hspace{1em} \dots \dots \dots \hspace{1em} & \Bigl[a_{n,1} b_{1,q} + a_{n,2} b_{2,q} \ + \ ... \ + \ a_{n,p} b_{p,q}\Bigr] \end{pmatrix} $$

Be careful, in a general way the matrices product does not have commutative law: \( (A \times B) \neq (B \times A) \) .

3. Multiplication of a matrix by a scalar \(\lambda\)

Let \(A \in \hspace{0.03em} \mathcal{M}_{n,p} (\mathbb{K})\) be a matrix.

When a matrix is multiplied by a scalar, it affects all its elements.

$$ \forall (i, j) \in [\![1, n]\!] \times [\![1, p]\!],$$
$$(\lambda A)_{i,j} = \lambda \ a_{i,j} $$

For example:

$$ A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} & \dots & a_{1, p} \\ a_{2,1} & a_{2,2} & a_{2,3} & \dots & a_{2, p} \\ \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \ddots & \hspace{0.5em} \vdots \\ a_{n,1} & a_{n,2} & a_{n,3} & \dots & a_{n, p} \end{pmatrix} $$
$$ \lambda A = \begin{pmatrix} \lambda \ a_{1,1} & \lambda \ a_{1,2} & \lambda \ a_{1,3} & \dots & \lambda \ a_{1, p} \\ \lambda \ a_{2,1} & \lambda \ a_{2,2} & \lambda \ a_{2,3} & \dots & \lambda \ a_{2, p} \\ \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \ddots & \hspace{0.5em} \vdots \\ \lambda \ a_{n,1} & \lambda \ a_{n,2} & \lambda \ a_{n,3} & \dots & \lambda \ a_{n, p} \end{pmatrix} $$
4. Linear combination of matrices

Let \((A,B) \in \hspace{0.03em} \mathcal{M}_{n,p} (\mathbb{K})^2\) be two matrices of the same size and \((\lambda, \mu) \in \hspace{0.04em} \mathbb{R}^2\).

With the previous properties of addition and multiplication by a scalar , we can create linear combinations and:

$$(\lambda A + \mu B)_{i,j} = \lambda \ a_{i,j} + \mu \ b_{i,j} $$
5. Matrix transposition

Let \(A \in \hspace{0.03em} \mathcal{M}_{n}(\mathbb{K})\) be a squared matrix of size \(n\).

Matrix transposition consists in reverse lines and columns indices for each elements. We note \(A^T\) (sometimes \(^t A\)) the transposed of the matrix \(A\).

$$ \forall (i, j) \in [\![1, n]\!]^2,$$
$$ \left(A^T \right)_{i,j} \hspace{0.03em} = a_{j,i} $$

For example:

$$ A = \begin{pmatrix} a_{1,1} & \textcolor{rgb(192 52 52)}{a_{1,2}} & \textcolor{rgb(192 52 52)}{a_{1,3}} & \textcolor{rgb(192 52 52)}{\dots} & \textcolor{rgb(192 52 52)}{a_{1, n}} \\ \textcolor{rgb(54 152 46)}{a_{2,1}} & a_{2,2} & \textcolor{rgb(192 52 52)}{a_{2,3}} & \textcolor{rgb(192 52 52)}{\dots} & \textcolor{rgb(192 52 52)}{a_{2, n}} \\ \textcolor{rgb(54 152 46)}{a_{3,1}} & \textcolor{rgb(54 152 46)}{a_{3,2}} & a_{3,3} & \textcolor{rgb(192 52 52)}{\dots} & \textcolor{rgb(192 52 52)}{a_{3, n}} \\ \hspace{1em} \textcolor{rgb(54 152 46)}{\vdots} & \hspace{1em} \textcolor{rgb(54 152 46)}{\vdots} & \hspace{1em} \textcolor{rgb(54 152 46)}{\vdots} & \ddots & \hspace{1em} \textcolor{rgb(192 52 52)}{\vdots} \\ \textcolor{rgb(54 152 46)}{a_{n,1}} & \textcolor{rgb(54 152 46)}{a_{n,2}} & \textcolor{rgb(54 152 46)}{a_{n,3}} & \textcolor{rgb(54 152 46)}{\dots} & a_{n, n} \\ \end{pmatrix} $$

So, its transposed is:

$$ A^T = \begin{pmatrix} a_{1,1} & \textcolor{rgb(54 152 46)}{a_{2,1}} & \textcolor{rgb(54 152 46)}{a_{3,1}} & \textcolor{rgb(54 152 46)}{\dots} & \textcolor{rgb(54 152 46)}{a_{n, 1}} \\ \textcolor{rgb(192 52 52)}{a_{1,2}} & a_{2,2} & \textcolor{rgb(54 152 46)}{a_{3,2}} & \textcolor{rgb(54 152 46)}{\dots} & \textcolor{rgb(54 152 46)}{a_{n, 2}} \\ \textcolor{rgb(192 52 52)}{a_{1,3}} & \textcolor{rgb(192 52 52)}{a_{2,3}} & a_{3,3} & \textcolor{rgb(54 152 46)}{\dots} & \textcolor{rgb(54 152 46)}{a_{n, 3}} \\ \hspace{0.8em} \textcolor{rgb(192 52 52)}{\vdots} & \hspace{0.8em} \textcolor{rgb(192 52 52)}{\vdots} & \hspace{0.8em} \textcolor{rgb(192 52 52)}{\vdots} & \ddots & \hspace{0.8em} \textcolor{rgb(54 152 46)}{\vdots} \\ \textcolor{rgb(192 52 52)}{a_{1,n}} & \textcolor{rgb(192 52 52)}{a_{2,n}} & \textcolor{rgb(192 52 52)}{a_{3,n}} & \textcolor{rgb(192 52 52)}{\dots} & a_{n, n} \\ \end{pmatrix} $$

Only the diagonal remains intact, because when \(i = j\), then \(a_{i,j} = a_{j,i}\).

6. Inversion of a matrix

Let \(A \in \hspace{0.03em} \mathcal{M}_{n,p} (\mathbb{K})\) be a matrix.

The inverse of the matrix \(A\) is the matrix written \(A^{-1}\) and the same size, such as: \(A A^{-1} = I_n\).




To check if a matrix can be inverted, we do have to compute its determinant, and:

$$ A \text{ is inversible } \Longleftrightarrow det(A) \neq 0 $$
7. Comatrix

Let \(A \in \hspace{0.03em} \mathcal{M}_{n}(\mathbb{K})\) be a squared matrix of size \(n\).

The comatrix of the matrix \(A\) is the matrix noted \(com(A)\), such as:

$$ \forall (i, j) \in [\![1, n]\!]^2,$$
$$ com(A)_{i,j} \hspace{0.03em} = C_{i,j} $$
$$ \text{où } \ \left \{ \begin{gather*} C_{i,j} : \text{cofactors of the element } a_{i, j} \\ M_{i, j} : \text{minor of the element } a_{i, j} \end{gather*} \right \} $$

\(C_{i, j}\) : cofactors of the element \(a_{i, j}\)

$$ C_{i,j} = (-1)^{i + j} \times det(M_{i, j}) $$

\(M_{i, j}\) : minor of the element \(a_{i, j}\)

The minor of \(a_{i, j}\) is the undermatrix of \(A\) without the line \(i\) and the column \(j\).

For example, starting from the following matrix \(A\), the minor \(\textcolor{rgb(85, 109, 229)}{M_{1,1}}\) appears in blue:

$$ A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} & \dots & a_{1, p} \\ a_{2,1} & \textcolor{rgb(85, 109, 229)}{a_{2,2}} & \textcolor{rgb(85, 109, 229)}{a_{2,3}} & \dots & \textcolor{rgb(85, 109, 229)}{a_{2, p}} \\ \hspace{0.5em} \vdots & \hspace{0.5em} \textcolor{rgb(85, 109, 229)}{\vdots} & \hspace{0.5em} \textcolor{rgb(85, 109, 229)}{\vdots} & \textcolor{rgb(85, 109, 229)}{\ddots} & \hspace{0.5em} \textcolor{rgb(85, 109, 229)}{\vdots} \\ a_{n,1} & \textcolor{rgb(85, 109, 229)}{a_{n,2}} & \textcolor{rgb(85, 109, 229)}{a_{n,3}} & \dots & \textcolor{rgb(85, 109, 229)}{a_{n, p}} \end{pmatrix} $$

So,

$$ \textcolor{rgb(85, 109, 229)}{ M_{1,1} = \begin{pmatrix} a_{2,2} & a_{2,3} & \dots & a_{2, p} \\ \vdots & \hspace{0.5em} \vdots & \ddots & \hspace{0.5em} \textcolor{rgb(85, 109, 229)}{\vdots} \\ a_{n,2} & a_{n,3} & \dots & a_{n, p} \end{pmatrix} } $$

For example, starting from the following matrix \(A\):

$$ A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \end{pmatrix} $$

Its comatrix is worth:

$$ com(A) = \begin{pmatrix} \textcolor{rgb(54 152 46)}{+}\begin{vmatrix} a_{2,2} & a_{2,3} \\ a_{3,2} & a_{3,3} \end{vmatrix} & \textcolor{rgb(192 52 52)}{-}\begin{vmatrix} a_{2,1} & a_{2,3} \\ a_{3,1} & a_{3,3} \end{vmatrix} & \textcolor{rgb(54 152 46)}{+}\begin{vmatrix} a_{2,1} & a_{2,2} \\ a_{3,1} & a_{3,2} \end{vmatrix} \\ \textcolor{rgb(192 52 52)}{-}\begin{vmatrix} a_{1,2} & a_{1,3} \\ a_{3,2} & a_{3,3} \end{vmatrix} & \textcolor{rgb(54 152 46)}{+}\begin{vmatrix} a_{1,1} & a_{1,3} \\ a_{3,1} & a_{3,3} \end{vmatrix} & \textcolor{rgb(192 52 52)}{-}\begin{vmatrix} a_{1,1} & a_{1,2} \\ a_{3,1} & a_{3,2} \end{vmatrix} \\ \textcolor{rgb(54 152 46)}{+}\begin{vmatrix} a_{1,2} & a_{1,3} \\ a_{2,2} & a_{2,3} \end{vmatrix} & \textcolor{rgb(192 52 52)}{-}\begin{vmatrix} a_{1,1} & a_{1,3} \\ a_{2,1} & a_{2,3} \end{vmatrix} & \textcolor{rgb(54 152 46)}{+}\begin{vmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{vmatrix} \end{pmatrix} $$
8. Matricial writing of a system of linear equations

A system of linear equations \( (S)\), where the unknown are the variables \(x_{i,j}\), can be written as a product matrix system :

$$ (S) \enspace \left \{ \begin{gather*} a_1 x_{1,1} + a_2 x_{1,2} + a_3 x_{1,3} + \hspace{0.1em}... \hspace{0.1em}+ a_n x_{1,p} = b_1 \\ a_1 x_{2,1} + a_2 x_{2,2} + a_3 x_{2,3} + \hspace{0.1em}... \hspace{0.1em}+ a_n x_{2,p} = b_2 \\ ........................ ............. \ = \ ..\\ a_1 x_{n,1} + a_2 x_{n,2} + a_3 x_{n,3} + \hspace{0.1em}... \hspace{0.1em}+ a_n x_{n,p} = b_n \\ \end{gather*} \right \} $$
$$ \Longleftrightarrow$$
$$ \underbrace{ \begin{pmatrix} x_{1,1} & x_{1,2} & x_{1,3} & \dots & x_{1, p} \\ x_{2,1} & x_{2,2} & x_{2,3} & \dots & x_{2, p} \\ \hspace{0.8em} \vdots & \hspace{0.8em} \vdots & \hspace{0.8em} \vdots & \ddots & \hspace{0.8em} \vdots \\ x_{n,1} & x_{n,2} & x_{n,3} & \dots & x_{n, p} \\ \end{pmatrix} } _\text{X} \times \underbrace{ \begin{pmatrix} a_1 \\ a_2 \\ \hspace{0.3em}\vdots \\ a_n \end{pmatrix} } _\text{A} = \underbrace{ \begin{pmatrix} b_1 \\ b_2 \\ \hspace{0.3em}\vdots \\ b_n \end{pmatrix} } _\text{B} \ \Longleftrightarrow \ MA = B, \ \text{with} \enspace \left \{ \begin{gather*} X \in \hspace{0.03em} \mathcal{M}_{n,p} (\mathbb{K}) \\ A \in \hspace{0.03em} \mathcal{M}_{1,p} (\mathbb{K}) \\ B \in \hspace{0.03em} \mathcal{M}_{1,p} (\mathbb{K}) \end{gather*} \right \} $$
9. Trace of a matrix

Let \(A \in \hspace{0.03em} \mathcal{M}_{n}(\mathbb{K})\) be a squared matrix of size \(n\).

