special.js

A collection of special math functions written in JavaScript with full complex support! I'm looking for contributors to help test and more functionality to the library so feel free to make an issue or puul request with you suggestions!

Usage

Grab the special.js or special.min.js from the repo and add it to your project to get started

Documentation

Documentation for the library lives here. All the functions are in a namespace and can be used using the special. prefix

Complex Numbers

Complex(a: Number, b: Number)

A constructor used to create a complex number. Can be used without the new keyword using special.complex(a, b)

Complex.add(a: Complex)

Returns the sum of a complex number

Complex.sub(a: Complex)

Returns the difference of a complex number

Complex.mult(a: Complex)

Returns the product of a complex number

Complex.div(a: Complex)

Returns the quotient of a complex number

Complex.get_conjugate()

Returns the conjugate of a complex number

Complex.magnitude()

Returns the magnitude (absolute value) of a complex number

add(a: Complex, b: Complex)

Returns the sum of 2 complex numbers

sub(a: Complex, b: Complex)

Returns the difference of 2 complex numbers

mult(a: Complex, b: Complex)

Returns the product of 2 complex numbers

div(a: Complex, b: Complex)

Returns the quotient of 2 complex numbers

exp(a: Complex)

Returns the exponential function of a complex number

sqrt(a: Complex, b: Complex)

Returns the square root of a complex number

polar(a: Complex)

Returns the polar form of a cartesian complex number

cartesian(a: Complex)

Returns the cartesian form of a polar complex number

pow(a: Complex, b: Complex)

Returns the power of a complex number to another

abs(a: Complex)

Returns the absolute value of a complex number

arg(a: Complex)

Returns the argument of a complex number

log(a: Complex)

Returns the natural logarithm of a complex number

sin(a: Complex)

Returns the sin of a complex number

cos(a: Complex)

Returns the cos of a complex number

tan(a: Complex)

Returns the tan of a complex number

sinh(a: Complex)

Returns the sinh of a complex number

cosh(a: Complex)

Returns the cosh of a complex number

tanh(a: Complex)

Returns the tanh of a complex number

asin(a: Complex)

Returns the arcsin of a complex number

acos(a: Complex)

Returns the arccos of a complex number

atan(a: Complex)

Returns the arctan of a complex number

Special Functions

gcd(a: Integer, b: Integer)

Returns the gcd of a 2 integers

lcm(a: Integer, b: Integer)

Returns the lcm of a 2 integers

isPrime(a: Integer)

Returns a boolean value determining the primality of an integer

mobius(a: Integer)

Returns the mobius mu function of an integer

nCr(a: Complex, b: Complex)

Returns nCr of 2 numbers making use of the gamma function to extend the definition to all of the complex plane

gamma(a: Complex)

Returns the gamma function of a complex number defined by $\displaystyle \Gamma(z)=\int_0^\infty e^{-t}t^{z-1}dt$ making use of the Lanczos Approximation. Precision is fixed.

bernoulli(n: Complex)

Returns the nth Bernoulli number making use of the equation $B_n=-n\zeta(1-n)$ where $\zeta(s)$ is the Riemann Zeta Function.

factorial(a: Integer)

Returns the factorial of a positive integer

riemann_zeta(s: Complex, prec: Number)

Returns the analytic continuation of the Riemann Zeta Function for the entire complex plane aside from $s=1$. Precision can be set and is 1e-3 at default. Computation makes use of the function $\displaystyle d_k = n\sum_{j=k}^n \frac{(n+j-1)!4^j}{(n-j)!(2j)!}$ where $n$ is chosen arbitrarily making use of the precision and uses $\displaystyle\zeta(s) = \frac{1}{d_0(1-2^{1-s})}\sum_{k=1}^n \frac{(-1)^{k-1}d_k}{k^s}$ for $\Re(s) > 1$ and uses the functional equation $\displaystyle \zeta(s) = 2^s\pi^{s-1}\sin\bigg(\frac{\pi s}{2}\bigg)\Gamma(1-s)\zeta(1-s)$ where $\Gamma(s)$ is the gamma function to extend it to the entire complex plane.

