S Chapter Contents : NAG Library Manual, Mark 27

Routine	Mark of Introduction	Purpose
s01baf Example Text Example Data	14	nagf_specfun_log_shifted $\ln (1 + x)$
s01eaf Example Text Example Data	14	nagf_specfun_exp_complex Complex exponential, $e^{z}$
s07aaf Example Text Example Data	1	nagf_specfun_tan $\tan x$
s09aaf Example Text Example Data	1	nagf_specfun_arcsin $\arcsin x$
s09abf Example Text Example Data	3	nagf_specfun_arccos $\arccos x$
s10aaf Example Text Example Data	3	nagf_specfun_tanh $\tanh x$
s10abf Example Text Example Data	4	nagf_specfun_sinh $\sinh x$
s10acf Example Text Example Data	4	nagf_specfun_cosh $\cosh x$
s11aaf Example Text Example Data	4	nagf_specfun_arctanh $arctanh x$
s11abf Example Text Example Data	4	nagf_specfun_arcsinh $arcsinh x$
s11acf Example Text Example Data	4	nagf_specfun_arccosh $arccosh x$
s13aaf Example Text Example Data Example Plot	1	nagf_specfun_integral_exp Exponential integral $E_{1} (x)$
s13acf Example Text Example Data Example Plot	2	nagf_specfun_integral_cos Cosine integral $Ci (x)$
s13adf Example Text Example Data Example Plot	5	nagf_specfun_integral_sin Sine integral $Si (x)$
s14aaf Example Text Example Data Example Plot	1	nagf_specfun_gamma Gamma function
s14abf Example Text Example Data Example Plot	8	nagf_specfun_gamma_log_real Log gamma function, real argument
s14acf Example Text Example Data Example Plot	14	nagf_specfun_polygamma $ψ (x) - \ln x$
s14adf Example Text Example Data	14	nagf_specfun_polygamma_deriv Scaled derivatives of $ψ (x)$
s14aef Example Text Example Data	20	nagf_specfun_psi_deriv_real Polygamma function $ψ^{(n)} (x)$ for real $x$
s14aff Example Text Example Data	20	nagf_specfun_psi_deriv_complex Polygamma function $ψ^{(n)} (z)$ for complex $z$
s14agf Example Text Example Data	21	nagf_specfun_gamma_log_complex Logarithm of the gamma function $\ln Γ (z)$ , complex argument
s14ahf Example Text Example Data Example Plot	23	nagf_specfun_gamma_log_scaled_real Scaled log gamma function
s14anf Example Text Example Data	27	nagf_specfun_gamma_vector Gamma function, vectorized $Γ (x)$
s14apf Example Text Example Data	27	nagf_specfun_gamma_log_real_vector Log gamma function, vectorized $\ln (Γ (x))$
s14baf Example Text Example Data Example Plot	14	nagf_specfun_gamma_incomplete Incomplete gamma functions $P (a, x)$ and $Q (a, x)$
s14bnf Example Text Example Data	27	nagf_specfun_gamma_incomplete_vector Incomplete gamma functions, vectorized $P (a, x)$ and $Q (a, x)$
s14cbf Example Text Example Data	24	nagf_specfun_beta_log_real Logarithm of the beta function $\ln B (a, b)$
s14ccf Example Text Example Data	24	nagf_specfun_beta_incomplete Regularized incomplete beta function $I_{x} (a, b)$ and its complement $1 - I_{x}$
s14cpf Example Text Example Data	27	nagf_specfun_beta_log_real_vector Logarithm of the beta function, vectorized $\ln B (a, b)$
s14cqf Example Text Example Data	27	nagf_specfun_beta_incomplete_vector Regularized incomplete beta function, vectorized $I_{x} (a, b)$ and its complement $1 - I_{x}$
s15abf Example Text Example Data	3	nagf_specfun_cdf_normal Cumulative Normal distribution function $P (x)$
s15acf Example Text Example Data	4	nagf_specfun_compcdf_normal Complement of cumulative Normal distribution function $Q (x)$
