3.5.2.5 Approximations of Special Functions from NAG

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.

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.

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.