### Chapter 3: Special Functions

 Chapter Introduction Module 3.1: nag_inv_hyp_fun - Inverse Hyperbolic Functions nag_arctanh Inverse hyperbolic tangent, arctanh x nag_arcsinh Inverse hyperbolic sine, arcsinh x nag_arccosh Inverse hyperbolic cosine, arccosh x Examples Module 3.2: nag_gamma_fun - Gamma Functions nag_gamma Gamma function nag_log_gamma Log gamma function nag_polygamma Polygamma functions nag_incompl_gamma Incomplete gamma functions Examples Module 3.3: nag_err_fun - Error Functions nag_erf Error function erf x nag_erfc Complementary error function erfc x nag_dawson Dawson's integral F(x) Examples Module 3.4: nag_bessel_fun - Bessel Functions nag_bessel_j0 Bessel function J0(x) nag_bessel_j1 Bessel function J1(x) nag_bessel_j Bessel function Jν(z) nag_bessel_y0 Bessel function Y0(x) nag_bessel_y1 Bessel function Y1(x) nag_bessel_y Bessel function Yν(z) nag_bessel_i0 Modified Bessel function I0(x) nag_bessel_i1 Modified Bessel function I1(x) nag_bessel_i Modified Bessel function Iν(z) nag_bessel_k0 Modified Bessel function K0(x) nag_bessel_k1 Modified Bessel function K1(x) nag_bessel_k Modified Bessel function Kν(z) Examples Module 3.5: nag_fresnel_intg - Fresnel Integrals nag_fresnel_s Fresnel integral S(x) nag_fresnel_c Fresnel integral C(x) Examples Module 3.6: nag_ell_intg - Elliptic Integrals nag_ell_rf Symmetrised elliptic integral of the first kind nag_ell_rc Degenerate form of elliptic integral of the first kind nag_ell_rd Symmetrised elliptic integral of the second kind nag_ell_rj Symmetrised elliptic integral of the third kind Examples Module 3.7: nag_ell_fun - Elliptic Functions nag_ell_jac Jacobian elliptic functions sn, cn and dn Examples Module 3.8: nag_airy_fun - Airy Functions nag_airy_ai Airy function Ai(z) nag_airy_bi Airy function Bi(z) Examples Module 3.9: nag_kelvin_fun - Kelvin Functions nag_kelvin_ber Kelvin function ber x nag_kelvin_bei Kelvin function bei x nag_kelvin_ker Kelvin function ker x nag_kelvin_kei Kelvin function kei x Examples