We call the trace of a matrix, the sum of all its diagonal elements:

$$ A = \begin{pmatrix} \textcolor{rgb(85, 109, 229)}{a_{1,1}} & a_{1,2} & a_{1,3} & \dots & a_{1,n} \\ a_{2,1} & \textcolor{rgb(85, 109, 229)}{a_{2,2}} & a_{2,3} & \dots & a_{2,n} \\ a_{3,1} & a_{3,2} & \textcolor{rgb(85, 109, 229)}{a_{3,3}} & \dots & a_{3,n} \\ \hspace{0.1em}\vdots & \hspace{0.1em} \vdots & \hspace{0.1em} \vdots & \textcolor{rgb(85, 109, 229)}{\ddots} & \hspace{0.1em} \vdots \\ a_{n,1} & a_{n,2} & a_{n,3} & \dots & \textcolor{rgb(85, 109, 229)}{a_{n,n}} \end{pmatrix} $$
$$Tr(A) = \sum_{k = 1}^n a_{k,k} = a_{1,1} + a_{2,2} \ + \ ... \ + a_{n,n}$$
2. Specific matrices
1. Diagonal matrix

A diagonal is a squared matrix where all the elements are \(0\) except on the main diagonal:

$$ D_n = \begin{pmatrix} \textcolor{rgb(85, 109, 229)}{d_{1,1}} & 0 & 0 & \dots & 0 \\ 0 & \textcolor{rgb(85, 109, 229)}{d_{2,2}} & 0 & \dots & 0 \\ 0 & 0 & \textcolor{rgb(85, 109, 229)}{d_{3,3}} & \dots & 0 \\ \hspace{0.1em}\vdots & \hspace{0.1em} \vdots & \hspace{0.1em} \vdots & \textcolor{rgb(85, 109, 229)}{\ddots} & \hspace{0.1em} \vdots \\ 0 & 0 & 0 & \dots & \textcolor{rgb(85, 109, 229)}{d_{n,n}} \end{pmatrix} $$
$$ \forall (i, j) \in [\![1, n]\!]^2, \ (i \neq j) \Longrightarrow d_{i,j} = 0$$

We also note the diagonal matrix \(D_n\) only in relation with its diagonal elements : \(D_n = diag(\lambda_1, \lambda_2, \ ..., \lambda_n)\).

2. Identity matrix

Identity matrix \(I_n\) are defined as follows:

$$ I_n = \begin{pmatrix} \textcolor{rgb(85, 109, 229)}{1} & 0 & 0 & \dots & 0 \\ 0 & \textcolor{rgb(85, 109, 229)}{1} & 0 & \dots & 0 \\ 0 & 0 & \textcolor{rgb(85, 109, 229)}{1} & \dots & 0 \\ \vdots & \vdots & \vdots & \textcolor{rgb(85, 109, 229)}{\ddots} & \vdots \\ 0 & 0 & 0 & \dots & \textcolor{rgb(85, 109, 229)}{1} \\ \end{pmatrix} $$

It is the square matrix of size \(n\) having the value \(1\) on its main diagonal, and \(0\) everywhere else. It's a specific case of diagonal matrix. For example,

$$ I_3 = \begin{pmatrix} \textcolor{rgb(85, 109, 229)}{1} & 0 & 0 \\ 0 & \textcolor{rgb(85, 109, 229)}{1} & 0 \\ 0 & 0 & \textcolor{rgb(85, 109, 229)}{1} \end{pmatrix} $$
3. Matrix of ones

Matrix of ones \(J_n\) is the square matrix of size \(n\) where all elements are worth \(1\):

$$ J_n = \begin{pmatrix} 1 & 1 & 1 & \dots & 1 \\ 1 & 1 & 1 & \dots & 1 \\ 1 & 1 & 1 & \dots & 1 \\ \vdots & \vdots & \vdots & \textcolor{rgb(85, 109, 229)}{\ddots} & \vdots \\ 1 & 1 & 1 & \dots & 1 \\ \end{pmatrix} $$

So, for exemple \(J_3\) is worth:

$$ J_3 = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{pmatrix} $$

Matrix product

Associativity

Distributivity

Bilinearity

Multiplication by the identity

Diagonal product matrix

1. Product of two diagonal matrix
$$ \forall \Bigl[ D_1 = diag(\lambda_1, \lambda_2, \ ..., \lambda_n), \ D_2 = diag(\mu_1, \mu_2, \ ..., \mu_n) \Bigr] \in \hspace{0.03em} \mathcal{M}_{n}(\mathbb{K})^2, $$
$$ D_1 \times D_2 = D_2 \times D_1 = diag \left(\lambda_1 \mu_1, \lambda_2 \mu_2, \ ..., \lambda_n \mu_n \right) $$
2. A diagonal matrix raised to the power of \(n\)
$$ \forall \Bigl[ D = diag(\lambda_1, \lambda_2, \ ..., \lambda_n) \Bigr] \in \hspace{0.03em} \mathcal{M}_{n}(\mathbb{K}), $$
$$ D^m = diag \left(\lambda_1^m, \lambda_2^m, \ ..., \lambda_n^m \right) $$

Matrix comatrix, transposed and determinant

Link between matrix, comatrix, transposed and determinant

Matrix transposition

Linearity of transposition

Transposed of a product

Inversion of matrix

Computation of l'inverse

Inverse of the inverse

Inverse of a transposed matrix

Inverse of a product

Both expressions \((9)\) and \((10)\) have the same behaviour:

So, the order of transposition or inversion has no importance,

Traces of matrix

Linearity of the trace

Trace of a product

Powers of matrices

Powers of the matrix of ones

Let \(J_n\) bet the matrix of ones of size \(n\).

Newton's binomial and Geometrical identity

Recap table of the properties of matrices

The properties of the binom\(: \binom{n}{p}\)

Let \((p,n) \in \hspace{0.04em}\mathbb{N}^2 \) be two natural numbers with \( p \leqslant n \).

We call \(\binom{n}{p} \) ("\( p \) among \( n \)") to number of ways to take \( p \) elements among a set of \( n \) elements.

We also call it the binom, and meets the following definition:

"0 among n" / "n among n"

"1 among n"

Symmetry

The pawn's formula

The Pascal's formula

Horizontal sum from 0 to n

Vertical sum from r to n

Recap table of the formulas of the binom

The properties of the powers of x (for natural exponents)

Let \( n\in \mathbb{N}\) be natural number and \( x \in \mathbb{R}\) a real number.

We call \(x^n\) a number \(x\) multiplied \(n\) times by itself:

All these formulas are demonstrated only for natural exponents \((n \in \mathbb{N})\).

Product/quotient of powers (having the same base)

Number to the power of zero

Inverse of a power

Power of a product/quotient

Power of a power

Recap table of the formulas of the powers of x

Analytical geometry in space

Let be \((O, \overrightarrow{i}, \overrightarrow{j}, \overrightarrow{k})\) an orthonormal coordinate system in space.

Lines and plans

Parametric equation of a straight line

The parametric equation of a straight line \(\mathcal{D}\) in space, passing through a point \(A\bigl(x_0, y_0, z_0\bigr)\) and directed by a vector \(\vec{u}\begin{pmatrix} a\\ b \\c \end{pmatrix} \) (with \(a, b, c \) all three non-zero) is:

Equation of a plan

The equation of a place \((\mathcal{P})\) in space, passing through a point \(A\bigl(x_0, y_0, z_0\bigr)\) and orthogonal to a vector \(\vec{n}\begin{pmatrix} a\\ b \\c \end{pmatrix}\) (with \(a, b, c \) all three non-zero) is:

Distance from a point to a plan

Let \(\mathcal{P}\) be a plan orthogonal to a vector \(\vec{n}\begin{pmatrix} a\\ b \\c \end{pmatrix}\) (with \(a, b, c \) all three non-zero), having as equation:

The distance from a point \(A\bigl(x_0, y_0, z_0\bigr)\) in relation to this plan \((\mathcal{P})\) orthogonally projecting on this plan at point \(H\bigl(x, y, z\bigr)\) is worth:

Projection of a vector on a plan

The projection of a vector \(\vec{u}\) on a plan \(\mathcal{P}\) orthogonal to a vector \(\vec{n}\) is worth:

Projection of a sum of vectors upon a plan

The projection of a sum of vectors in a plan is the sum of each vector's projection

Geometrical figures

Equation of a sphere

The sphere \((\mathcal{S})\) having a radius \(R\) and centered at point \(A\bigl(x_0, y_0, z_0\bigr)\) has for equation in space:

Equation of a cylinder

The cylinder\((\mathcal{C})\) having a radius \(r\) and centered at point \(A\bigl(x_0, y_0, z_0\bigr)\) has for equation in space:

Equation of a cone

The cone \((\mathcal{C})\) having a half-angle \( \theta\), and centered at point \(A\bigl(x_0, y_0, z_0\bigr)\) has for equation in space:

Coordinates system

From cartesian to spherical coordinates

1. Longitude-latitude coordinates

$$ \Biggl \{ \begin{gather*} x = R \ \cos(\varphi) \ \cos(\theta) \\ y = R \ \cos(\varphi) \ \sin(\theta) \\ z = R \ \sin(\varphi) \end{gather*} \qquad (\theta : longitude- \varphi : latitude) $$
$$\text{with} \enspace \left \{ \begin{gather*} R =\sqrt{x^2 +y^2 +z^2 } \\ \theta = \operatorname{Arctan} \left( \frac{y}{x} \right) \\ \varphi = \operatorname{Arcsin} \left( \frac{z}{R} \right) \end{gather*} \right \}$$
2. Longitude-colatitude coordinates

$$ \Biggl \{ \begin{gather*} x = R \ \sin(\psi) \ \cos(\theta) \\ y = R \ \sin(\psi) \ \sin(\theta) \\ z = R \ \cos(\psi) \end{gather*} \qquad (\theta : longitude- \psi : colatitude) $$
$$\text{with} \enspace \left \{ \begin{gather*} R =\sqrt{x^2 +y^2 +z^2 } \\ \theta = \operatorname{Arctan} \left( \frac{y}{x} \right) \\ \psi = \operatorname{Arccos} \left( \frac{z}{R} \right) \end{gather*} \right \}$$

The calculation of Pi \( (\pi) \) by geometrical method

Let us calculate the value of \(\pi \) using a geometrical method.

To do this, we will process by successive iterations.

The number \(\pi \) corresponds to the ration between the half-perimeter of a circle having as radius \( R = 1 \).

The theoretical formula that we will proove is:

(with \( n\) the number of \( 2\) inside the second square root)

Calculations of surfaces and volumes by integration

Surfaces

The calculation of the surface of a circle

The surface of the circle of radius \( R\) is worth:

The calculation of the surface of a sphere

The surface of a sphere of radius \( R\) is worth:

The calculation of the surface of a cone

The cone is a pyramid with a circular base. It is characterized by its height \( h\), its radius \( r\) from its base and its apothem \(a\).

Volumes

The calculation of the volume of a sphere

The volume of a sphere of radius \( R\) is worth:

The calculation of the volume of a cone

The volume of a cone of height \( h\) with a circular base of radius \( r\) is worth:

The Euler's trigonometric formulas

Euler's formula: exponential form of a complex number

Euler's trigonometric formulas

The geometric laws of the triangle

In the context of an unspecified triangle \(\{a, b, c\}\), with each angle in front of its respective length, such as:

And such as the following figure:

Relationships between lengths and angles

Law of \sines

Al-Kashi's theorem

Methods of calculating areas

Area formula

Héron's formula

Héron's formula tells us that:

For any triplet of lengths \((a, b, c)\) of a triangle:

The Moivre's formula

Let \( n \in \mathbb{Z}\) be an integer, and \( z \in \mathbb{C}\) be a complex number having as a module \( |z|= 1\) under its trigonometric form, such as:

Moivre's formula tells us that:

The potency of a point in relation to a circle

Case 1: Internal point of a circle

If a point \( O\) is the intersection between two ropes \( (AB )\) and \( (CD )\) of a circle inside it, and such as the following figure:

So, the two triangles \( AOC \) and \( DOB \) are similar , and the product between the lengths respectively proportional starting from the point \(O\) is worth:

Case 2: External point of a circle

If a point \( O\) is the intersection between two ropes \( (AC )\) and \( (DB )\) of a circle outside it, and such as the following figure:

So, the two triangles \( ABO \) and \( DCO \) are similar , and the product between the lengths respectively proportional starting from the point \(O\) is worth:

Case 3: External point of a circle with one tangent

If a point \( O\) is the intersection between a rope \( (AC )\) of a circle and a tangent to the same circle passing through \( T\), and such as the following figure:

So, the two triangles \( OAT \) and \( CTO \) are similar , and the product between the lengths respectively proportional starting from the point \(O\) is worth:

Case 4: External point of a circle with two tangents

If a point \( O\) is the intersection between two tangents to the circle passing respectively through \( T\) and \( T'\), and such as the following figure:

So, the triangle \( OTT' \) is an isosceles triangle, and in this case:

The properties of complex numbers

Modules\(:|z|\)

We write \( |z| \) the module of a complex number \( z \).

Let be \( (x, y) \in \hspace{0.04em} \mathbb{R}^2, \enspace z \in \mathbb{C}, \enspace \Biggl \{ \begin{gather*} z = x + iy \\ |z| = \sqrt{x^2 + y^2 } \end{gather*} \)

Modules of the opposite and the conjugate

Module of a product

In the same way, we will have:

Module of a complex number raised to an integer power

Arguments\(: \arg(z) \)

Let be \( (x, y) \in \hspace{0.04em} \mathbb{R}^2, \enspace z \in \mathbb{C}, \)

We write \( \arg(z) \) the argument of a cmplex number \( z \).

Arguments of conjugate and opposite

Argument of a product

Argument of an inverse

Argument of a quotient

Argument of a complex number raised to an integer power

Conjugates\(: \overline{z}\)

We write \( \overline{z} \) the conjugate of a complex number \( z \).

Let be \( (x, y) \in \hspace{0.04em} \mathbb{R}^2, \enspace z \in \mathbb{C}, \enspace \Biggl \{ \begin{gather*} z = x + iy \\ \overline{z} = x -iy \end{gather*} \)

Conjugate of a sum

In the same way,

Conjugate of a product

Conjugate of a quotient

Complex number multiplied by its conjugate

Conjugate of a complex number raised to an integer power

Recap table of the properties of the complex numbers formulas

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The properties of the scalar product

Let \( \vec{u}\) and \( \vec{v}\) be two vectors.