s15adf Example Text Example Data	4	nagf_specfun_erfc_real Complement of error function $erfc (x)$
s15aef Example Text Example Data	4	nagf_specfun_erf_real Error function $\erf (x)$
s15aff Example Text Example Data	7	nagf_specfun_dawson Dawson's integral
s15agf Example Text Example Data	22	nagf_specfun_erfcx_real Scaled complement of error function, $erfcx (x)$
s15apf Example Text Example Data	27	nagf_specfun_cdf_normal_vector Cumulative Normal distribution function, vectorized $P (x)$
s15aqf Example Text Example Data	27	nagf_specfun_compcdf_normal_vector Complement of cumulative Normal distribution function, vectorized $Q (x)$
s15arf Example Text Example Data	27	nagf_specfun_erfc_real_vector Complement of error function, vectorized $erfc (x)$
s15asf Example Text Example Data	27	nagf_specfun_erf_real_vector Error function, vectorized $\erf (x)$
s15atf Example Text Example Data	27	nagf_specfun_dawson_vector Dawson's integral, vectorized
s15auf Example Text Example Data	27	nagf_specfun_erfcx_real_vector Scaled complement of error function, vectorized $erfcx (x)$
s15ddf Example Text Example Data	14	nagf_specfun_erfc_complex Scaled complex complement of error function, $\exp (- z^{2}) erfc (- i z)$
s15drf Example Text Example Data	27	nagf_specfun_erfc_complex_vector Scaled complex complement of error function, vectorized $\exp (- z^{2}) erfc (- i z)$
s17acf Example Text Example Data Example Plot	1	nagf_specfun_bessel_y0_real Bessel function $Y_{0} (x)$
s17adf Example Text Example Data Example Plot	1	nagf_specfun_bessel_y1_real Bessel function $Y_{1} (x)$
s17aef Example Text Example Data Example Plot	5	nagf_specfun_bessel_j0_real Bessel function $J_{0} (x)$
s17aff Example Text Example Data Example Plot	5	nagf_specfun_bessel_j1_real Bessel function $J_{1} (x)$
s17agf Example Text Example Data Example Plot	8	nagf_specfun_airy_ai_real Airy function $Ai (x)$
s17ahf Example Text Example Data Example Plot	8	nagf_specfun_airy_bi_real Airy function $Bi (x)$
s17ajf Example Text Example Data Example Plot	8	nagf_specfun_airy_ai_deriv Airy function ${Ai}^{'} (x)$
s17akf Example Text Example Data Example Plot	8	nagf_specfun_airy_bi_deriv Airy function ${Bi}^{'} (x)$
s17alf Example Text Example Data	20	nagf_specfun_bessel_zeros Zeros of Bessel functions $J_{α} (x)$ , $J_{α}^{'} (x)$ , $Y_{α} (x)$ or $Y_{α}^{'} (x)$
s17aqf Example Text Example Data	24	nagf_specfun_bessel_y0_real_vector Bessel function vectorized $Y_{0} (x)$
s17arf Example Text Example Data	24	nagf_specfun_bessel_y1_real_vector Bessel function vectorized $Y_{1} (x)$
s17asf Example Text Example Data	24	nagf_specfun_bessel_j0_real_vector Bessel function vectorized $J_{0} (x)$
s17atf Example Text Example Data	24	nagf_specfun_bessel_j1_real_vector Bessel function vectorized $J_{1} (x)$
s17auf Example Text Example Data	24	nagf_specfun_airy_ai_real_vector Airy function vectorized $Ai (x)$
s17avf Example Text Example Data	24	nagf_specfun_airy_bi_real_vector Airy function vectorized $Bi (x)$
s17awf Example Text Example Data	24	nagf_specfun_airy_ai_deriv_vector Derivatives of the Airy function, vectorized ${Ai}^{'} (x)$
s17axf Example Text Example Data	24	nagf_specfun_airy_bi_deriv_vector Derivatives of the Airy function, vectorized ${Bi}^{'} (x)$
