3.5.1.3 Special Math

1 Airy
2 Bessel
3 Beta
4 Error
5 Gamma
6 Integral
7 Kelvin
8 Miscellaneous

Airy

Name	Brief
airy_ai	Evaluates an approximation to the Airy function, Ai(x).
airy_ai_deriv	Evaluates an approximation to the derivative of the Airy function Ai(x).
airy_bi	Evaluates an approximation to the Airy function Bi(x).
airy_bi_deriv	Evaluates an approximation to the derivative of the Airy function Bi(x).

Bessel

Name	Brief
Bessel_i_nu	Evaluates an approximation to the modified Bessel function of the first kind I $\nu$ /4 (x)
Bessel_i_nu_scaled	Evaluates an approximation to the modified Bessel function of the first kind $e^{-x}I_{\frac \nu 4}(x)$
Bessel_i0	Evaluates an approximation to the modified Bessel function of the first kind, I0(x).
Bessel_i0_scaled	Evaluates an approximation to $e^{-\|x\|}I_0(x)$
Bessel_i1	Evaluates an approximation to the modified Bessel function of the first kind, $I_1(x)$ .
Bessel_i1_scaled	Evaluates an approximation to $e^{-\|x\|}I_1(x)$
Bessel_j0	Evaluates the Bessel function of the first kind, $J_0(x)$
Bessel_j1	Evaluates an approximation to the Bessel function of the first kind $J_1(x)$
Bessel_k_nu	Evaluates an approximation to the modified Bessel function of the second kind $K_{\upsilon /4}(x)$
Bessel_k_nu_scaled	Evaluates an approximation to the modified Bessel function of the second kind $e^{-x}K_{\upsilon /4}(x)$
Bessel_k0	Evaluates an approximation to the modified Bessel function of the second kind, $K_0\left( x\right)$
Bessel_k0_scaled	Evaluates an approximation to $e^xK_0\left( x\right)$
Bessel_k1	Evaluates an approximation to the modified Bessel function of the second kind, $K_1\left( x\right)$
Bessel_k1_scaled	Evaluates an approximation to $e^xK_1\left( x\right)$
Bessel_y0	Evaluates the Bessel function of the second kind, $Y_0$ , x > 0.
Bessel_y1	Evaluates the Bessel function of the second kind, $Y_1$ , x > 0.
Jn(x, n)	Bessel function of order n
Yn(x, n)	Bessel Function of Second Kind
J1(x)	First Order Bessel Function
Y1(x)	First order Bessel function of second kind has the following form: Y1(x)
J0(x)	Zero Order Bessel Function
Y0(x)	Zero Order Bessel Function of Second Kind

Beta

Name	Brief
beta(a, b)	Beta Function
incbeta(x, a, b)	Incomplete Beta Function

Error

Name	Brief
Erf	An error function calculated by $\mathrm{erf}(x)=\frac{2}{\sqrt\pi}\int_{0}^{x}e^{-u^2}du$
Erfc	Calculates an approximate value for the complement of the error function $erfc(x)=\frac 1{\sqrt{\pi }}\int_x^\infty e^{\frac{-u^2}2}du=1-{erf(x)}$
Erfcinv	Computes the value of the inverse complementary error function for specified y
Erfcx	An scaled complementary error function calculated by $erfcx(x) = e^{x^2}\cdot erfc(x)$
Erfinv	Calculates the inverse of error function $erf$

Gamma

Name	Brief
Gamma	Evaluates $\Gamma (x)=\int_0^\infty t^{x-1}e^{-t}dt$
Incomplete_gamma	Evaluates the incomplete gamma functions in the normalized form $P(a,x)=\frac 1{\Gamma (a)}\int_0^xt^{a-1}e^{-t}dt$
Log_gamma	Evaluates $\ln \Gamma (x)$ , x > 0.
Real_polygamma	Evaluates an approximation to the kth derivative of the psi function $\psi (x)$ by $\Psi ^k(x)=\frac{d^k}{dx^k}\Psi (x)=\frac{d^k}{dx^k}(\frac d{dx^k}\log _e\Gamma (x))$ where x is real with x≠0, -1, -2, ... and k=0,1,......6.
incomplete_gamma(a, x)	Incomplete gamma functions
gammaln(x)	Natural Log of the Gamma Function
Incgamma	Calculate the incomplete Gamma function

Integral

Name	Brief
Cos_integral	Evaluates an approximation of $C_i\left( x\right) =y+\ln x+\int_0^x\frac{\cos u-1}udu$ .
Cumul_normal	Evaluates the cumulative Normal distribution function $P(x)=\frac 1{\sqrt{2\pi }}\int_{-\infty }^xe^{\frac{-u^2}2}du$
Cumul_normal_complem	Evaluates an approximate value for the complement of the cumulative normal distribution function $Q(x)=\frac 1{\sqrt{2\pi }}\int_x^\infty e^{\frac{-u^2}2}du$
Elliptic_integral_rc	calculates an approximate value for the integral $R_c(x,y)=\frac 12\int_0^\infty \frac{dt}{\sqrt{t+x}(t+y)}$ where x ≥ 0 and y ≠ 0.
Elliptic_integral_rd	Calculates an approximate value for the integral $R_D(x,y,z)=\frac 32\int_0^\infty \frac{dt}{\sqrt{(t+z)(t+y)(t+z)^3}}$ .
Elliptic_integral_rf	Calculates an approximation to the integral $R_F(x,y,z)=\frac 12\int_0^\infty \frac{dt}{\sqrt{(t+x)(t+y)(t+z)}}$ .
Elliptic_integral_rj	Calculates an approximation to the integral $R_J(x,y,z,\rho )=\frac 32\int_0^\infty \frac{dt}{(t+\rho )\sqrt{(t+x)(t+y)(t+z)}}$ .
Exp_integral	Evaluates $E_1(x)=\int_x^\infty \frac{e^{-u}}udu$ , x>0.
Fresnel_c	Evaluates an approximation to the Fresnel Integral $S(x)=\int_0^x\cos (\frac \pi 2t^2)dt$ .
Fresnel_s	Evaluates an approximation to the Fresnel Integral $S(x)=\int_0^x\sin (\frac \pi 2t^2)dt$ .
Sin_integral	Evaluates the approximation of the formula $Si(x)=\int_0^x\frac{\sin u}udu$

Kelvin

Name	Brief
Kelvin_bei	Evaluates an approximation to the Kelvin function bei x.
Kelvin_ber	Evaluates an approximation to the Kelvin function ber x.
Kelvin_kei	Evaluates an approximation to the Kelvin function kei x.
Kelvin_ker	Evaluates an approximation to the Kelvin function ker x.

Miscellaneous

Name	Brief
Jacobian_theta	Evaluates an approximation to the Jacobian theta functions.
LambertW	Evaluates an approximate value for the real branches of Lambert’s W function.
Boltzmann	Boltzmann Function
Dhyperbl	Double Rectangular Hyperbola Function
ExpAssoc	Exponential Associate Function
ExpDecay2	Exponential Decay 2 with Offset Function
ExpGrow2	Exponential Growth 2 with Offset Function
Gauss	Gaussian Function
Hyperbl	Hyperbola Function
Logistic	Logistic Dose Response Function
Lorentz	Lorentzian Function
Poly	Polynomial Function
Pulse	Pulse Function
LambertW	Lambert’s W function (sometimes known as the ‘product log’ or ‘Omega’ function)
Erfcx	Complementary error function