We represent \(||\vec{u}|| \) and \(||\vec{v}|| \) as the respective magnitudes of \( \vec{u}\) and \( \vec{v}\), and \( (\vec{u} , \vec{v})\) the forming angle by these two vectors.

We call the scalar product \( \vec{u}\cdot\vec{v} \), the real number resulting from:

This is the magnitude of the orthogonal projection of the vector \( \vec{u}\) onto the vector \( \vec{v}\), multiplied by the magnitude of the vector \( \vec{v}\).

Commutative law

Orthogonal vectors

Squared vector

Scalar product by the vectors coordinates

Bilinearity

Furthermore, performing the scalar product \( (\lambda\vec{u})\cdot(\mu\vec{v})\), we do obtain:

Distributive law in relation to the addition

And also the distributive law to the left:

Remarkable identities

These are the same formulas as remarkable identities .

Expression depending on norms

Projection of a vector onto a vector

The vectorial projection of \(\vec{u}\) onto \(\vec{v}\) is worth:

Recap table of the properties of the scalar product

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The properties of the vector product

Let \( \vec{u}\) and \( \vec{v}\) be two non-null vectors.

We call vector product (or cross product of two vectors) \( \vec{u} \land \vec{v} \), a new vector extracted from \( \vec{u}\) and \( \vec{v}\) such as:

The vector product \( \vec{u} \land \vec{v} \) is orthogonal to both vectors \( \vec{u}\) and \( \vec{v}\).

Cartesian coordinates

Norm

Lagrange's identity

Collinearity of two vectors

Anticommutative law

Distributive law in relation to the addition

And also the distributive law to the left:

Freedom of the constant

Gibbs' formula

Jacobi's identity

Recap table of the properties of the vector product

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The Pythagorean theorem and its reciprocal

The Pythagorean theorem

The Pythagorean theorem tells us that:

In a right-angle triangle, the square of the hypotenuse is equal to the sum of the square on the sides of the right angle.

Let us take any triangle \(\{a, b, c\}\), right-angled between \( a\) and \( b\) such as the following figure:

We do have the following equation:

The Pythagorean theorem reciprocal

Its reciprocal tells us the contrary:

In any triangle \(\{a, b, c\}\), where \( c\) is the longest side:

The Pythagorean theorem equivalence

Both previous implications then form the following equivalence:

The right-angled triangle inscribed in a circle

In a circle, if an inscribed triangle has the diameter of this circle as its longest side, then this triangle will be right-angled.

The similarity of two triangles

Two triangles are said to be similar when they have their respective lengths proportional and their respective angles equal.

This implies a ratio \(k\) between the respective lengths, corresponding to an enlargement, a shrinkage or a preservation (if \(k = 1\)).

There are mainly three cases of triangle similarities.

Case 1: Three respective proportional sides

Two triangles are similar if their three respective sides are proportional .

Case 2: Two pairwise equal angles

Two triangles are similar if they have at least two pairwise equal angles .

Case 3: A common angle and two proportional lengths

Two triangles are similar if they have a common angle and their two respective lengths are proportional .

The Thales' theorem and its reciprocal

In an triangle, The Thales' theorem implies proportionality ratios between sides length.

Let be an ordinary triangle, in which we have drew a parallel of one of the sides, and such as the following figure:

The Thales' theorem

The Thales' theorem tells us that, in a triangle \(ABD\), if it exists any straight line \(BC\) intersecting \(AD\) and \(AE\) respectively on \(B\) and \(C\) such as \(BC \parallel DE\), then it implies lengths ratios between sides:

As well, it is possible to apply it in a reversed triangle, such as the following figure:

1. Extension of the theorem

Finally, by extension of the Thales theorem , if we have established these following equalities between ratios:

$$ \frac{AB}{AD} = \frac{AB'}{AD'} = \frac{BB'}{DD'} $$

These ratios will apply for all projections (not necessarily orthogonal) on the side \( DD' \).


$$ \frac{AB}{AD} = \frac{AB_1}{AD_1} = \frac{AB_2}{AD_2}= \frac{AB'}{AD'} $$

The Thales' theorem reciprocal

The Thales' theorem reciprocal tells us that if in any triangle \( ADE \), with any straight line \( BC \) intersecting \( AD \) and \( AE \) respectively on \( B \) and \( C \), such as the following figure:

Then, it implies a parallelism relation:

Furthermore, only one over three ratio equalities is enough to imply the parallelism, and:

The Thales' theorem equivalence

Both previous implications then form the following equivalence:

The trigonometric operations formulas

The duplication formulas

The sines and cosines function

As well, hare are their respective expressions depending on \( \tan(\alpha)\) :

The tangent function

The addition of parameters formulas

The sines and cosines function

The tangent function

The binomial's formulas

The product towards sum transformation formulas

The sines and cosines function

The sum towards product transformation formulas

The sines and cosines function

Furthermore,

The tangent function

Recap table of the trigonometric duplication and addition formulas

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The antiderivatives of trigonometric functions

The basic trigonometric functions\(:\sin(x), \cos(x), \tan(x)\)

The sines function\(: {\displaystyle \int^x} \sin(t) \ dt \)

The \( \sin(x) \) function is defined as follows:

Its general antiderivative is:

The cosines function\(: {\displaystyle \int^x} \cos(t) \ dt \)

The \( \cos(x) \) function is defined as follows:

Its general antiderivative is:

The tangent function\(: {\displaystyle \int^x} \tan(t) \ dt \)

The \( \tan(x) \) function is defined as follows:

Its general antiderivative is:

The basic trigonometric reciprocal functions\(:\operatorname{Arcsin}(x)\), \(\operatorname{Arccos}(x)\), \( \operatorname{Arctan}(x)\)

The arcsines function\(: {\displaystyle \int^x} \operatorname{Arcsin}(t) \ dt \)

The \( \operatorname{Arcsin}(x) \) is the reciprocal function of the \( \sin(x) \) function , it is defined as follows:

Its general antiderivative is:

The arccosines function\(: {\displaystyle \int^x} \operatorname{Arccos}(t) \ dt \)

The \( \operatorname{Arccos}(x) \) function is the reciprocal function of the \( \cos(x) \) function , it is defined as follows:

Its general antiderivative is:

The arctangent function\(: {\displaystyle \int^x} \operatorname{Arctan}(t) \ dt \)

The \( \operatorname{Arctan}(x) \) function is the reciprocal function of the \( \tan(x) \) function , it is defined as follows:

Its general antiderivative is:

The secant trigonometric functions: \(\csc(x), \sec(x), \cot(x)\)

The cosecant function\(: {\displaystyle \int^x} \csc(t) \ dt \)

The \( \csc(x) \) function is defined as follows:

Its general antiderivatives are:

1. By the secant trigonometric functions
$$ \forall k \in \mathbb{Z}, \ \forall x \in \Bigl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \bigl \{ k\pi \bigr \} \Bigr],$$
$$\int^x \csc(t) \ dt = \ln \left|\csc(x) -\cot(x) \right|$$
2. By applying Bioche's rules
$$ \forall k \in \mathbb{Z}, \ \forall x \in \Bigl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \bigl \{ k\pi \bigr \} \Bigr],$$
$$\int^x \csc(t) \ dt = \ln \left| \tan \left( \frac{x}{2}\right) \right| $$

The secant function\(: {\displaystyle \int^x} \sec(t) \ dt \)

The \( \sec(x) \) function is defined as follows:

Its general antiderivatives are:

1. By the secant trigonometric functions
$$ \forall k \in \mathbb{Z}, \enspace \forall x \in \biggl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \Bigl \{ \frac{\pi}{2} + k\pi \Bigr \} \biggr], $$
$$\int^x \sec(t) \ dt = \ln \left|\sec(x) + \tan(x) \right|$$
2. By applying Bioche's rules
$$ \forall k \in \mathbb{Z}, \enspace \forall x \in \biggl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \Bigl \{ \frac{\pi}{2} + k\pi \Bigr \} \biggr], $$
$$\int^x \sec(t) \ dt = \ln \left| \tan\left( \frac{x}{2} + \frac{\pi}{4} \right) \right| $$

The cotangent function\(: {\displaystyle \int^x} \cot(t) \ dt \)

The \( \cot(x) \) function is defined as follows:

Its general antiderivative is:

The secant trigonometric reciprocal functions: \(\operatorname{Arccsc}(x)\), \(\operatorname{Arcsec}(x)\), \( \operatorname{Arccot}(x)\)

The arccosecant function\(: {\displaystyle \int^x} \operatorname{Arccsc}(x)(t) \ dt \)

The \( \operatorname{Arccsc}(x) \) is the reciprocal function of the \( \csc(x) \) function , it is defined as follows:

Its general antiderivative is:

The arcsecant function\(: {\displaystyle \int^x} \operatorname{Arcsec}(x)(t) \ dt \)

The \( \operatorname{Arcsec}(x) \) is the reciprocal function of the \( \sec(x) \) function, it is defined as follows:

Its general antiderivative is:

The arccotangent function\(: {\displaystyle \int^x} \operatorname{Arccot}(x)(t) \ dt \)

The \( \operatorname{Arccot}(x) \) is the reciprocal function of the \( \cot(x) \) function , it is defined as follows:

Its general antiderivative is:

The hyperbolic functions: \(\sinh(x), \cosh(x), \tanh(x)\)

The hyperbolic sines function\(: {\displaystyle \int^x} \sinh(t) \ dt \)

The \( \sinh(x) \) function is defined as follows:

Its general antiderivative is:

The hyperbolic cosines function\(: {\displaystyle \int^x} \cosh(t) \ dt \)

The \( \cosh(x) \) function is defined as follows:

Its general antiderivative is:

The hyperbolic tangent function\(: {\displaystyle \int^x} \tanh(t) \ dt \)

The \( \tanh(x) \) function is defined as follows:

Its general antiderivative is:

The hyperbolic reciprocal functions: \( \operatorname{Arcsinh}(x)\), \(\operatorname{Arccosh}(x)\), \( \operatorname{Arctanh}(x)\)

The hyperbolic arcsines function\(: {\displaystyle \int^x} \operatorname{Arcsinh}(t) \ dt \)

The \( \operatorname{Arcsinh}(x) \) is the reciprocal function of the \( \sinh(x) \) function , it is defined as follows:

In addition, we can define it more explicitly by :

(\(\Longrightarrow\) see demonstration of it )

Its general antiderivative is:

The hyperbolic arccosines function\(: {\displaystyle \int^x} \operatorname{Arccosh}(t) \ dt \)

The \( \operatorname{Arccosh}(x) \) is the reciprocal function of the \( \cosh(x) \) function , it is defined as follows:

In addition, we can define it more explicitly by :

(\(\Longrightarrow\) see demonstration of it )

Its general antiderivative is:

The hyperbolic arctangent function\(: {\displaystyle \int^x} \operatorname{Arctanh}(t) \ dt \)

The \( \operatorname{Arcsinh}(x) \) is the reciprocal function of the \( \sinh(x) \) function , it is defined as follows:

In addition, we can define it more explicitly by :

(\(\Longrightarrow\) see demonstration of it )

Its general antiderivative is:

The hyperbolic secant functions: \(\operatorname{csch}(x), \operatorname{sech}(x), \operatorname{coth}(x)\)

The hyperbolic cosecant function\(: {\displaystyle \int^x} \operatorname{csch}(t) \ dt \)

The \( \operatorname{csch}(x) \) function is defined as follows:

Its general antiderivatives are:

1. By the secant trigonometric functions
$$ \forall x \in \hspace{0.04em} \mathbb{R}^*, $$
$$\int^x \operatorname{csch}(t) \ dt = \ln \left|\operatorname{csch}(x) -\operatorname{coth}(x) \right|$$
2. By using the change of variable \(u = e^t\)
$$ \forall x \in \hspace{0.04em} \mathbb{R}^*,$$
$$\int^x \operatorname{csch}(t) \ dt = \ln \left| \operatorname{coth}\left(\frac{x}{2} \right) \right|$$

The hyperbolic secant function\(: {\displaystyle \int^x} \operatorname{sech}(t) \ dt \)

The \( \operatorname{sech}(x) \) function is defined as follows:

Its general antiderivatives are:

1. By the secant trigonometric functions
$$ \forall x \in \mathbb{R}, $$
$$\int^x \operatorname{sech}(t) \ dt = \operatorname{Arctan}(\sinh(x)) $$
2. By using the change of variable \(u = e^t\)
$$ \forall x \in \mathbb{R}, $$
$$ \int^x \operatorname{sech}(t) \ dt = 2 \ \operatorname{Arctan}(e^x) $$

The hyperbolic cotangent function\(: {\displaystyle \int^x} \operatorname{coth}(t) \ dt \)

The \( \operatorname{coth}(x) \) function is defined as follows:

Its general antiderivative is:

The hyperbolic secant reciprocal functions: \(\operatorname{Arccsch}(x)\), \(\operatorname{Arcsech}(x)\), \( \operatorname{Arccoth}(x)\)

The hyperbolic arccosecant function\(: {\displaystyle \int^x} \operatorname{Arccsch}(t) \ dt \)

The \( \operatorname{Arccsch}(x) \) is the reciprocal function of the \( \operatorname{csch}(x) \) function , it is defined as follows:

Its general antiderivative is:

The hyperbolic arcsecant function\(: {\displaystyle \int^x} \operatorname{Arcsech}(t) \ dt \)

The \( \operatorname{Arcsech}(x) \) is the reciprocal function of the \( \operatorname{sech}(x) \) function , it is defined as follows:

Its general antiderivative is:

The hyperbolic arccotangent function\(: {\displaystyle \int^x} \operatorname{Arccoth}(t) \ dt \)

The \( \operatorname{Arccoth}(x) \) is the reciprocal function of the \( \operatorname{coth}(x) \) function , it is defined as follows:

Its general antiderivative is:

Antiderivatives of trigonometric functions recap table

Convexity of a function

We call the convexity of a function \(f\) its general shape, rather bowl-shaped ( convex ) or rather cellar-shaped ( concave ).

