3.5.1.3 Special Math
Contents
Airy
| Name | Brief |
|---|---|
| Evaluates an approximation to the Airy function, Ai(x). | |
| Evaluates an approximation to the derivative of the Airy function Ai(x). | |
| Evaluates an approximation to the Airy function Bi(x). | |
| Evaluates an approximation to the derivative of the Airy function Bi(x). |
Bessel
| Name | Brief |
|---|---|
| Evaluates an approximation to the modified Bessel function of the first kind I\(\nu\)/4 (x) | |
| Evaluates an approximation to the modified Bessel function of the first kind \(e^{-x}I_{\frac \nu 4}(x)\) | |
| Evaluates an approximation to the modified Bessel function of the first kind, I0(x). | |
| Evaluates an approximation to \(e^{-|x|}I_0(x)\) | |
| Evaluates an approximation to the modified Bessel function of the first kind,\(I_1(x)\). | |
| Evaluates an approximation to \(e^{-|x|}I_1(x)\) | |
| Evaluates the Bessel function of the first kind,\(J_0(x)\) | |
| Evaluates an approximation to the Bessel function of the first kind \(J_1(x)\) | |
| Evaluates an approximation to the modified Bessel function of the second kind \(K_{\upsilon /4}(x)\) | |
| Evaluates an approximation to the modified Bessel function of the second kind \(e^{-x}K_{\upsilon /4}(x)\) | |
| Evaluates an approximation to the modified Bessel function of the second kind,\(K_0\left( x\right)\) | |
| Evaluates an approximation to \(e^xK_0\left( x\right)\) | |
| Evaluates an approximation to the modified Bessel function of the second kind,\(K_1\left( x\right) \) | |
| Evaluates an approximation to \(e^xK_1\left( x\right)\) | |
| Evaluates the Bessel function of the second kind,\(Y_0\) , x > 0. | |
| Evaluates the Bessel function of the second kind,\(Y_1\) , x > 0. | |
| Modified Bessel function of first kind | |
| Bessel function of first kind | |
| Modified Bessel function of second kind | |
| Bessel function of second kind | |
| Bessel function of order n | |
| Bessel Function of Second Kind | |
| First Order Bessel Function | |
| First order Bessel function of second kind has the following form: Y1(x) | |
| Zero Order Bessel Function | |
| Zero Order Bessel Function of Second Kind |
Beta
| Name | Brief |
|---|---|
| Beta Function | |
| Incomplete Beta Function |
Error
| Name | Brief |
|---|---|
| An error function calculated by \(\mathrm{erf}(x)=\frac{2}{\sqrt\pi}\int_{0}^{x}e^{-u^2}du\) | |
| Calculates an approximate value for the complement of the error function \(erfc(x)=\frac 1{\sqrt{\pi }}\int_x^\infty e^{\frac{-u^2}2}du=1-{erf(x)}\) | |
| Computes the value of the inverse complementary error function for specified y | |
| An scaled complementary error function calculated by \(erfcx(x) = e^{x^2}\cdot erfc(x)\) | |
| Calculates the inverse of error function \(erf\) |
Gamma
| Name | Brief |
|---|---|
| Evaluates \(\Gamma (x)=\int_0^\infty t^{x-1}e^{-t}dt\) | |
| Evaluates the incomplete gamma functions in the normalized form \(P(a,x)=\frac 1{\Gamma (a)}\int_0^xt^{a-1}e^{-t}dt\) | |
| Evaluates \(\ln \Gamma (x)\), x > 0. | |
| 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 functions | |
| Natural Log of the Gamma Function | |
| Calculate the incomplete Gamma function |
Integral
| Name | Brief |
|---|---|
| Evaluates an approximation of \(C_i\left( x\right) =y+\ln x+\int_0^x\frac{\cos u-1}udu\). | |
| Evaluates the cumulative Normal distribution function \(P(x)=\frac 1{\sqrt{2\pi }}\int_{-\infty }^xe^{\frac{-u^2}2}du\) | |
| Evaluates an approximate value for the complement of the cumulative normal distribution function \(Q(x)=\frac 1{\sqrt{2\pi }}\int_x^\infty e^{\frac{-u^2}2}du\) | |
| 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. | |
| Calculates an approximate value for the integral \(R_D(x,y,z)=\frac 32\int_0^\infty \frac{dt}{\sqrt{(t+z)(t+y)(t+z)^3}}\) . | |
| Calculates an approximation to the integral \(R_F(x,y,z)=\frac 12\int_0^\infty \frac{dt}{\sqrt{(t+x)(t+y)(t+z)}}\) . | |
| Calculates an approximation to the integral \(R_J(x,y,z,\rho )=\frac 32\int_0^\infty \frac{dt}{(t+\rho )\sqrt{(t+x)(t+y)(t+z)}}\). | |
| Evaluates \(E_1(x)=\int_x^\infty \frac{e^{-u}}udu \), x>0. | |
| Evaluates an approximation to the Fresnel Integral \(S(x)=\int_0^x\cos (\frac \pi 2t^2)dt\). | |
| Evaluates an approximation to the Fresnel Integral \(S(x)=\int_0^x\sin (\frac \pi 2t^2)dt\). | |
| Evaluates the approximation of the formula \(Si(x)=\int_0^x\frac{\sin u}udu\) |
Kelvin
| Name | Brief |
|---|---|
| Evaluates an approximation to the Kelvin function bei x. | |
| Evaluates an approximation to the Kelvin function ber x. | |
| Evaluates an approximation to the Kelvin function kei x. | |
| Evaluates an approximation to the Kelvin function ker x. |
Miscellaneous
| Name | Brief |
|---|---|
| Evaluates an approximation to the Jacobian theta functions. | |
| Evaluates an approximate value for the real branches of Lambert’s W function. | |
| Boltzmann Function | |
| Double Rectangular Hyperbola Function | |
| Exponential Associate Function | |
| Exponential Decay 2 with Offset Function | |
| Exponential Growth 2 with Offset Function | |
| Gaussian Function | |
| Hyperbola Function | |
| Logistic Dose Response Function | |
| Lorentzian Function | |
| Polynomial Function | |
| Pulse Function | |
| Lambert’s W function (sometimes known as the ‘product log’ or ‘Omega’ function) | |
| Complementary error function |