# 3.5.2.5 Approximations of Special Functions from NAG

## 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.

### 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.