We define it more precisely according to the position of its tangents and the chords of the function.

Let \(f\) be a function of class \( \mathbb{C}^{2}\) on its definition set \( D_f \) and any interval \(I\).

The sign of the second derivative indicates convexity

Just as the derivative indicates the direction of variation of a function, the sign of the second derivative indicates convexity.

And,

The inequality of tangents

A function \( f \) is said to be convex on an interval \( I \), if any tangent at a point lies below the curve. A contrario , it is concave if any tangent is located above the curve.

The inequality of convexity

A function \( f \) is said to be convex on an interval \( I \), if any rope which connects two points of this interval lies above the curve. A contrario , it is concave if any rope is located below the curve.

The derivatives of operations on functions

Let be by default two functions \( f, g \), depending on \( x \) such as:

Function multiplied by a constant\(: (\lambda f )' \)

When we derive a function mutliplied by a constant \(\lambda \in \mathbb{R}\), we can take it out and derive the function separately.

Sum of two functions\(: (f+g )' \)

In the same way,

Linear combination of two functions\(: (\lambda f+ \mu g )' \)

The derivative of a linear combination is the linear combination of each derivative function.

1. Generalization
$$ \forall n \in \mathbb{N}, \enspace \forall (f,g), \enspace \forall k \in [\![ 1, n ]\!], \enspace \forall \lambda_k \in \hspace{0.04em} \mathbb{R}^n,$$
$$ \Biggl( \sum_{k=0}^n \lambda_k f_k \Biggl)' = \sum_{k=0}^n \lambda_k f'_k $$

Product of two functions\(: (fg )' \)

1. Simple derivation
$$ \forall (f,g),$$
$$ \left ( f g \right)' = f'g + g'f $$
2. Multiple derivation: the Leibniz's formula

Let be \(f, g\) two functions of class \( \mathbb{C}^{\infty}\) on an interval \(I\). We note \(f^{(n)}\) the \(n\)-th derivative of \(f\).

The Leibniz's formula tells us that::

$$ \forall n\in \hspace{0.04em} \mathbb{N}, \enspace \forall (f,g), $$
$$ (fg)^{(n)} = \sum_{p = 0}^n \binom{n}{p} f^{(n-p)} \hspace{0.1em} g^{(p)} \qquad \text{(Leibniz's formula)} $$

Inverse of a function\(: (1 / g )' \)

Quotient of two functions\(: (f / g )' \)

Composite functions\(: (f \circ g )' \)

1. Two-functions composite

Let be \( f, g \) two functions.

$$ g : I \longmapsto J , \enspace x \longmapsto g(x) $$
$$ f : J \longmapsto K, \enspace y = g(x) \longmapsto f(y) = f \left(g(x)\right) $$

We define a composite function \( (f \circ g) \) as:

$$ (f \circ g)(x) = f \left(g(x)\right) $$

Its derivative is:

$$ \forall (f,g),$$
$$ (f \circ g)' = g'(f' \circ g) $$

So:

$$ (f \circ g)' = g'.f' \left(g\right) $$

We also call it a chain derivation.

2. Generalization: any composite function

Let us define a new operator of composition:

$$ \forall n \in \mathbb{N}, \ \Biggl( \overset{n}{\underset{k=1}{\bigcirc f_k}} \ \Biggr )(x) = \Bigl(f_1 \circ f_2 \circ f_3 \circ \ ... \ \circ f_{n-1} \circ f_{n}\Bigr)(x) $$

We now define a new function \(\Psi_n (x) \):

$$ \forall n \in \mathbb{N}, \ \Psi_n (x) = \ \Biggl( \overset{n}{\underset{k=1}{\bigcirc f_k}} \ \Biggr ) (x) $$

Thus we can model it as,

$$ \forall n \in \mathbb{N}, \ \forall k \in [\![1, n ]\!], \ \forall f_k \in \hspace{0.04em} f^k,$$
$$ \Psi_n' = f'_n \times \prod_{k=1}^{n-1}\Biggl[ f'_{n-k} \circ \Biggl( \overset{n}{\underset{j= (n - k) + 1}{\bigcirc f_j}} \ \Biggr ) \Biggr] \\ $$
$$ \text{with} \enspace \Psi_n (x) = \Biggl( \overset{n}{\underset{k=1}{\bigcirc f_k}} \ \Biggr ) (x) = \Bigl(f_1 \circ f_2 \circ f_3 \circ \ ... \ \circ f_{n-1} \circ f_{n}\Bigr)(x) $$

And under its developped form,

$$ \forall n \in \mathbb{N}, \ \forall k \in [\![1, n ]\!], \ \forall f_k \in \hspace{0.04em} f^k,$$
$$ \Psi_n' = \Biggl( \overset{n}{\underset{k=1}{\bigcirc f_k}} \ \Biggr )'= f_n' \times \bigl( f_{n-1}' \circ f_n \bigr) \times \ ... \ \times \bigl(f_1' \circ f_2 \circ \ ... \ \circ f_{n-1} \circ f_n \bigr)$$

Recap table of the derivatives of composite functions

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Reciprocal function\(: (f^{-1} )' \)

Let be \( f \) a function such as:

We define its reciprocal function by:

The reciprocal function derivative \( (f^{-1})' \) is:

Recap table of the derivatives of operations on functions

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The derivatives of standard functions

The constant function: \( (\lambda )' \)

A constant function is defined as followed:

Its derivative is:

The affine function: \( (ax + b )' \)

An affine function is defined as followed:

Its derivative is:

The absolute value function\(: |x|' \)

The absolute value function is defined as followed:

Its derivative is:

The square function: \( (x^2 )' \)

The square function is defined as followed:

Its derivative is:

The powers of x functions: \( (x^n)' \)

In this part, there will be many cases where \(x\) will be found in the denominator, then for simplicity we will remove the case where \((x = 0)\).

Then, we specifically define the power of \(x\) function as follows:

Its derivative is:

The powers of n functions: \( (n^x)'\)

In this part, there will be many cases where \(n\) will be found under a logarithm , then for simplicity we will remove the case where \((n = 0)\).

Then, we specifically define a power of \(n\) function as followed:

Its derivative is:

The square root function: \( (\sqrt{x})' \)

The square root function is defined as followed:

Its derivative is:

The inverse function \( : \hspace{0.03em} \left(\frac{1}{x}\right)' \)

The inverse function is defined as followed:

Its derivative is:

The Napierian logarithm function: \(\Bigl[ \ln(x) \Bigr]'\)

1. The \(\ln(x)\) function

The the Napierian logarithm function \((\ln(x))\) is defined as the reciprocal function of the exponential function \((e^x)\) :

$$ \forall x \in \mathbb{R_+^*}, \enspace f(x) = \ln(x) = (e^x)^{-1}$$

It can be defined by an integral :

$$ \forall x \in \mathbb{R_+^*}, \enspace \ln(x) = \int^x_1 \frac{dt}{t}$$
$$ \forall x \in \mathbb{R_+^*}, $$
$$ \Bigl[ \ln(x) \Bigr]' = \frac{1}{x} $$
2. The \(\ln|x|\) function

The Napierian logarithm function \((\ln(x))\) in absolute value is defined as:

$$ \forall x \in \mathbb{R^*}, \enspace f(x) = \ln|x| = \Biggl \{ \begin{gather*} \forall x \in \mathbb{R_-^*}, \ f(x) = \ln(-x) \\ \forall x \in \mathbb{R_+^*}, \ f(x) = \ln(x) \end{gather*} $$
$$ \forall x \in \mathbb{R^*}, $$
$$ \Bigl[ \ \ln|x| \ \Bigr]' = \frac{1}{x} $$

The logarithm to the base n function\(: \Bigl[ log_n(x) \Bigr] '\)

The logarithm to the base \(n\) function is defined as followed:

Its derivative is:

The exponential function: \((e^x)'\)

The exponential function is defined as the the derivative of a reciprocal function of the Napierian logarithm function \((\ln(x))\) :

Its derivative is itself:

Recap table of the standard functions derivatives

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The derivatives of trigonometric functions

For all these trigonometric functions, we will also have their respective reciprocal function .

Between a function and its reciprocal function , we do have the following relation:

Here is an example with \(\sin(x)\) and \(\operatorname{Arcsin}(x)\) :

Be careful not to be confused with the notation "\( f^{-1} \)" of reciprocal functions with that of an inverse .

Indeed, we note "\( \cos^{-1}, \ \sin^{-1}, \ \tan^{-1}... \)" for trigonometric reciprocal functions \( (arcsin, \ arccos, \ \operatorname{Arctan}...) \), but this is a different notation from "\( f^{-1} \)" which generally means the inverse function \( (x^{-1} = \frac{1}{x}) \) .

The basic trigonometric functions\(:\sin(x), \cos(x), \tan(x)\)

Applying the Thales' theorem , we definitely see the following relation:

The sines function\(: \sin(x)\)

The \( \sin(x) \) function is defined as follows:

Its derivative is:

The cosines function\(: \cos(x)\)

The \( \cos(x) \) function is defined as follows:

Its derivative is:

The tangent function\(: \tan(x)\)

The \( \tan(x) \) function is defined as follows:

Its derivative is:

The basic trigonometric reciprocal functions: \(\operatorname{Arcsin}(x), \operatorname{Arccos}(x), \operatorname{Arctan}(x)\)

The arcsines function\(: \operatorname{Arcsin}(x)\)

The \( \operatorname{Arcsin}(x) \) is the reciprocal function of the \( \sin(x) \) function , it is defined as follows:

Its derivative is:

The arccosines function\(: \operatorname{Arccos}(x)\)

The \( \operatorname{Arccos}(x) \) function is the reciprocal function of the \( \cos(x) \) function , it is defined as follows:

Its derivative is:

The arctangent function\(: \operatorname{Arctan}(x)\)

The \( \operatorname{Arctan}(x) \) function is the reciprocal function of the \( \tan(x) \) function , it is defined as follows:

Its derivative is:

The secant trigonometric functions: \(\csc(x), \sec(x), \cot(x)\)

The three secant trigonometric functions are the \( \csc(x), \sec(x) \) and \( \cot(x) \) functions.

They are respectively the inverses of the \( \sin(x), \cos(x) \) and \( \tan(x) \) functions.

Applying the Thales' theorem , we definitely see the following relation:

The cosecant function\(: \csc(x)\)

The \( \csc(x) \) function is defined as follows:

Its derivative is:

Furthermore, we do notice that:

The secant function\(: \sec(x)\)

The \( \sec(x) \) function is defined as follows:

Its derivative is:

Furthermore, we do notice that:

The cotangent function\(: \cot(x)\)

The \( \cot(x) \) function is defined as follows:

Its derivative is:

The secant trigonometric reciprocal functions: \(\operatorname{Arccsc}(x)\), \(\operatorname{Arcsec}(x)\), \( \operatorname{Arccot}(x)\)

The arccosecant function\(: \operatorname{Arccsc}(x)\)

The \( \operatorname{Arccsc}(x) \) is the reciprocal function of the \( \csc(x) \) function , it is defined as follows:

Its derivative is:

The arcsecant function\(: \operatorname{Arcsec}(x)\)

The \( \operatorname{Arcsec}(x) \) is the reciprocal function of the \( \sec(x) \) function, it is defined as follows:

Its derivative is:

The arccotangent function\(: \operatorname{Arccot}(x)\)

The \( \operatorname{Arccot}(x) \) is the reciprocal function of the \( \cot(x) \) function , it is defined as follows:

Its derivative is:

The hyperbolic functions: \(\sinh(x), \cosh(x), \tanh(x)\)

The three hyperbolic functions are the \( \sinh(x), \cosh(x) \) and \( \tanh(x) \) functions.

They are the twins of the \( \sin(x), \cos(x) \) and \( \tan(x) \) functions, and notably concerning their properties.

The hyperbolic sines function\(: \sinh(x)\)

The \( \sinh(x) \) function is defined as follows:

Its derivative is:

The hyperbolic cosines function\(: \cosh(x)\)

The \( \cosh(x) \) function is defined as follows:

Its derivative is:

The hyperbolic tangent function\(: \tanh(x)\)

The \( \tanh(x) \) function is defined as follows:

Its derivative is:

The hyperbolic reciprocal functions: \( \operatorname{Arcsinh}(x)\), \(\operatorname{Arccosh}(x)\), \( \operatorname{Arctanh}(x)\)

The hyperbolic arcsines function\(: \operatorname{Arcsinh}(x)\)

The \( \operatorname{Arcsinh}(x) \) is the reciprocal function of the \( \sinh(x) \) function , it is defined as follows:

In addition, we can define it more explicitly by :

(\(\Longrightarrow\) see demonstration of it )

Its derivative is:

The hyperbolic arccosines function\(: \operatorname{Arccosh}(x)\)

The \( \operatorname{Arccosh}(x) \) is the reciprocal function of the \( \cosh(x) \) function , it is defined as follows:

In addition, we can define it more explicitly by :

(\(\Longrightarrow\) see demonstration of it )

Its derivative is:

The hyperbolic arctangent function\(: \operatorname{Arctanh}(x)\)

The \( \operatorname{Arctanh}(x) \) is the reciprocal function of the \( \tanh(x) \) function , it is defined as follows:

In addition, we can define it more explicitly by :

(\(\Longrightarrow\) see demonstration of it )

Its derivative is:

The hyperbolic secant functions: \(\operatorname{csch}(x), \operatorname{sech}(x), \operatorname{coth}(x)\)

The three hyperbolic secant functions are the \( \operatorname{csch}(x), \operatorname{sech}(x) \) and \( \operatorname{coth}(x) \) functions.

