All the following special functions are available in the static SpecialFunctions class:
Factorial(x)
$$$ x \mapsto x! = \prod_{k=1}^{x} k = \Gamma(x+1)
Code Sample:
[lang=csharp]
double x = SpecialFunctions.Factorial(14); // 87178291200.0
double y = SpecialFunctions.Factorial(31); // 8.2228386541779224E+33
FactorialLn(x)
$$$ 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
$$$ \binom{n}{k_1,k_2,\dots,k_r} = \frac{n!}{k_1! k_2! \cdots k_r!} = \frac{n!}{\prod_{i=1}^{r}k_i!}
ExponentialIntegral(x,n)
Generalized Exponential Integral
$$$ E_n(x) = \int_1^\infty t^{-n} e^{-xt},\mathrm{d}t
Gamma(a)
$$$ \Gamma(a) = \int_0^\infty t^{a-1} e^{-t},\mathrm{d}t
GammaLn(a)
$$$ \ln\Gamma(a)
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
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
$$$ \mathrm{P}^{-1}(a,y)
DiGamma(x)
$$$ \psi(x) = \frac{\mathrm{d}}{\mathrm{d}x}\ln\Gamma(x)
DiGammaInv(p)
Inverse
$$$ \psi^{-1}(p)
Beta(a,b)
$$$ \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)
$$$ \ln\mathrm{B}(a,b) = \ln\Gamma(a) + \ln\Gamma(b) - \ln\Gamma(a+b)
BetaIncomplete(a,b,x)
Lower incomplete Beta function (unregularized).
$$$ \mathrm{B}_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1},\mathrm{d}t
BetaRegularized(a,b,x)
Lower incomplete regularized Beta function.
$$$ \mathrm{I}_x(a,b) = \frac{\mathrm{B}(a,b,x)}{\mathrm{B}(a,b)}
Erf(x)
$$$ \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2},\mathrm{d}t
ErfInv(z)
Inverse
$$$ z \mapsto \mathrm{erf}^{-1}(z)
Erfc(x)
$$$ \mathrm{erfc}(x) = 1-\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}}\int_x^\infty e^{-t^2},\mathrm{d}t
ErfcInv(z)
Inverse
$$$ z \mapsto \mathrm{erfc}^{-1}(z)
Code Sample:
[lang=csharp]
double erf = SpecialFunctions.Erf(0.9); // 0.7969082124
Logistic(x)
$$$ x \mapsto \frac{1}{1+e^{-x}}
Logit(y)
Inverse of the Logistic function, for
$$$ y \mapsto \ln \frac{y}{1-y}
Harmonic(t)
The n-th Harmonic number is the sum of the reciprocals of the first n natural numbers.
With
$$$ \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.
$$$ \mathrm{H}{n,m} = \sum{k=1}^{n}\frac{1}{k^m}
Bessel functions are canonical solutions
$$$ 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'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 are solutions
$$$ x^2\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+x\frac{\mathrm{d}y}{\mathrm{d}x}+(x^2-\alpha^2)y = \frac{4(\frac{x}{2})^{\alpha+1}}{\sqrt{\pi}\Gamma(\alpha+\frac{1}{2})}
Modified equation:
$$$ 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.
$$$ x \mapsto \mathrm{L}_1(x)
BesselI0MStruveL0(x)
Difference between the Bessel
$$$ x \mapsto I_0(x) - L_0(x)
BesselI1MStruveL1(x)
Difference between the Bessel
$$$ x \mapsto I_1(x) - L_1(x)
ExponentialMinusOne(power)
$$$ x \mapsto e^x - 1
Hypotenuse(a, b)
$$$ (a,b) \mapsto \sqrt{a^2 + b^2}
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
- Conversion routines between radian, degree and grad.