s17dcf Example Text Example Data	13	nagf_specfun_bessel_y_complex Bessel functions $Y_{ν + a} (z)$ , real $a \geq 0$ , complex $z$ , $ν = 0, 1, 2, \dots$
s17def Example Text Example Data	13	nagf_specfun_bessel_j_complex Bessel functions $J_{ν + a} (z)$ , real $a \geq 0$ , complex $z$ , $ν = 0, 1, 2, \dots$
s17dgf Example Text Example Data	13	nagf_specfun_airy_ai_complex Airy functions $Ai (z)$ and ${Ai}^{'} (z)$ , complex $z$
s17dhf Example Text Example Data	13	nagf_specfun_airy_bi_complex Airy functions $Bi (z)$ and ${Bi}^{'} (z)$ , complex $z$
s17dlf Example Text Example Data	13	nagf_specfun_hankel_complex Hankel functions $H_{ν + a}^{(j)} (z)$ , $j = 1, 2$ , real $a \geq 0$ , complex $z$ , $ν = 0, 1, 2, \dots$
s17gaf Example Text Example Data Example Plot	26.1	nagf_specfun_struve_h0 Struve function of order $0$ , $H_{0} (x)$
s17gbf Example Text Example Data Example Plot	26.1	nagf_specfun_struve_h1 Struve function of order $1$ , $H_{1} (x)$
s18acf Example Text Example Data Example Plot	1	nagf_specfun_bessel_k0_real Modified Bessel function $K_{0} (x)$
s18adf Example Text Example Data Example Plot	1	nagf_specfun_bessel_k1_real Modified Bessel function $K_{1} (x)$
s18aef Example Text Example Data Example Plot	5	nagf_specfun_bessel_i0_real Modified Bessel function $I_{0} (x)$
s18aff Example Text Example Data Example Plot	5	nagf_specfun_bessel_i1_real Modified Bessel function $I_{1} (x)$
s18aqf Example Text Example Data	24	nagf_specfun_bessel_k0_real_vector Modified Bessel function vectorized $K_{0} (x)$
s18arf Example Text Example Data	24	nagf_specfun_bessel_k1_real_vector Modified Bessel function vectorized $K_{1} (x)$
s18asf Example Text Example Data	24	nagf_specfun_bessel_i0_real_vector Modified Bessel function vectorized $I_{0} (x)$
s18atf Example Text Example Data	24	nagf_specfun_bessel_i1_real_vector Modified Bessel function vectorized $I_{1} (x)$
s18ccf Example Text Example Data	10	nagf_specfun_bessel_k0_scaled Scaled modified Bessel function $e^{x} K_{0} (x)$
s18cdf Example Text Example Data	10	nagf_specfun_bessel_k1_scaled Scaled modified Bessel function $e^{x} K_{1} (x)$
s18cef Example Text Example Data	10	nagf_specfun_bessel_i0_scaled Scaled modified Bessel function $e^{- \|x\|} I_{0} (x)$
s18cff Example Text Example Data	10	nagf_specfun_bessel_i1_scaled Scaled modified Bessel function $e^{- \|x\|} I_{1} (x)$
s18cqf Example Text Example Data	24	nagf_specfun_bessel_k0_scaled_vector Scaled modified Bessel function vectorized $e^{x} K_{0} (x)$
s18crf Example Text Example Data	24	nagf_specfun_bessel_k1_scaled_vector Scaled modified Bessel function vectorized $e^{x} K_{1} (x)$
s18csf Example Text Example Data	24	nagf_specfun_bessel_i0_scaled_vector Scaled modified Bessel function vectorized $e^{- \|x\|} I_{0} (x)$
s18ctf Example Text Example Data	24	nagf_specfun_bessel_i1_scaled_vector Scaled modified Bessel function vectorized $e^{- \|x\|} I_{1} (x)$
s18dcf Example Text Example Data	13	nagf_specfun_bessel_k_complex Modified Bessel functions $K_{ν + a} (z)$ , real $a \geq 0$ , complex $z$ , $ν = 0, 1, 2, \dots$
s18def Example Text Example Data	13	nagf_specfun_bessel_i_complex Modified Bessel functions $I_{ν + a} (z)$ , real $a \geq 0$ , complex $z$ , $ν = 0, 1, 2, \dots$
s18gaf Example Text Example Data Example Plot	26.1	nagf_specfun_struve_l0 Modified Struve function of order $0$ , $L_{0} (x)$