They are respectively the inverses of the \( \sinh(x), \cosh(x) \) and \( \tanh(x) \) functions.

The hyperbolic cosecant function\(: \operatorname{csch}(x)\)

The \( \operatorname{csch}(x) \) function is defined as follows:

Its derivative is:

Furthermore, we do notice that:

The hyperbolic secant function\(: \operatorname{sech}(x)\)

The \( \operatorname{sech}(x) \) function is defined as follows:

Its derivative is:

The hyperbolic cotangent function\(: \operatorname{coth}(x)\)

The \( \operatorname{coth}(x) \) function is defined as follows:

Its derivative is:

The hyperbolic secant reciprocal functions: \(\operatorname{Arccsch}(x)\), \(\operatorname{Arcsech}(x)\), \( \operatorname{Arccoth}(x)\)

The hyperbolic arccosecant function\(: \operatorname{Arccsch}(x)\)

The \( \operatorname{Arccsch}(x) \) is the reciprocal function of the \( \operatorname{csch}(x) \) function , it is defined as follows:

Its derivative is:

The hyperbolic arcsecant function\(: \operatorname{Arcsech}(x)\)

The \( \operatorname{Arcsech}(x) \) is the reciprocal function of the \( \operatorname{sech}(x) \) function , it is defined as follows:

Its derivative is:

The hyperbolic arccotangent function\(: \operatorname{Arccoth}(x)\)

The \( \operatorname{Arccoth}(x) \) is the reciprocal function of the \( \operatorname{coth}(x) \) function , it is defined as follows:

Its derivative is:

Recap table of the trigonometric functions derivatives

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Derivativity of functions

Derivativity is key notion in analysis, because it underlies all physics and science in general.

The idea of derivativity

Let \( f :x \longmapsto f(x) \) be a continuous function.

1. Derived number

We do call \( f'(a) \) the derived number from the function \( f \) at point \( (x=a )\) such as

$$ f'(a) = \lim_{h \to 0} \enspace \frac{f(a+h) - f(a)}{h}$$

If (and only if) this number is defined, we then say that \( f \) is derivable at point \( a\).

$$ f \text{ is derivable at point } a \Longleftrightarrow \lim_{h \to 0} \enspace \frac{f(a+h) - f(a)}{h}= f'(a) \in \mathbb{R} $$

Determining the general expression of the derivative function \(f'\), we will define where the function \(f\) is derivable.

2. Derivative function

Function \(f'\), derivated from fucntion \( f \) is expressed as follows:

$$ f'(x) = \lim_{h \to 0} \enspace \frac{f(x+h) - f(x)}{h} $$

The is the limit of the rate variation when \( h \to 0 \).

It can also be found in this form:

$$ f'(x) = \lim_{t \to x} \enspace \frac{f(t) - f(x)}{t - x} $$

At this point, it will be the limit of rate variation when \( x \to a \).

Derivativity implies continuity

The sign of the derivative indicates the variation

Furthermore, if and only if \(f'\) changes sign between before and after a certain point \(a\), the \(f\) function admits a local extremum at this point.

Equation of the tangent to the curve at point a

We saw in the definition of the derivative that the derived number correspond to the slope of the tangent to the curve of a function.

This line admits for equation at the point of abscissa \(a\):

Furthermore, in the case of a convex function (resp. concave) , this tangent is always below (resp. above) the curve.

Link between Taylor series of order 1 and derivativity

Fourier Series

Let \(f(t)\) be a \(T\)-periodic function.

The trigonometric notation

Fourier's hypothesis is that this function can be reconstructed as a sum of cosines and sines:

With the first coefficient \(a_0\):

And the two coefficients \((a_n, \ b_n)\):

The exponential notation

By switching to exponential notation, we get everything under a single sum:

As well as a single coefficient \((c_n)\):

Integration methods for rational fractions

With a second degree denominateur only\(: {\displaystyle \int^x} \frac{1}{Q_2(t)} \ dt \)

• \( \alpha) \) Discriminant is positive\(: \Delta > 0 \)
$$\forall x \in \Bigl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \bigl \{x_1, x_2 \bigr \} \Bigr],$$
$$ \int^x \frac{1}{t^2 + pt + q} \ dt = \frac{1}{(x_1 - x_2)} \ \ln \left| \frac{x-x_1}{x-x_2} \right|, \hspace{2em} \text{with } \left \{ \begin{gather*} x_1 = \frac{- p - \sqrt{p^2 - 4q}}{2}\\ x_2 = \frac{- p + \sqrt{p^2 - 4q}}{2} \end{gather*} \right \} \qquad \bigl(if \ \Delta = p^2 - 4q > 0 \bigr) $$

In the specific case where \((p=0, \ q-1)\), we do have as a bonus an explicite definition of the \(\operatorname{Arctanh}\) function:

$$ \forall x \in \hspace{0.04em} ]-1, \hspace{0.1em} 1[, $$
$$\operatorname{Arctanh}(x) = \frac{1}{2} \ln \left|\frac{x+1}{x-1} \right|$$
• \( \beta) \) Discriminant is zero\(: \Delta = 0 \)
$$\forall x \in \hspace{0.04em} \biggl[ \mathbb{R} \hspace{0.04em} \backslash \hspace{0.2em} \Bigl \{ -\frac{p}{2} \Bigr \} \biggr],$$
$$ \int^x \frac{1}{t^2 + pt + q} \ dt = - \frac{1}{ x + \frac{p}{2} } \qquad \bigl(if \ \Delta = p^2 - 4q = 0 \bigr) $$
• \( \gamma) \) Discriminant is negative\(: \Delta < 0 \)
$$\forall x \in \mathbb{R},$$
$$ \int^x \frac{1}{t^2 + pt + q} \ dt = \frac{1}{ \sqrt{ q - \frac{p^2}{4} } } \ \operatorname{Arctan} \left (\frac{x + \frac{p}{2}}{ \sqrt{ q - \frac{p^2}{4} } } \right ) \qquad \bigl(if \ \Delta = p^2 - 4q < 0 \bigr) $$

With a numerator of the first degree and a denominator of the second degree\(: {\displaystyle \int^x} \frac{P_1(t)}{Q_2(t)} \ dt \)

Integration methods and antiderivatives recap table

Integration methods for rational fractions with square roots

Let \(a \in \hspace{0.04em} \mathbb{R}\) be a real number.

Integrals containing \(\sqrt{a^2 - t^2}\)

1. Fraction with a root at the denominator\(: \frac{1}{\sqrt{a^2 - t^2}}\)
$$ \forall a \in \hspace{0.04em} \mathbb{R}^*, \ \forall x \in \hspace{0.04em} \Bigl [-|a|, \hspace{0.2em} |a| \Bigr], $$
$$ \int^x \frac{dt}{\sqrt{a^2 - t^2}} = \operatorname{Arcsin}\left(\frac{x}{|a|}\right) $$
2. Fraction with \(t\) and a root in the denominator\(: \frac{1}{t\sqrt{a^2 - t^2}}\)
$$ \forall a \in \hspace{0.04em} \mathbb{R}, \ \forall x \in \hspace{0.04em} \Bigl]-|a|, \hspace{0.2em} 0 \Bigr[ \hspace{0.04em} \cup \hspace{0.04em} \Bigl]0, \hspace{0.2em} |a| \Bigr[ , $$
$$ \int^x \frac{dt}{t\sqrt{a^2 - t^2}} = \frac{1}{2a} \ln\left|\sqrt{a^2 - x^2} - a \right| + \frac{1}{2a} \ln\left|\sqrt{a^2 - x^2} + a\right| $$
3. Fraction with \(t^2\) and a root in the denominator\(: \frac{1}{t^2\sqrt{a^2 - t^2}}\)
$$ \forall a \in \hspace{0.04em} \mathbb{R}, \ \forall x \in \hspace{0.04em} \Bigl]-|a|, \hspace{0.2em} 0 \Bigr[ \hspace{0.04em} \cup \hspace{0.04em} \Bigl]0, \hspace{0.2em} |a| \Bigr[ ,$$
$$ \int^x \frac{dt}{t^2\sqrt{a^2 - t^2}} = - \frac{1}{a^2} \frac{\sqrt{a^2 - x^2}}{x} $$
4. Simple root\(: \sqrt{a^2 - t^2} \)
$$ \forall a \in \hspace{0.04em} \mathbb{R}^*, \ \forall x \in \hspace{0.04em} \Bigl [-|a|, \hspace{0.2em} |a| \Bigr], $$
$$ \int^x \sqrt{a^2 - t^2} \ dt = \frac{a^2}{2} \operatorname{Arcsin}\left(\frac{x}{|a|}\right) + \frac{x}{2} \sqrt{a^2 - x^2} $$

Integrals containing \(\sqrt{a^2 + t^2}\)

1. Fraction with a root at the denominator\(: \frac{1}{\sqrt{a^2 + t^2}}\)
1. Setting down \( t = |a| \ \sinh(u)\)
$$\forall a \in \hspace{0.04em} \mathbb{R}^*, \ \forall x \in \hspace{0.04em} \mathbb{R}, $$
$$ \int^x \frac{dt}{\sqrt{a^2 + t^2}} = \operatorname{Arcsinh}\left(\frac{x}{|a|} \right) $$
2. Setting down \( t = |a| \ \tan(u)\)
$$ \Bigl[ \forall (a,x) \in \hspace{0.04em} \mathbb{R}^2 \Bigr] \hspace{0.2em} \backslash \hspace{0.2em} \Bigl \{ (0,0) \Bigr \} , $$
$$ \int^x \frac{dt}{\sqrt{a^2 + t^2}} = \ln \left|\sqrt{ a^2 + x^2 } + x\right|$$

Both expressions having a common member, they are both equal up to a constant and we do obtain as a bonus an explicit definition of the \(\operatorname{Arcsinh}\) function with \(a = 1\):

$$ \forall x \in \mathbb{R}, $$
$$ \operatorname{Arcsinh}(x) = \ln \left| x + \sqrt{ 1 + x^2 } \right| $$
2. Fraction with \(t\) and a root in the denominator\(: \frac{1}{t\sqrt{a^2 + t^2}}\)
$$\forall a \in \hspace{0.04em} \mathbb{R}^+, \ \forall x \in \hspace{0.04em} \mathbb{R}^*,$$
$$ \int^x \frac{dt}{t\sqrt{a^2 + t^2}} = \frac{1}{2a} \ln\left|\sqrt{a^2 + x^2}-a\right| - \frac{1}{2a} \ln \left|\sqrt{a^2 + x^2} + a\right| $$
3. Fraction with \(t^2\) and a root in the denominator\(: \frac{1}{t^2\sqrt{a^2 + t^2}}\)
$$\forall a \in \hspace{0.04em} \mathbb{R}^+, \ \forall x \in \hspace{0.04em} \mathbb{R}^*,$$
$$ \int^x \frac{dt}{t^2\sqrt{a^2 + t^2}} =- \frac{1}{a^2} \frac{\sqrt{a^2+x^2}}{x}$$
4. Simple root\(: \sqrt{a^2 + t^2} \)
$$ \Bigl[ \forall (a,x) \in \hspace{0.04em} \mathbb{R}^2 \Bigr] \hspace{0.2em} \backslash \hspace{0.2em} \Bigl \{ (0,0) \Bigr \} , $$
$$ \int^x \sqrt{a^2 + t^2} \ dt = \frac{x \ \sqrt{a^2 +x^2} + a^2 \ \ln\left|\sqrt{a^2 +x^2} + x \right|}{2} $$

Integrals containing \(\sqrt{t^2 - a^2}\)

1. Fraction with a root at the denominator\(: \frac{1}{\sqrt{t^2 - a^2}}\)
1. Setting down \(t = |a| \ \cosh(u) \)
$$ \forall a \in \hspace{0.04em} \mathbb{R}^+, \ \forall x \in \hspace{0.04em} ]a, +\infty[, $$
$$ \int^x \frac{dt}{\sqrt{t^2 - a^2}} = \operatorname{Arccosh}\left( \frac{x}{a}\right) $$
2. Setting down \(t = |a| \ \sec(u) \)
$$\Bigl[ \forall a \in \hspace{0.04em} \mathbb{R}^*, \ \forall x \in \Bigl] -\infty, -|a| \Bigr[ \hspace{0.04em} \cup \hspace{0.04em} \Bigl]|a|, +\infty \Bigr[ \Bigr] \lor \Bigl[ (x = 0) \land (x > 0) \Bigr],$$
$$\int^x \frac{dt}{\sqrt{t^2 - a^2}} = \ln \left| \sqrt{ x^2 - a^2 } + x \right| $$

Both expressions having a common member, they are both equal up to a constant and we do obtain as a bonus an explicit definition of the \(\operatorname{Arccosh}\) function with \(a = 1\):

$$ \forall x \in [1, +\infty[,$$
$$ \operatorname{Arccosh}(x) = \ln \left| x + \sqrt{ x^2 - 1} \right| $$
2. Fraction with \(t\) and a root in the denominator\(: \frac{1}{t\sqrt{t^2 - a^2}}\)
$$ \forall a \in \hspace{0.04em} \mathbb{R}^+, \ \forall x \in \Bigl] -\infty, -|a| \Bigr[ \hspace{0.04em} \cup \hspace{0.04em} \Bigl]|a|, +\infty \Bigr[,$$
$$ \int^x \frac{dt}{t\sqrt{t^2 - a^2}}dt = \frac{1}{a} \operatorname{Arctan} \left(\frac{\sqrt{x^2 - a^2}}{a} \right) $$
3. Fraction with \(t^2\) and a root in the denominator\(: \frac{1}{t^2\sqrt{t^2 - a^2}}\)
$$ \forall a \in \hspace{0.04em} \mathbb{R}^+, \ \forall x \in \Bigl] -\infty, -|a| \Bigr[ \hspace{0.04em} \cup \hspace{0.04em} \Bigl]|a|, +\infty \Bigr[, $$
$$ \int^x \frac{dt}{t^2\sqrt{t^2 - a^2}} =\frac{1}{a^2} \frac{\sqrt{ x^2 - a^2 } }{x}$$
4. Simple root\(: \sqrt{ t^2 - a^2} \)
$$ \Bigl[ \forall a \in \hspace{0.04em} \mathbb{R}^*, \ \forall x \in \Bigl] -\infty, -|a| \Bigr[ \hspace{0.04em} \cup \hspace{0.04em} \Bigl]|a|, +\infty \Bigr[ \Bigr] \lor \Bigl[ (x = 0) \land (x > 0) \Bigr], $$
$$ \int^x \sqrt{t^2 - a^2} \ dt = \frac{ x\sqrt{x^2 - a^2} - a^2 \ \ln \left| x+ \sqrt{x^2 - a^2} \ \right| }{2} $$

Integration methods and antiderivatives recap table

L'Hôpital's rule: a way to remove indeterminacy

Let \( f, g\) be two functions defined on an interval \( I = \bigl[a,b \bigr]\) and their respective derivative \( f', g'\) defined on \(]a , b [\).

