# Factorial function ## Definition The [factorial function](https://en.wikipedia.org/wiki/Factorial) is defined as: $$ n! = \prod_{i=1}^{n}i $$ ## I

# Computes the vector norm We follow the convention of the l2 norm in this implementation. ``` # use scilib::coordinate::cartesian::Cartesian; let v

# Raising polynomial to an integer power ## Definition We compute the new polynomial that corresponds to the power it is raised to. ## Inputs - `n`:

# Binomial theorem ## Definition The [binomial theorem](https://en.wikipedia.org/wiki/Binomial_theorem) is defined as: $$ \binom{n}{k} = \frac{n!}{k!

# Mean value of a series ## Definition We follow the mathematical definition of the mean: $$ m = \frac{1}{n} \sum^{n}_{i = 1} x_i $$ ## Inputs - `va

# $J_nu$ Bessel function (real order) ## Definition $$ J_\alpha(x) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha+1)}\left( \frac{x}{2} \right)

# Gamma function ## Definition The [gamma function](https://en.wikipedia.org/wiki/Gamma_function) is a generalization of the factorial, and is define

# Computing for a real value ## Definition Since there can be behaviors associated to some polynomials, we may need to call a specific computation fu

# Y Bessel function, real index ## Definition The $Y$ Bessel represent the second kind of Bessel function, and are solution to Bessel differential eq

# Error function ## Definition The [error function](https://en.wikipedia.org/wiki/Error_function) is defined as: $$ \mathrm{erf}(z) = \frac{2}{\sqrt{

# $I$ modified Bessel function (real order) ## Definition The $I$ modified First Bessel function represent another kind of solution to the Bessel dif

# Object irradiance ## Definition Computes the irradiance at a certain distance from an object, using its luminosity. The `luminosity` is in Watts an

# Finds the maximum value in a slice ## Definition Looks for the largest number in the given set. ## Inputs - `val`: the slice with the numbers to c

# First spherical Bessel function (array): j ## Inputs Computes the first kind of spherical bessel function by recurrence to get an array of function

# First spherical Bessel function: $j_n$ (real order) ## Definition We follow the definition based on the $J$ Bessel functions: $$ j_n(x) = \sqrt{\fr

# Standard deviation of a series ## Definition We follow the mathematical definition of the standard deviation: $$ \sigma = \sqrt{ \frac{1}{n} \sum^{

# Second spherical Bessel function (array): y ## Inputs Computes the second kind of spherical bessel function by recurrence to get an array of functi

# Second spherical Bessel function: $y$ ## Definition We follow the definition based on the $Y$ Bessel functions: $$ y_n(x) = \sqrt{\frac{\pi}{2x}}Y_

# Generalized binomial function ## Definition The [generalized binomial theorem](https://en.wikipedia.org/wiki/Binomial_theorem) is defined as: $$ \b

# Polynomial derivation ## Definition Computes the mth derivative of the polynomial. ## Inputs - `m` the derivative order to compute ## Example ```

# Distance between two points ``` # use std::f64::consts::SQRT_2; # use scilib::coordinate::spherical::Spherical; let s1 = Spherical::from_degree(SQR

# Distance squared between two points Computes the square of the distance between two points. As the distance is the L2 norm, it requires taking the

# Falling factorial ## Definition The [falling factorial](https://en.wikipedia.org/wiki/Falling_and_rising_factorials) is a polynomial, which can be

# First Hankel function: $H^{(1)}$ ## Definition Computes the first kind of Hankel function, which is defined as: $$ H_\alpha^{(1)}(x) = J_\alpha(x)

# Second Hankel function: $H^{(2)}$ ## Definition Computes the first kind of Hankel function, which is defined as: $$ H_\alpha^{(2)}(x) = J_\alpha(x)

# The actual Hill radius equation

# Hill radius ## Definition The  gives an estimate of the sphere of influence of an object gi

# Creating a range from a number of points Creating a range with the start, stop, and number of points. If the range is invalid, an empty Vec is retu

# Finds the minimum value in a slice ## Definition Looks for the smallest number in the given set. ## Inputs - `val`: the slice with the numbers to

# Hyperbolic secant ## Definition The hyperbolic secant is defined as: $$ \mathrm{sech}(x) = \frac{1}{\cosh(x)} $$ ## Inputs - `x`: the value at whi

# Variance of a series ## Definition The definition of the variance is: $$ V = \frac{1}{n - 1} \sum^{n}_{i = 1} (x_i - m)^2 $$ Where $m$ is the mean

# Absolute magnitude ## Definition Same approach that the apparent magnitude, but set at a distance of 10pc. $$ M = -2.5\log\left( \frac{L}{L_\circ}

# Apparent magnitude ## Definition Uses the standard reference luminosity to compute the apparent magnitude of an object. $$ m = -2.5\log\left( \frac

# Bernoulli polynomials ## Definition The [Bernoulli polynomial](https://en.wikipedia.org/wiki/Bernoulli_polynomials) are present in a great variety

