Special Math


Contents

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.
besseli Modified Bessel function of first kind
besselj Bessel function of first kind
besselk Modified Bessel function of second kind
bessely Bessel function of second kind
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