Now, let \( \alpha \in \overset{-}{I}\cup{\{\infty\}} \) be a point or an extremity of \( I\); \( \alpha \) be a point or an extremity, it can be whatever value, including \( -\infty \) or \( +\infty \).

In order to calculate the limit in \( \alpha\) of a function under the form of a quotient presenting an indeterminate form of type \( \left[ \frac{0}{0} \right]\) or \( \left[ \frac{\pm \infty}{\pm \infty} \right]\), there may be an interest in using this method, with the aim of removing indeterminacy.

L'Hôpital's rule tells us that:

The Lagrange interpolating polynomial

Let \( n \hspace{0.1em} \in \hspace{0.04em} \mathbb{N} \) be a natural number, and a given dataset of antecedents/images \( \Bigl \{ \bigl \{x_i, y_i \bigr \}_{ i \hspace{0.1em} \in \hspace{0.04em} n} \Bigr \}\).

This method allows you to find the expression of a unique polynomial function , passing through to each points of this series of values.

Lagrange interpolating polynomial tells us that it is possible to construct this polynomial .

There will be two ways to generate this polynomial :

By the direct construction of a polynomial \(L(X) = \sum L_j(X)\)

By constructing a polynomial \(P_n(X)\), by finding the only solution for the coefficients \(\bigl \{a_0, \ a_1, \ ..., \ a_n \bigr \} \) unknown to the system \( (S) \)

Given the following system \( (S) \):

We can resolve it at least by two methods:

1. By directly solving the system \( (S) \) by substitution or by a Gaussian pivot
$$ P_n(X) = a_0 + a_1 X + a_2 X^2 + \hspace{0.1em}... \hspace{0.1em}+ a_n X^n $$

With \( \bigl \{a_0, \ a_1, \ ..., \ a_n \bigr \} \) as the unique solution of the system \( (S)\)

2. By solving the equivalent matricial system \( (S^*) \)
$$ (S^*) \enspace \Longleftrightarrow \enspace \underbrace{ \begin{pmatrix} 1 & x_0 & x_0^2 & \dots & x_0^n \\ 1 & x_1 & x_1^2 & \dots & x_1^n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_n & x_n^2 & \dots & x_n^n \\ \end{pmatrix} } _\text{X} \times \underbrace{ \begin{pmatrix} a_0 \\ a_1 \\ a_2 \\ . \\ a_n \\ \end{pmatrix} } _\text{A} = \underbrace{ \begin{pmatrix} y_0 \\ y_1 \\ y_2 \\ . \\ y_n \\ \end{pmatrix} } _\text{Y} $$

And at that moment:

$$ P_n(X) = a_0 + a_1 X + a_2 X^2 + \hspace{0.1em}... \hspace{0.1em}+ a_n X^n $$

$$ \forall k \in [\![ 0, n]\!], \enspace a_k = \frac{det(X_k)}{det(X)}$$

Where \(X_k\) represents the square matrix formed by \(X\) in which we inverted its \(k\)-th column with the column matrix \(Y\).

The length of a curve over an interval

Let be \(f\) a continuous function over an interval \(I = \bigl[a, b \bigr]\), and \( n \in \mathbb{N}\) be a natural number.

The length of the curve of \(f\) over this interval \(I\) is worth:

Moreover, we can express this length as a Riemann integral :

The link between integrals and antiderivatives

Let be \(f\) a continous function on an interval \(I = \bigl[a, b \bigr]\).

Fundamental theorem of calculus

From a definite integral, it is possible to determine an antiderivative \(S \) of \(f\).

This antiderivative will be the antiderivative of \(f\) which vanishes at point \(a\):

We can then define a primitive using this integral:

Conversely, from an antiderivative \(F\) of function \(f\), it is possible to determine the integral between two bounds \(a\) and \(x\).

Or more precisely, for two fixed bounds \(a\) and \(b\) :

The logarithm and exponential functions

The logarithm functions\(: \ln(x), log_a(x)\)

Both functions \(log_a(x)\) and \(\ln(x)\) being defined as similar up to a coefficient \(\ln(a)\), the following properties will be true for both logarithmic functions.

Definition of the natural logarithm

As well we do have:

Logarithm of a product \(: \ln(ab)\)

Logarithm of an inverse\(: \ln \left(\frac{1}{a} \right)\)

Logarithm of a quotient\(: \ln \left(\frac{a}{b} \right)\)

Logarithm of a power\(: \ln \left( a^n \right)\)

Link between logarithm to the base a, b and natural logarithm\( : log_a(x), log_b(x), \ln(x)\)

In the general way, for two logarithms respectively to the base \(a\), \(b\) or \(e\) (natural logarithm),

The exponential functions\(: e^x, a^x \)

Exponential functions, being defined as the reciprocal function of the logarithm (respectively to its base), have all the reciprocal properties of logarithms.

The definition of the exponential function

Product of an exponential

Inverse of an exponential

Quotient of an exponential

Exponential to the power of n

Value of \(e\)

Link between exponential to the base a, b or e\(: a^x, b^x, e^x\)

Recap table of logarithm and exponential functions

The mean value theorem

This theorem is a direct consequence of Rolle's theorem .

Let be \(f(x)\) a continuous function on an interval \(\bigl[a,b \bigr]\), and derivable on \(\bigl ]a,b \bigr[\).

The Newton's method

This method consists of calculating a certain value that is more or less known, but which we would like to approximate more precisely. The method is done by successive iteration.

Let \( f \) be a convex function (resp. concave) and strictly increasing (resp. decreasing) and positive on a interval \( I \) and \( a_0 \) a real number belonging to \( I \) such as \( f(a_0) > 0 \) (resp. \( f(a_0) < 0 \)).

The method consists of determining the real number \( x_0 \in I \), such as: \( f(x_0) = 0 \)

In a general way, we will determine the value of \( x_0 \) for which \( f \) is worth \( 0 \) on \( I \).

And this value is worth:

The properties of integrals

Let be \(f, g\) two functions of class \(\mathcal{C}^1\) on an interval \(I = \bigl ]a,b \bigr[\), and a function \(F\) which is an antiderivative of \(f\).

The integral from \(a\) to \(b\) is the area located between the curve of \(f\) and the abscissa axis , from \(a\) to \(b\) (see link between integrals and antiderivatives ).

Function which is defined by an integral

Integral from a to a

Integral from b to a

Chasles' relation

Linearity of the integral

The integral of a linear combination is the linear combination of the integral of each function

Positivity of the integral

In the same way, if \( f(x) \leqslant 0 \) on \( \bigl[a, b \bigr] \),

Growth of the integral

Mean value

Mean value theorem for definite integrals

The extreme value theorem

Property of the king

Recap table of the properties of integrals

Solving 1st order lin. diff. equations \( (LDE_1) \) with continuous coefficient

Let \( y \) be a function of class \( \mathbb{C}^{1}\) on an interval \( I\).

As well, let \( a(x)\) be a continuous function and \( f(x)\) any function.

Let \( (E) \) be a linear differential equation or order \( 1 \) with continuous coefficient, and \( (H) \) its associated homogeneous equation:

Solving the homogeneous equation \( (H) \)

\( y_h = 0 \) is solution, but considering \( y_h \) as a non-zero function, the following function \(y_h\) will be an homogeneous solution of \( (H) \) :

Solving the general equation \( (E) \)

Function \(y_s\) will be specific solution of \( (E) \) :

We will have as a total solution of \( (E) \), the addition of these two solutions:

Solving second order lin. diff. equations \( (EDL_2) \) with constants coefficients

Let \( y \) be a function of class \( \mathbb{C}^{2}\) on an interval \( I\).

As well, let \( (a, b) \in \hspace{0.04em} \mathbb{R}^2\) be two coefficients and \( f(x)\) any function.

Let \( (E) \) be a linear differential equation of order \( 2 \), and \( (H) \) its associated homogeneous equation:

Solving the homogeneous equation \( (H) \)

If \( y = 0 \), thus \( y_h = 0 \) is an obvious solution, but considering the \( y \) function as a non-zero function, we will have on the first hand to calculate the discriminant \( \Delta \) of the characteristic equation \( (E_c) \):

After examining the discriminant \( (\Delta = a^2 -4b) \), depending on the case the function \(y_h\) will be an homogeneous solution of \( (H) \) :

• \( \alpha) \ \Delta > 0\): two distinct roots \( \alpha, \beta \)
$$ \left \{ \begin{gather*} \alpha = \frac{-a - \sqrt{\Delta}}{2} \\ \beta = \frac{-a + \sqrt{\Delta}}{2} \end{gather*} \right \} $$
$$ y_h =c_1\hspace{0.1em} e^{\alpha x} +c_2 e^{\beta x} \qquad (\text{with} \enspace (c_1,c_2) \in \hspace{0.04em} \mathbb{R}^2) $$
• \( \beta) \ \Delta = 0\): a double root \( \alpha \)
$$\alpha = \frac{-a}{2} $$
$$ y_h =c_1\hspace{0.1em} e^{\alpha x} +c_2 \hspace{0.2em} x \hspace{0.2em} e^{\alpha x} \qquad (\text{with} \enspace (c_1,c_2) \in \hspace{0.1em} \mathbb{R}^2) $$
• \( \gamma) \ \Delta < 0\): two conjugate complex roots \( \alpha, \overline{\alpha} \)
$$ \left \{ \begin{gather*} \alpha = \frac{-a - i\sqrt{|\Delta}|}{2} \\ \overline{\alpha} = \frac{-a + i\sqrt{|\Delta}|}{2} \end{gather*} \right \} $$
$$ y_h = e^{Ax} \Biggl[c_1\hspace{0.1em} \cos(Bx) +c_2 \hspace{0.1em} \sin(Bx) \Biggr] \qquad \text{avec} \enspace \left \{ \begin{gather*} (c_1,c_2)\in \hspace{0.04em}\mathbb{R}^2 \\ A = -\frac{a}{2} \enspace et \enspace B = \frac{ \sqrt{|\Delta|} }{2} \end{gather*} \right \} $$

Solving the general equation \( (E) \)

Having determinded the constants \((c_1, c_2) \) and the solutions \((y_1, y_2) \) according to the result of the calculation of \(\Delta\), we had for the solution of \((H) \) that:

To determine a specific solution for \( (E) \), we solve the system \( (S) \):

If the determinant \(det(Y) \neq 0\), then the system \( (S) \) admits solutions which are:

The function \(y_s\) will be a specific solution of \( (E) \):

We will have as a total solution of \( (E) \), the addition of these two solutions:

Solving third-degree equations (cubic equations)

A third degree equation is of the form:

Existence of obvious root(s)

If there are one or more obvious roots, then the polynomial can be factored and reduced to a lower degree.

Determination of the number of real roots according to the cases

After calculating the discriminant \(\Delta_3\) :

We do have the following number of real roots depending on the cases:

Resolution

1. if \( (\Delta_3 > 0) \)
• a simple real root
$$ X_1 = \sqrt[3]{\frac{-q + \sqrt{\Delta_3}}{2}} + \sqrt[3]{\frac{-q - \sqrt{\Delta_3}}{2}} - \frac{b}{3a} $$
• two complex roots
$$ C_2 = e^{\frac{2i\pi}{3}} \ \ \sqrt[3]{\frac{-q - \sqrt{\Delta_3}}{2}} + e^{-\frac{2i\pi}{3}} \ \ \sqrt[3]{\frac{-q + \sqrt{\Delta_3}}{2}} - \frac{b}{3a} $$
$$ C_3 = e^{\frac{2i\pi}{3}} \ \ \sqrt[3]{\frac{-q + \sqrt{\Delta_3}}{2}} + e^{-\frac{2i\pi}{3}} \ \ \sqrt[3]{\frac{-q - \sqrt{\Delta_3}}{2}} - \frac{b}{3a} $$

And \( P_3(X) \) can be factorized as follows:

$$ P_3(X) = a(X - X_1)(X - C_2)(X - C_3) $$
2. if \( (\Delta_3 = 0) \)
• a simple real root
$$ X_1 = 2\sqrt[3]{-\frac{q}{2}} - \frac{b}{3a} $$
• a double real root
$$ X_2 = X_3 = -\sqrt[3]{-\frac{q}{2}} - \frac{b}{3a} $$

And \( P_3(X) \) can be factorized as follows:

$$ P_3(X) = a(X - X_1)(X - X_2)^2 $$



Moreover, these three solutions can merely be written as:

• a simple real root
$$ X_1 = \frac{3q}{p} - \frac{b}{3a} $$
• a double real root
$$ X_2 = X_3 = \frac{3q}{2p} - \frac{b}{3a} $$
3. if \( (\Delta_3 < 0) \)
• three real roots
$$ X_1 = \sqrt[3]{\frac{-q \ + \ i\sqrt{-\Delta_3}}{2}} + \sqrt[3]{\frac{-q \ - \ i\sqrt{-\Delta_3}}{2}} - \frac{b}{3a} $$
$$ X_2 = e^{\frac{2i\pi}{3}}\sqrt[3]{\frac{-q \ + \ i\sqrt{-\Delta_3}}{2}} + e^{-\frac{2i\pi}{3}}\sqrt[3]{\frac{-q \ - \ i\sqrt{-\Delta_3}}{2}} - \frac{b}{3a} $$
$$ X_3 = e^{\frac{2i\pi}{3}}\sqrt[3]{\frac{-q \ - \ i\sqrt{-\Delta_3}}{2}} + e^{-\frac{2i\pi}{3}}\sqrt[3]{\frac{-q \ + \ i\sqrt{-\Delta_3}}{2}} - \frac{b}{3a} $$

Even though they appear in a complex form , after solving these solutions definitely become real .

And \( P_3(X) \) can be factorized as follows:

$$ P_3(X) = a(X - X_1)(X - X_2)(X - X_3) $$

Solving second-degree equations (quadratic equations)

A quadratic equation is under the form:

Solving by finding roots and factorizing

Solutions for \( X \) which \( P_2(X) = 0 \) are called the roots of the polynomial .

They make it possible to obtain a factorized form.

Three cases should be considered after calculating the discriminant \( \Delta \):

• \( \alpha) \) if \( \textcolor{#456E3D}{\Delta > 0}\):two distinct roots \( X_1, \ X_2 \)
$$ X_1 = \frac{- b - \sqrt{\Delta}}{2a} $$
$$ X_2 = \frac{- b + \sqrt{\Delta}}{2a} $$

And \( P_2(X) \) can be factorized like this:

$$ P_2(X) = a(X - X_1)(X - X_2) $$
• \( \beta) \) if \( \Delta =0\): one double root \( X_0 \)
$$ X_0 = \frac{- b}{2a} $$

And \( P_2(X) \) can be factorized like this:

$$ P_2(X) = a(X - X_0)^2 $$
• \( \gamma) \) if \( \Delta < 0\): two conjugate complex roots \( C_1, \ C_2 \)
$$ C_1 = \frac{- b - i\sqrt{-\Delta}}{2a} $$
$$ C_2 = \frac{- b + i\sqrt{-\Delta}}{2a} $$

And \( P_2(X) \) can be factorized like this:

$$ P_2(X) = a(X - C_1)(X - C_2) $$

Link between coefficients and roots

Furthermore, in the general case we will also have these two relationships between the coefficients and the roots:

Standard antiderivatives and general integration methods

Since each antiderivative of a given function is equal up to a constant, we will use the unified notation \({\displaystyle \int^x} f(t) \ dt \),meaning this family of primitives (or general antiderivative ).

We will sometimes use the capital letter of a function to signify its antiderivative .

Recall on standard antiderivatives

A certain amount of antiderivative can be directly calculated by taking the reverse path of the derivative or by an integration by parts .

For others, each case must be considered as a specific case.

This table resumes antiderivatives directly determined from the inverse operation of the derivative , namey:

General integration methods

Integration by parts

Integration by substitution

Let \(\phi : t \longmapsto \phi(t) \) be a function of class \(\mathcal{C}^1\) on an interval \( J \subset f(I) \).

The integration by substitution is the transposition of the derivative of a composite function , but applied to integral calculus .

General integration methods recap table

The superposition principle

Let \( (m,n) \in \hspace{0.04em} \mathbb{N}^2 \) be two natural numbers and:

• a series of continuous functions \( a_1(x), a_2(x), \hspace{0.2em} ... \hspace{0.2em}, a_n(x) \)

• a series of functions \( f_1(x), f_2(x), \hspace{0.2em} ... \hspace{0.2em}, f_m(x) \)

Let \( y(x) \) be a function of class \( \mathbb{C}^{n}\) on an interval \(I\). We note \(y^{(n)}\) its \(n\)-th derivative .

In the context of solving a linear differential equation of order \(n\) where the right hand side is a linear combination of functions such that \( (E)\):

with for all \( k \in [\![ 1, m ]\!] \), the function \( (y_k) \) as a specific solution of the equation \( (E_k) \) :

The superposition principle tells us that:

The Taylor-Young and Taylor-Laplace Formulas

Let be \( f : x \longmapsto f(x) \) a function of class \( \mathcal{C}^n \) and \( f^{(n)} \) its \( n \)-th derivative .

Taylor-Young's formula

The Taylor-Young formula tells us that any function \( f \), centered at \( x = a \), can be written as a Taylor series \( (TS_n(a)) \) with a remainder \(R_n\), such as:

Here is the decomposed form:

Where the notation \( o\bigl((x - a)^n\bigr) \) represents the Landau remainder (a little rate), meaning that the remainder is negligible compared to \( (x - a)^n \) as \( x \to a \):

Furthermore, setting down \( (x = a + h) \), we do obtain a new form of this formula:

Taylor-Laplace's formula

With the Taylor-Laplace's formula, the remainder is quantified by an integral.

Thus,

Recap of the main Taylor series

Another notation used to characterize the remainder of an Taylor series is the Landau notation \(o(x^n)\).

In our specific case, we study Taylor series at the neighbourhood of \(( a = 0 ) \), so:

Convergence criteria for series

Series with positive terms

Fundamental criteria of convergence

Let \((u_n)_{n \in \mathbb{N}}\) be a numerical sequence.

So, if a sequence has a non-zero limit, we directly deduce that it is divergent .

Comparison with a reference series

Let \((u_n)_{n \in \mathbb{N}}\) be a numerical sequence whose nature wish to know.

We can compare this sequence term by term with another sequence \((U_n)_{n \in \mathbb{N}}\) whose nature is known.

Symmetrically, we have the complement:

D'Alembert's rule

Cauchy's rule

Rule of \(n^{\alpha} u_n\)

Let there be two series with positive terms \(\Bigl(\sum u_n, \sum v_n \Bigr)\).

Integrals-series comparison

1. Case of an increasing function

Let \((u_n)_{n \in \mathbb{N}}\) be a numerical sequence with positive terms and \(f(n)\) its associated increasing piecewise continuous function.



We can bound a series by integrals to approximate it such as:

For any function increasing piecewise continuous \(f\),

$$ \int_{0}^n f(t) \ dt \leqslant \sum_{k = 0}^n f(k) \leqslant \int_{1}^{n + 1} f(t) \ dt $$
$$ (\text{with } u_n = f(n)) $$

2. Case of a decreasing function

If on the other hand, the associated function \(f(n)\) is decreasing , then positions switch:



For any function decreasing piecewise continuous \(f\),

$$ \int_{1}^{n + 1} f(t) \ dt \leqslant \sum_{k = 0}^n f(k) \leqslant \int_{0}^n f(t) \ dt $$
$$ (\text{avec } u_n = f(n)) $$

Alternating series

Specific criteria for alternating series

For any decreasing sequence with positive terms \((u_n)_{n \in \mathbb{N}}\) and its alternating series \((A_n)\):

The Fibonacci sequence \((F_n)\) and the golden ratio \((\varphi)\)

The Fibonacci sequence is expressed in a recurrent way as:

Here are its first terms:

Binet's formula

The general term of the Fibonacci sequence \((F_n)\) is worth:

Golden ratio \((\varphi)\): quotient of two consecutive term of the Fibonacci sequence

Retrieving the previous Binet's formula :

We determine that the golden ratio \((\varphi)\) is the the limit of two consecutive terms of the Fibonacci sequence is worth:

Expressions of powers of the golden ratio under a recurrent form

1. With \(\varphi\) only
$$ \forall n \in \mathbb{N}, $$
$$ \varphi^{n + 2} = \varphi^{n + 1} + \varphi^{n} $$
2. With \(\varphi\) and \((F_n)\)
$$ \forall n \in \mathbb{N}, $$
$$ \varphi^{n + 1} = \varphi F_{n + 1} + F_n $$

Recurisve expressions of the golden ration

1. With square roots

Starting from the following expression:

$$ \varphi = \sqrt{\varphi + 1} \qquad (1) $$

We deduce the two following two.

1. Under the form of infinite square roots
$$ \varphi = \sqrt{\sqrt{\sqrt{\sqrt{1 + \dots} + 1} + 1} + 1} $$
2. Under the form of a recurrent sequence
$$ \varphi = \lim_{n \to \infty} a_{n} $$
$$ \text{where } \Biggl \{ \begin{gather*} \text{the sequence } a_n \text{ is expressed under a recurrent form } : a_{n + 1} = \sqrt{1 + a_{n}} \\ \text{its first term } : a_0 = 0 \end{gather*} $$
2. With fractions

Starting from the following expression:

$$ \varphi = 1 + \frac{1}{\varphi} \qquad (2) $$

We deduce the two following two.

1. Under the form of infinite fractions
$$ \varphi = 1 - \frac{1}{1 - \frac{1}{1 - \frac{1}{1 - \frac{1}{1 - \dots}}}} $$
2. Under the form of a recurrent sequence
$$ \varphi = \lim_{n \to \infty} b_{n} $$
$$ \text{where } \Biggl \{ \begin{gather*} \text{the sequence } b_n \text{ is expressed under a recurrent form } : b_{n + 1} = 1 + \frac{1}{b_n} \\ \text{its first term } : b_1 = 1 \end{gather*} $$

The properties of converge series

Multiplication by a reel number

Addition

1. If both series converge
$$ \forall \bigl(a_n, b_n\bigr),$$
$$ \sum a_n \text{ and } \sum b_n \text{ converges } \Longrightarrow \sum (a_n + b_n) \text{ converges } $$
$$ \Longrightarrow $$
$$ \sum_{k = 0}^{+ \infty} (a_k + b_k) = \sum_{k = 0}^{+ \infty} a_k + \sum_{k = 0}^{+ \infty} b_k $$
2. If one of the series diverge
$$ \forall \bigl(a_n, b_n\bigr),$$
$$ \sum a_n \text{ converges and } \sum b_n \text{ diverges } \Longrightarrow \sum (a_n + b_n) \text{ diverges } $$

Identification of series having the same nature

Let \((c_n, d_n)\) be two sequences having terms with a constant sign.

Moreover, if \((l = 1)\) both sequences are equivalent , then:

The properties of numerical sequences

Cesàro's theorem

Let \( (u_n)_{n \in \mathbb{N}} \) be a numerical sequence.

Then, if \((u_n)\) tends towards a certain limit, its average tends towards that same limit.

General term of a sequence of order 2, linear combination of its preceding terms

Let \( (u_n)_{n \in \mathbb{N}} \) be a non-zero recurrent numerical sequence of order 2, and linear combination of its preceding terms such as:

Then, we cand define the general term of this sequence:

The properties of sum

Let \( \bigl(a_n, b_n\bigr) \) be two numerical sequences.

Linearity

Change of index

Let \( n \in \hspace{0.04em} \mathbb{N} \) be a natural number.

1. Simple sum

1. Reversing direction

$$ \forall n \in \hspace{0.04em} \mathbb{N},$$
$$ \sum_{k = 0}^n a_k = \sum_{i = 0}^n a_{n - i} $$

2. Shift to zero

$$ \forall (p, n) \in \hspace{0.04em} \mathbb{N}^2, \enspace p \leqslant n, $$
$$ \sum_{k = p}^n a_k = \sum_{i = 0}^{n - p} a_{p + i} $$
2. Double sum

1. Sum which not depend of indexes

In the case of a double sum indexed by a rectangle \( [\![1,m]\!] \times [\![1,n]\!] \):

$$ \forall (m,n) \in \hspace{0.04em} \mathbb{N}^2,$$
$$ \sum_{i = 1}^m \sum_{j = 1}^n a_{i,j} = \sum_{j = 1}^n \sum_{i = 1}^m a_{i, j} $$

2. Sum which do depend of indexes

In the case of a double triangular sum :

$$ \forall n \in \hspace{0.04em} \mathbb{N},$$
$$ \sum_{1 \leqslant i \leqslant j \leqslant n}^n \Bigl[ a_{i,j} \Bigr] = \sum_{i = 1}^n \sum_{j = i}^n a_{i, j} = \sum_{j = 1}^n \sum_{i = 1}^j a_{i, j} $$

Telescoping of terms of a recurring numerical series

When subtracting two consecutive terms within a sum, we can perform a telescoping of terms :

The usual sums\(: \sum k, \ \sum k^2, \ \sum k^3, \ \sum (2k+1)... \)

Let \((a,n) \in \hspace{0.04em} \mathbb{N}^2\) be two natural numbers.