# Bernoulli number generator ## Definition Following the $B_m^{-}$ convention, we use the formula: $$ B_m^{-} = \sum_{k=0}^{m}\sum_{v=0}^{k}(-1)^v \bi

# Bessel Polynomials ## Definition The [Bessel polynomials](https://en.wikipedia.org/wiki/Bessel_polynomials) are an orthogonal sequence of polynomia

# Euler Beta function ## Definition The [beta function](https://en.wikipedia.org/wiki/Beta_function) is an integral similar to the gamma function, an

# Creating a range by increment Creating a range with the start, stop, and increment. If the range is invalid, an empty Vec is returned. The points o

# Cauchy distribution ## Definition The [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution) is a continuous probability distribu

# Cumulative function of the Cauchy distribution ## Definition The cumulative function for Cauchy is defined by: $$ F_\gamma(x_0, x) = \frac{1}{\pi}\

# Compute the polynomial for a real number

# Compute the polynomial for a complex number

# Computing for a complex value ## Definition Since there can be behaviors associated to some polynomials, we may need to call a specific computation

# Computation for Legendre

# Computation for Legendre

# Convolution Computes the convolution of two vectors, including the edges. ``` # use scilib::signal::convolve; // Creating two vectors to convolve

# Exact convolution Computes the convolution of two vectors, only for points where both vectors are completely covered. ``` # use scilib::signal::co

# Hyperbolic cotangent ## Definition The hyperbolic cotangent is defined as: $$ \coth(x) = \frac{\cosh(x)}{\sinh(x)} = \frac{\exp(2x) + 1}{\exp(2x) -

# Hyperbolic cosecant ## Definition The hyperbolic cosecant is defined as: $$ \mathrm{csch}(x) = \frac{1}{\sinh(x)} $$ ## Inputs - `x`: the value at

# Distance between two points Computes the distance between two points in space. ``` # use scilib::coordinate::cartesian::Cartesian; let f = Cartesi

# Distance between two points ``` # use std::f64::consts::SQRT_2; # use scilib::coordinate::cylindrical::Cylindrical; let s1 = Cylindrical::from_degr

# Distance modulus ## Definition Computes the distance of an object based on its apparent and absolute magnitudes: $$ d = 10^{1 + \frac{m-M}{5}} $$ W

# Distance in space between two objects ## Definition Computes the actual distance in space between two objects, using the spherical coordinates ##

# Equilibrium temperature ## Definition Knowing the stellar effective temperature, its radius, the distance from the star and the albedo of the consi

# Complementary error function ## Definition The [complementary error function](https://en.wikipedia.org/wiki/Error_function) is defined as: $$ \math

# Imaginary error function ## Definition The [imaginary error function](https://en.wikipedia.org/wiki/Error_function) is defined as: $$ \mathrm{erfi}

# Euler polynomials ## Definition The [Euler polynomial](https://en.wikipedia.org/wiki/Bernoulli_polynomials) are present in a great variety of parti

# Euler number generator ## Definition We use the formula: $$ E_k = \sum_{i=1}^{k}(-1)^{i}\frac{1}{2^i}\sum_{l=0}^{2i}(-1)^l\binom{2i}{l}(i-l)^{k} $$

# Falling factorial polynomials ## Definition The [falling factorial](https://en.wikipedia.org/wiki/Falling_and_rising_factorials) is defined as: $$

# Rising factorial polynomials ## Definition The [rising factorial](https://en.wikipedia.org/wiki/Falling_and_rising_factorials) is defined as: $$ x^

# Fast Fourier transform Computes the FFT for a one-dimensional array, based on the discrete Fourier transform. This function accepts complex input.

# Frequency Planck's law ## Definition This computes the following equation: $$ B_\nu(\nu,T) = \frac{2h\nu^3}{c^2} \frac{1}{\exp\left(\frac{h\nu}{k_BT

# Frequency Planck's law (spectrum) ## Definition This computes the following equation: $$ B_\nu(\nu,T) = \frac{2h\nu^3}{c^2} \frac{1}{\exp\left(\frac

# From the point Creates a Spherical struct from three given points in space. ``` # use scilib::coordinate::spherical::Spherical; let m = Spherical

# From each point Creates a Cartesian struct from three given points in space. ``` # use scilib::coordinate::cartesian::Cartesian; let m = Cartesian

# From the point Creates a Cylindrical struct from three given points in space. ``` # use scilib::coordinate::cylindrical::Cylindrical; let m = Cyli

# Creates a new polynomial ## Definition We create a polynomial with integer order powers, and real coefficients, such that: $$ p = c_0 + c_1x + c_2x

# From another coordinate system Creates an Spherical struct from another coordinate system. Calls the `Into<Spherical>` method, which is verified in

# From another coordinate system Creates an Cartesian struct from another coordinate system. Calls the `Into<Cartesian>` method, which is verified in

# From another coordinate system Creates an Cylindrical struct from another coordinate system. Calls the `Into<Cylindrical>` method, which is verifie

# Radec coordinates from angles in degrees Generates the struct from angles given in degrees. ## Inputs - `ra`: right ascension, in degrees - `dec`: