# Special Functions

All the following special functions are available in the static SpecialFunctions class:

## Factorial

• Factorial(x)

$x \mapsto x! = \prod_{k=1}^{x} k = \Gamma(x+1)$

Code Sample:

 double x = SpecialFunctions.Factorial(14); // 87178291200.0 double y = SpecialFunctions.Factorial(31); // 8.2228386541779224E+33 
• FactorialLn(x)

$x \mapsto \ln x! = \ln\Gamma(x+1)$

• Binomial(n,k)

Binomial Coefficient

$\binom{n}{k} = \mathrm{C}_n^k = \frac{n!}{k! (n-k)!}$

• BinomialLn(n,k)

$\ln \binom{n}{k} = \ln n! - \ln k! - \ln(n-k)!$

• Multinomial(n,k[])

Multinomial Coefficient

$\binom{n}{k_1,k_2,\dots,k_r} = \frac{n!}{k_1! k_2! \cdots k_r!} = \frac{n!}{\prod_{i=1}^{r}k_i!}$

## Exponential Integral

• ExponentialIntegral(x,n)

Generalized Exponential Integral

$E_n(x) = \int_1^\infty t^{-n} e^{-xt}\,\mathrm{d}t$

## Gamma functions

#### Gamma

• Gamma(a)

$\Gamma(a) = \int_0^\infty t^{a-1} e^{-t}\,\mathrm{d}t$

• GammaLn(a)

$\ln\Gamma(a)$

#### Incomplete Gamma

• GammaLowerIncomplete(a,x)

Lower incomplete Gamma function, unregularized.

$\gamma(a,x) = \int_0^x t^{a-1} e^{-t}\,\mathrm{d}t$

• GammaUpperIncomplete(a,x)

Upper incomplete Gamma function, unregularized.

$\Gamma(a,x) = \int_x^\infty t^{a-1} e^{-t}\,\mathrm{d}t$

#### Regularized Gamma

• GammaLowerRegularized(a,x)

Lower regularized incomplete Gamma function.

$\mathrm{P}(a,x) = \frac{\gamma(a,x)}{\Gamma(a)}$

• GammaUpperRegularized(a,x)

Upper regularized incomplete Gamma function.

$\mathrm{Q}(a,x) = \frac{\Gamma(a,x)}{\Gamma(a)}$

• GammaLowerRegularizedInv(a, y)

Inverse $$x$$ of the lower regularized Gamma function, such that $$\mathrm{P}(a,x) = y$$.

$\mathrm{P}^{-1}(a,y)$

#### Psi: Derivative of Logarithmic Gamma

• DiGamma(x)

$\psi(x) = \frac{\mathrm{d}}{\mathrm{d}x}\ln\Gamma(x)$

• DiGammaInv(p)

Inverse $$x$$ of the DiGamma function, such that $$\psi(x) = p$$.

$\psi^{-1}(p)$

## Euler Beta functions

#### Euler Beta

• Beta(a,b)

$\mathrm{B}(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1}\,\mathrm{d}t = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$

• BetaLn(a,b)

$\ln\mathrm{B}(a,b) = \ln\Gamma(a) + \ln\Gamma(b) - \ln\Gamma(a+b)$

#### Incomplete Beta

• BetaIncomplete(a,b,x)

Lower incomplete Beta function (unregularized).

$\mathrm{B}_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1}\,\mathrm{d}t$

#### Regularized Beta

• BetaRegularized(a,b,x)

Lower incomplete regularized Beta function.

$\mathrm{I}_x(a,b) = \frac{\mathrm{B}(a,b,x)}{\mathrm{B}(a,b)}$

## Error functions

#### Error Function

• Erf(x)

$\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,\mathrm{d}t$

• ErfInv(z)

Inverse $$x$$ of the Error function, such that $$\mathrm{erf}(x) = z$$.

$z \mapsto \mathrm{erf}^{-1}(z)$

#### Complementary Error function.

• Erfc(x)

$\mathrm{erfc}(x) = 1-\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}}\int_x^\infty e^{-t^2}\,\mathrm{d}t$

• ErfcInv(z)

Inverse $$x$$ of the complementary Error function, such that $$\mathrm{erfc}(x) = z$$.

$z \mapsto \mathrm{erfc}^{-1}(z)$

Code Sample:

 double erf = SpecialFunctions.Erf(0.9); // 0.7969082124 

## Sigmoid: Logistic function

• Logistic(x)

$x \mapsto \frac{1}{1+e^{-x}}$

• Logit(y)

Inverse of the Logistic function, for $$y$$ between 0 and 1 (where the function is real-valued).

$y \mapsto \ln \frac{y}{1-y}$

## Harmonic Numbers

• Harmonic(t)

The n-th Harmonic number is the sum of the reciprocals of the first n natural numbers. With $$\gamma$$ as the Euler-Mascheroni constant and the DiGamma function:

$\mathrm{H}_n = \sum_{k=1}^{n}\frac{1}{k} = \gamma - \psi(n+1)$

• GeneralHarmonic(n, m)

Generalized harmonic number of order n of m.