beta(a: Complex, b: Complex)

Returns the beta function of 2 complex numbers defined by $\displaystyle B(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$

lower_incomplete_gamma(s: Complex, z: Complex, prec: Number)

Returns the lower incomplete gamma function defined by $\displaystyle \int_0^z t^{s-1}e^{-t}dt$ to a specified precision with default 1e-3

incomplete_gamma(s: Complex, z: Complex, prec: Number)

Returns the incomplete gamma function defined by $\displaystyle \int_x^\infty t^{s-1}e^{-t}dt$ using $\displaystyle \gamma(s,z)+\Gamma(s,z) = \Gamma(s)$ where $\gamma(s,z)$ is the lower incomplete gamma function computed to a specified precision with default 1e-3

Li(s: Complex, prec: Number)

Returns the Logarithmic Integral function defined by $\displaystyle \int_0^x \frac{dt}{\ln(x)}$ using a sum given by Ramanujan defined by $\displaystyle Li(x)= \gamma + \ln\ln x + \sqrt{x}\sum_{n=1}^\infty \frac{(-1)^{n-1}(\ln x)^n}{n! \ 2^{n-1}} \sum_{k=0}^{\lfloor \frac{n-1}{2}\rfloor} \frac{1}{2k+1}$ where $\gamma$ is the Euler-Mascheroni constant $0.57221...$. Precision is set to 1e-3 by default

Ei(s: Complex, prec: Number)

Returns the Exponential Integral function defined by $\displaystyle \int_{-\infty}^x \frac{e^t}{t}dt$ using a sum by Ramanujan defined by $\displaystyle Ei(x)= \gamma + \ln x + \exp{\frac{x}{2}}\sum_{n=1}^\infty \frac{(-1)^{n-1}(x)^n}{n! \ 2^{n-1}} \sum_{k=0}^{\lfloor \frac{n-1}{2}\rfloor} \frac{1}{2k+1}$ where $\gamma$ is the Euler-Mascheroni constant $0.57221...$. Precision is set to 1e-3 by default

Si(s: Complex, prec: Number)

Returns the Sine Integral function defined by $\displaystyle \int_{0}^x \frac{\sin t}{t}dt$ using a sum by Ramanujan defined by $\displaystyle Si(x)=\sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)(2n+1)!}$ Precision is set to 1e-3 by default

Ci(s: Complex, prec: Number)

Returns the Cosine Integral function defined by $\displaystyle \int_{0}^x \frac{\cos t}{t}dt$ using a sum by Ramanujan defined by $\displaystyle Si(x)=\gamma + \ln x + \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)(2n)!}$ where $\gamma$ is the Euler-Mascheroni constant $0.57221...$. Precision is set to 1e-3 by default

erf(s: Complex, prec: Number)

Returns the Error function defined by $\displaystyle \frac{2}{\sqrt \pi}\int_{0}^x e^{-t^2}dt$ using and indentity defined by $\displaystyle \text{erf}(x) = \pi ^ {-\frac{1}{2}} \gamma\bigg(\frac{1}{2}, x^2\bigg)$ where $\gamma$ is the lower incomplete gamma function. Precision is set to 1e-3 by default

LambertW(x: Complex, prec: Number)

Returns the Lambert W function defined by begin the inverse of $xe^x$ using the Halley's Iteration method $\displaystyle w_{j+1} = w_j - \frac{w_je^{w_j}-z}{e^{w_j}(w_j+1)-\frac{(w_j+2)(w_je^{w_j}-z)}{2w_j+2}}$ with a default precision of 1e-3

AGM(a: Complex, b: Complex, prec: Number)

Returns the Arithmetic-Geometric Mean of 2 numbers using the iteration formula $\displaystyle a_{n+1} = \frac{1}{2}(a_n + b_n)$ and $\displaystyle b_{n+1} = \sqrt{a_nb_n}$ until the difference between $a_n$ and $b_n$ is less than the precision. Default precision is 1e-3