s18gbf Example Text Example Data Example Plot	26.1	nagf_specfun_struve_l1 Modified Struve function of order $1$ , $L_{1} (x)$
s18gcf Example Text Example Data Example Plot	26.1	nagf_specfun_struve_i0ml0 The function $I_{0} (x) - L_{0} (x)$ , where $I_{0} (x)$ is a modified Bessel function and $L_{0} (x)$ is a Struve function
s18gdf Example Text Example Data Example Plot	26.1	nagf_specfun_struve_i1ml1 The function $I_{1} (x) - L_{1} (x)$ , where $I_{1} (x)$ is a modified Bessel function and $L_{1} (x)$ is a Struve function
s18gkf Example Text Example Data	21	nagf_specfun_bessel_j_seq_complex Bessel function of the 1st kind $J_{α \pm n} (z)$
s19aaf Example Text Example Data	11	nagf_specfun_kelvin_ber Kelvin function $ber x$
s19abf Example Text Example Data	11	nagf_specfun_kelvin_bei Kelvin function $bei x$
s19acf Example Text Example Data	11	nagf_specfun_kelvin_ker Kelvin function $\ker x$
s19adf Example Text Example Data	11	nagf_specfun_kelvin_kei Kelvin function $kei x$
s19anf Example Text Example Data	24	nagf_specfun_kelvin_ber_vector Kelvin function vectorized $ber x$
s19apf Example Text Example Data	24	nagf_specfun_kelvin_bei_vector Kelvin function vectorized $bei x$
s19aqf Example Text Example Data	24	nagf_specfun_kelvin_ker_vector Kelvin function vectorized $\ker x$
s19arf Example Text Example Data	24	nagf_specfun_kelvin_kei_vector Kelvin function vectorized $kei x$
s20acf Example Text Example Data Example Plot	5	nagf_specfun_fresnel_s Fresnel integral $S (x)$
s20adf Example Text Example Data Example Plot	5	nagf_specfun_fresnel_c Fresnel integral $C (x)$
s20aqf Example Text Example Data	24	nagf_specfun_fresnel_s_vector Fresnel integral vectorized $S (x)$
s20arf Example Text Example Data	24	nagf_specfun_fresnel_c_vector Fresnel integral vectorized $C (x)$
s21baf Example Text	8	nagf_specfun_ellipint_symm_1_degen Degenerate symmetrised elliptic integral of 1st kind $R_{C} (x, y)$
s21bbf Example Text	8	nagf_specfun_ellipint_symm_1 Symmetrised elliptic integral of 1st kind $R_{F} (x, y, z)$
s21bcf Example Text	8	nagf_specfun_ellipint_symm_2 Symmetrised elliptic integral of 2nd kind $R_{D} (x, y, z)$
s21bdf Example Text	8	nagf_specfun_ellipint_symm_3 Symmetrised elliptic integral of 3rd kind $R_{J} (x, y, z, r)$
s21bef Example Text Example Plot	22	nagf_specfun_ellipint_legendre_1 Elliptic integral of 1st kind, Legendre form, $F (ϕ ∣ m)$
s21bff Example Text Example Plot	22	nagf_specfun_ellipint_legendre_2 Elliptic integral of 2nd kind, Legendre form, $E (ϕ ∣ m)$
s21bgf Example Text	22	nagf_specfun_ellipint_legendre_3 Elliptic integral of 3rd kind, Legendre form, $Π (n; ϕ ∣ m)$
s21bhf Example Text	22	nagf_specfun_ellipint_complete_1 Complete elliptic integral of 1st kind, Legendre form, $K (m)$
s21bjf Example Text	22	nagf_specfun_ellipint_complete_2 Complete elliptic integral of 2nd kind, Legendre form, $E (m)$
s21caf Example Text Example Data	15	nagf_specfun_jacellip_real Jacobian elliptic functions sn, cn and dn of real argument
s21cbf Example Text Example Data	20	nagf_specfun_jacellip_complex Jacobian elliptic functions sn, cn and dn of complex argument
s21ccf Example Text Example Data	20	nagf_specfun_jactheta_real Jacobian theta functions $θ_{k} (x, q)$ of real argument