The sum of natural numbers\( : \sum k\)

1. The first terms

The sum of the \( (n + 1) \) first natural numbers, that is to say from \(0\) to \(n\), is worth:

$$ \forall n \in \mathbb{N}, $$
$$ \sum_{k = 0}^n k = \frac{n(n+1)}{2} $$
2. Generalization: sum from a to n

In the general way, this sum going from \((k = a) \) until \(n\) is worth:

$$ \forall (a,n) \in \hspace{0.04em} \mathbb{N}^2, $$
$$ \sum_{k =a}^{n} k = \frac{(n + a)(n+1- a)}{2} $$

The sum of natural squares\( : \sum k^2\)

1. The first terms

The sum of the \( (n + 1) \) first natural squares, that is to say from \(0\) to \(n\), is worth:

$$ \forall n \in \mathbb{N}, $$
$$ \sum_{k = 0}^n k^2 = \frac{n(n+1)(2n+1)}{6} $$
2. Generalization: sum from a to n

In the general way, this sum going from \((k = a) \) until \(n\) is worth:

$$ \forall (a,n) \in \hspace{0.04em} \mathbb{N}^2, $$
$$ \sum_{k =a}^{n} k^2 = \frac{1}{6} \Bigl( n+1-a\Bigr) \biggl( n(2n+1) + a(2n +2a -1)\biggr) $$

The sum of natural cubes\( : \sum k^3\)

1. The first terms

The sum of the \( (n + 1) \) first natural cubes, that is to say from \(0\) to \(n\), is worth:

$$ \forall n \in \mathbb{N}, $$
$$ \sum_{k = 0}^n k^3 = \frac{n^2 (n+1)^2 }{4} $$

Moreover,

$$ \forall n \in \mathbb{N}, $$
$$ \sum_{k = 0}^n k^3 = \Biggl( \hspace{0.1em} \sum_{k = 0}^n k \hspace{0.1em} \Biggr)^2$$
2. Generalization: sum from a to n

In the general way, this sum going from \((k = a) \) until \(n\) is worth:

$$ \forall (a,n) \in \hspace{0.04em} \mathbb{N}^2, $$
$$ \sum_{k =a}^{n} k^3 = \frac{1}{4} \Bigl( n+1-a\Bigr) \biggl( n^2(n+1) + a(n^2 + an +a^2 - a)\biggr) $$

The sum of odd numbers\( : \sum (2k +1) \)

1. The first terms

The sum of the \( (n + 1) \) first odd numbers, that is to say from \(0\) to \(n\), is worth:

$$ \forall n \in \mathbb{N}, $$
$$ \sum_{k = 0}^n (2k +1) = (n+1)^2 $$
2. Generalization: sum from a to n

In the general way, this sum going from \((k = a) \) until \(n\) is worth:

$$ \forall (a,n) \in \hspace{0.04em} \mathbb{N}^2, $$
$$ \sum_{k =a}^{n} (2k +1) = (n + 1+ a)(n+1- a) $$

The sum of even numbers\( : \sum (2k) \)

1. The first terms

The sum of the \( (n + 1) \) first even numbers, that is to say from \(0\) to \(n\), is worth:

$$ \forall n \in \mathbb{N}, $$
$$ \sum_{k = 0}^n (2k) = n(n+1) $$
2. Generalization: sum from a to n

In the general way, this sum going from \((k = a) \) until \(n\) is worth:

$$ \forall (a,n) \in \hspace{0.04em} \mathbb{N}^2, $$
$$ \sum_{k =a}^{n} (2k) = (n + a)(n+1- a) $$

The sum of the terms of an arithmetical sequence\( : \sum (u_0 + kr) \)

1. The first terms

The sum of the \( (n + 1) \) first terms of an arithmetical sequence, that is to say from \(0\) to \(n\), is worth:

$$ \forall n \in \mathbb{N}, \enspace \forall (u_0, r) \in \hspace{0.04em} \mathbb{R}^2, $$
$$ \sum_{k = 0}^n (u_0 + kr) = \Bigl(n + 1 \Bigr) \Biggl( \frac{u_0 + u_n}{2} \Biggr)$$
$$ (\text{with} \enspace u_n = u_0 + nr) $$
2. Generalization: sum from a to n

In the general way, this sum going from \((k = a) \) until \(n\) is worth:

$$ \forall (a,n) \in \hspace{0.04em} \mathbb{N}^2, $$
$$ \sum_{k = a}^n (u_0 + kr) = \Bigl(n + 1 - a\Bigr) \Bigg( \frac{u_0 + u_{n+a}}{2} \Biggr)$$
$$ (\text{with} \enspace u_n = u_0 + nr) $$

The sum of the natural powers of a real number\( : \sum q^k\)

1. The first terms

The sum of the \( (n + 1) \) first natural powers of a real number, that is to say from \(0\) to \(n\), is worth:

$$ \forall n \in \mathbb{N}, \enspace \forall q \in \hspace{0.04em} \Bigl[ \mathbb{R} \ \backslash \ \bigl \{0,1 \bigr \} \Bigr] , $$
$$ \sum_{k = 0}^n q^k = \frac{q^{n+1} - 1}{q-1} $$
2. Generalization: sum from a to n

In the general way, this sum going from \((k = a) \) until \(n\) is worth:

$$ \forall (a,n) \in \hspace{0.04em} \mathbb{N}^2, \enspace \forall q \in \hspace{0.04em} \Bigl[ \mathbb{R} \ \backslash \ \bigl \{0,1 \bigr \} \Bigr] , $$
$$ \sum_{k =a}^{n} q^k = \frac{q^{n+1} - q^{a}}{q-1} $$

The sum of the terms of a geometrical sequence\( : \sum (v_0 . q^k) \)

1. The first terms

The sum of the \( (n + 1) \) first terms of a geometrical sequence, that is to say from \(0\) to \(n\), is worth:

$$ \forall n \in \mathbb{N}, \enspace \forall v_0 \in \hspace{0.04em} \mathbb{R}, \ \forall q \in \hspace{0.04em} \Bigl[ \mathbb{R} \ \backslash \ \bigl \{0,1 \bigr \} \Bigr] , $$
$$ \sum_{k = 0}^n (v_0.q^k ) = v_0.\frac{q^{n+1} - 1}{q-1} $$
2. Generalization: sum from a to n

In the general way, this sum going from \((k = a) \) until \(n\) is worth:

$$ \forall (a,n) \in \hspace{0.04em} \mathbb{N}^2, \enspace \forall v_0 \in \hspace{0.04em} \mathbb{R}, \ \forall q \in \hspace{0.04em} \Bigl[ \mathbb{R} \ \backslash \ \bigl \{0,1 \bigr \} \Bigr] , $$
$$ \sum_{k =a}^{n} v_0.q^k = v_0.\frac{q^{n+1} - q^{a}}{q-1} $$

Sum of a binomial \( : \sum \binom{n}{p} \)

1. Horizontal sum from 0 to n
$$\forall n \in \mathbb{N}, $$
$$ \sum_{p = 0}^n \binom{n}{p} = \hspace{0.2em} 2^n $$
2. Vertical sum from r to n
$$\forall (r, n) \in \hspace{0.04em}\mathbb{N}^2, \enspace r \leqslant n, $$
$$ \sum_{k=r}^n \binom{k}{r} = \binom{n+1}{r +1} $$

Trigonometric sums \( : \sum \bigl[\cos(k \theta) + i.\sin(kx)\bigr] \)

Hence, as:

Thus:

The infinite sum of the inverse of natural squares \( : \sum \frac{1}{k^2} \)

Equivalent of the sum of square roots \( : \sum \sqrt{k} \)

Recap table of usual sums

The Bézout's theorem and its corollary

The Bézout's theorem

Bézout's theorem tells us that:

Corollary of the Bézout's theorem

Bézout's theorem corollary tells us that::

As well, for a simple product:

And it is possible to generalize it for any product of integers:

The Euclid's algorithm

Let be two natural numbers \( (a, b) \in \hspace{0.04em}\mathbb{N}^2, \enspace a > b \).

Euclid's algorithm tells us that:

1. If \( b \) divides \( a \)
   $$ \exists q \in \mathbb{N}, $$
   $$ b / a \ \Longrightarrow \ a = bq \ \Longrightarrow \ PGCD(a, b) = b $$
2. If \( b\) does not divide \( a\)
   $$ \exists q \in \mathbb{N}, \enspace \exists R \in [\![0, (b-1) ]\!] , $$
   $$ b \nmid a \ \Longrightarrow \ a = bq + R $$

   The \( GCD(a, b) \) is the last non-zero \( R_n \) of the Euclidian division of \( a \) by \( b \) such as:

   $$ a = \textcolor{rgb(192 52 52)}{b} q_0 + \textcolor{rgb(54 152 46)}{R_0} $$

   As long as the remainder does not divide the previous division divisor , we continue the algorithm.

   We then decompose the new element at each step:

   $$ b = \textcolor{rgb(192 52 52)}{R_0} q_1 + \textcolor{rgb(54 152 46)}{R_1} \ \Longrightarrow \ R_0 = \textcolor{rgb(192 52 52)}{R_1} q_2 + \textcolor{rgb(54 152 46)}{R_2} \ \Longrightarrow \ R_k = \textcolor{rgb(192 52 52)}{R_{k + 1}} q_{k + 2} + \textcolor{rgb(54 152 46)}{R_{k + 2}} $$
   $$ up \ to \ \Longrightarrow R_{n - 3} = q_{n -1 }\textcolor{rgb(192 52 52)}{R_{n - 2}} + \textcolor{rgb(54 152 46)}{R_{n -1}} $$

   As long a the remainder \( R_{n - 1} \) still does not divide \( R_{n - 2} \), the remainder \( R_{n - 2} \) can be written:

   $$ R_{n - 2} = q_{n}\textcolor{rgb(192 52 52)}{R_{n - 1}} + \textcolor{rgb(54 152 46)}{R_{n}} $$

   Until the moment where finally \( R_n \) divides \( R_{n - 1} \), so \( R_{n - 1} \) will be written:

   $$ R_{n - 1} = q_{n + 1}\textcolor{rgb(192 52 52)}{R_{n}} $$

   At this stage, as we trace back the algorithm injecting each previous step's values until \( a \) and \( b \), we realize that we can always factorize by \(R_{n} \). Thus:

   $$ GCD(a, b) = R_{n} $$

   \( \forall n \in \mathbb{N}\), \(\forall k \in [\![0, n ]\!] \), as long as the remainder \(R_{k} \) does not divide its predecessor \(R_{k -1 } \):

   $$ a \nmid b \Longrightarrow GCD(a, b) = GCD(b, R_0) = GCD(R_0, R_1) = \enspace ... \enspace = GCD(R_{n - 1}, R_n) = R_n \neq 0$$

The fermat's little theorem

Let \(a \in \mathbb{Z}\) be an integer and \(p \in \mathbb{P}\) a prime number.

Fermat's small theorem tells us that:

The Gauss' theorem and its corollary

The Gauss' theorem

Gauss' theorem tells us that :

Corollary of the Gauss' theorem

Gauss' theorem corollary tells us that :

The properties of congruences

With congruences, even if we have to deal with negative numbers in an equation, we will try to get back to positive numbers.

Let \((a, b) \in \hspace{0.04em} \mathbb{Z}^2\) be two natural numbers, with \( a > b\).

We say that \(a\) and \(b\) are congruent modulo \(n\), if they have the same remainder \(R\) in the Euclidian division by \(n\).

Congruence equivalence

Congruence between a number and zero

Reflexivity

Symmetry

Transitivity

Simplification of the divisor

We can simplify a congruence by a number on both sides, only if this number and the divisor are coprime.

Addition

As well, we can also see this property under the form of a linear combination,

Multiplication

Multiplication by a number

Simplification by a coprime number with the divisor

Powers

Recap table of the properties of congruences

The properties of divisibility

Simplification

Transitivity

Addition of the dividends

Linear combination of the dividends

Recap table of the properties of divisibility

The properties of prime numbers

The set \(\mathbb{P}\) is the set of prime numbers:

We call a prime number, a number \(p \in \mathbb{P}\) which has only itself and \(1\) as a divisor.

Likewise, we will say that two numbers \((a, b) \in \hspace{0.04em} \mathbb{Z}^2\) are coprime if their unique common divisor is \(1\).

Two prime numbers are coprime

Two prime numbers are coprime.

Breakdown in prime factors

All natural number \(n \geqslant 2\) uniquely decomposes into a prime factors product.

Every number higher than 2 owns at least one prime divisor

All natural number \(n \geqslant 2\) has at least one prime divisor.

Every non-prime number higher than 4 owns at least one strict divisor

All non-prime natural number \(n \geqslant 4 \) has at least one strict divisor \(d_0 \) such as \( d_0 \leqslant \sqrt{n} \).

Coprimity link between a prime number and an integer

Euclid's lemma

Euclid's lemma corollary

Recap table of the prime numbers properties

The properties of the GCD of two natural numbers

For all \( (a, b) \in \hspace{0.04em}\mathbb{N}^2, \enspace a > b \), we notice:

• \( \delta = a \wedge b = GCD(a, b) \);

\( GCD(a, b) \) is the greatest common divisor of \( a \) and \( b \).

It is the last non-zero remainder \( R_n \) of the Euclidian division of \( a \) by \( b \) in the Euclid's algorithm .

• \( \mathcal{D}(a, b)\) the set of common divisors of \( a \) and \( b \);

• \( \mathcal{D}(\delta)\) the set of all divisors of \( GCD(a, b) \).

Equality between the divisors of a and b and the divisors of their GCD

The set of common divisors of \( a \) and \( b \) is the set of divisors of \( GCD(a, b) \).

Equality between the successives GCD of the Euclid's algorithm

Breakdown in prime factors

Let be the breakdown of \(a \) and \(b \) in prime factors :

So,

The Bézout's identity

Breakdown of two numbers in relation to their GCD

Linearity

Link between GCD and LCM

Recap table of the GCD properties