$\mathrm{H}_{n,m} = \sum_{k=1}^{n}\frac{1}{k^m}$

## Bessel and Struve Functions

#### Bessel functions

Bessel functions are canonical solutions $$y(x)$$ of Bessel's differential equation

$x^2\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+x\frac{\mathrm{d}y}{\mathrm{d}x}+(x^2-\alpha^2)y = 0$

#### Modified Bessel functions

Modified Bessel's equation:

$x^2\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+x\frac{\mathrm{d}y}{\mathrm{d}x}-(x^2+\alpha^2)y = 0$

Modified Bessel functions:

\begin{align} \mathrm{I}_\alpha(x) &= \imath^{-\alpha}\mathrm{J}_\alpha(\imath x) = \sum_{m=0}^\infty \frac{1}{m!\Gamma(m+\alpha+1)}\left(\frac{x}{2}\right)^{2m+\alpha} \\ \mathrm{K}_\alpha(x) &= \frac{\pi}{2} \frac{\mathrm{I}_{-\alpha}(x)-\mathrm{I}_\alpha(x)}{\sin(\alpha\pi)} \end{align}

• BesselI0(x)

Modified or hyperbolic Bessel function of the first kind, order 0.

$x \mapsto \mathrm{I}_0(x)$

• BesselI1(x)

Modified or hyperbolic Bessel function of the first kind, order 1.

$x \mapsto \mathrm{I}_1(x)$

• BesselK0(x)

Modified or hyperbolic Bessel function of the second kind, order 0.

$x \mapsto \mathrm{K}_0(x)$

• BesselK0e(x)

Exponentionally scaled modified Bessel function of the second kind, order 0.

$x \mapsto e^x\mathrm{K}_0(x)$

• BesselK1(x)

Modified or hyperbolic Bessel function of the second kind, order 1.

$x \mapsto \mathrm{K}_1(x)$

• BesselK1e(x)

Exponentially scaled modified Bessel function of the second kind, order 1.

$x \mapsto e^x\mathrm{K}_1(x)$

#### Struve functions

Struve functions are solutions $$y(x)$$ of the non-homogeneous Bessel's differential equation

$x^2\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+x\frac{\mathrm{d}y}{\mathrm{d}x}+(x^2-\alpha^2)y = \frac{4(\frac{x}{2})^{\alpha+1}}{\sqrt{\pi}\Gamma(\alpha+\frac{1}{2})}$

#### Modified Struve functions

Modified equation:

$x^2\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+x\frac{\mathrm{d}y}{\mathrm{d}x}-(x^2+\alpha^2)y = \frac{4(\frac{x}{2})^{\alpha+1}}{\sqrt{\pi}\Gamma(\alpha+\frac{1}{2})}$

Modified Struve functions:

$\mathrm{L}_\alpha(x) = \left(\frac{x}{2}\right)^{\alpha+1}\sum_{k=0}^\infty \frac{1}{\Gamma(\frac{3}{2}+k)\Gamma(\frac{3}{2}+k+\alpha)}\left(\frac{x}{2}\right)^{2k}$

• StruveL0(x)

Modified Struve function of order 0.

$x \mapsto \mathrm{L}_0(x)$

• StruveL1(x)

Modified Struve function of order 1.

$x \mapsto \mathrm{L}_1(x)$

#### Misc

• BesselI0MStruveL0(x)

Difference between the Bessel $$I_0$$ and the Struve $$L_0$$ functions.

$x \mapsto I_0(x) - L_0(x)$

• BesselI1MStruveL1(x)

Difference between the Bessel $$I_1$$ and the Struve $$L_1$$ functions.

$x \mapsto I_1(x) - L_1(x)$

## Numeric Stability

• ExponentialMinusOne(power)

$$\exp x-1$$ is a typical case where a subtraction can be fatal for accuracy. For example, at $$10^{-13}$$ the naive expression is 0.08% off, at $$10^{-15}$$ roughly 11% and at $$10^{-18}$$ it just returns 0.

$x \mapsto e^x - 1$

• Hypotenuse(a, b)

$(a,b) \mapsto \sqrt{a^2 + b^2}$

## Trigonometry

The Trig class provides the complete set of fundamental trigonometric functions for both real and complex arguments.

• Trigonometric: Sin, Cos, Tan, Cot, Sec, Csc
• Trigonometric Inverse: Asin, Acos, Atan, Acot, Asec, Acsc
• Hyperbolic: Sinh, Cosh, Tanh, Coth, Sech, Csch
• Hyperbolic Area: Asinh, Acosh, Atanh, Acoth, Asech, Acsch
• Sinc: Normalized sinc function $$x \mapsto \frac{\sin\pi x}{\pi x}$$