s21daf Example Text Example Data	20	nagf_specfun_ellipint_general_2 General elliptic integral of 2nd kind $F (z, k^{'}, a, b)$ of complex argument
s22aaf Example Text Example Data	20	nagf_specfun_legendre_p Legendre functions of 1st kind $P_{n}^{m} (x)$ or $\bar{P_{n}^{m}} (x)$
s22baf Example Text	24	nagf_specfun_hyperg_confl_real Real confluent hypergeometric function ${}_{1}F_{1} (a; b; x)$
s22bbf Example Text	24	nagf_specfun_hyperg_confl_real_scaled Real confluent hypergeometric function ${}_{1}F_{1} (a; b; x)$ in scaled form
s22bef Example Text	25	nagf_specfun_hyperg_gauss_real Real Gauss hypergeometric function ${}_{2}F_{1} (a, b; c; x)$
s22bff Example Text	25	nagf_specfun_hyperg_gauss_real_scaled Real Gauss hypergeometric function ${}_{2}F_{1} (a, b; c; x)$ in scaled form
s22caf Example Text Example Data Example Plot	27	nagf_specfun_mathieu_ang_periodic_real Calculates values of real periodic angular Mathieu functions
s30aaf Example Text Example Data	22	nagf_specfun_opt_bsm_price Black–Scholes–Merton option pricing formula
s30abf Example Text Example Data	22	nagf_specfun_opt_bsm_greeks Black–Scholes–Merton option pricing formula with Greeks
s30baf Example Text Example Data	22	nagf_specfun_opt_lookback_fls_price Floating-strike lookback option pricing formula in the Black-Scholes-Merton model
s30bbf Example Text Example Data	22	nagf_specfun_opt_lookback_fls_greeks Floating-strike lookback option pricing formula with Greeks in the Black-Scholes-Merton model
s30caf Example Text Example Data	22	nagf_specfun_opt_binary_con_price Binary option, cash-or-nothing pricing formula
s30cbf Example Text Example Data	22	nagf_specfun_opt_binary_con_greeks Binary option, cash-or-nothing pricing formula with Greeks
s30ccf Example Text Example Data	22	nagf_specfun_opt_binary_aon_price Binary option, asset-or-nothing pricing formula
s30cdf Example Text Example Data	22	nagf_specfun_opt_binary_aon_greeks Binary option, asset-or-nothing pricing formula with Greeks
s30faf Example Text Example Data	22	nagf_specfun_opt_barrier_std_price Standard barrier option pricing formula
s30jaf Example Text Example Data	22	nagf_specfun_opt_jumpdiff_merton_price Jump-diffusion, Merton's model, option pricing formula
s30jbf Example Text Example Data	22	nagf_specfun_opt_jumpdiff_merton_greeks Jump-diffusion, Merton's model, option pricing formula with Greeks
s30naf Example Text Example Data	22	nagf_specfun_opt_heston_price Heston's model option pricing formula
s30nbf Example Text Example Data	23	nagf_specfun_opt_heston_greeks Heston's model option pricing formula with Greeks
s30ncf Example Text Example Data	25	nagf_specfun_opt_heston_term Heston's model option pricing with term structure
s30qcf Example Text Example Data	22	nagf_specfun_opt_amer_bs_price American option, Bjerksund and Stensland pricing formula
s30saf Example Text Example Data	22	nagf_specfun_opt_asian_geom_price Asian option, geometric continuous average rate pricing formula
s30sbf Example Text Example Data	22	nagf_specfun_opt_asian_geom_greeks Asian option, geometric continuous average rate pricing formula with Greeks

NAG FL InterfaceS (Specfun)Approximations of Special Functions

NAG FL Interface
S (Specfun)
Approximations of Special Functions