S Chapter Contents : NAG Library CL Interface Manual, Mark 27

Function	Mark of Introduction	Purpose
s01bac Example Text Example Data	7	nag_specfun_log_shifted $\ln (1 + x)$
s10aac Example Text Example Data	1	nag_specfun_tanh Hyperbolic tangent, tanh $x$
s10abc Example Text Example Data	1	nag_specfun_sinh Hyperbolic sine, sinh $x$
s10acc Example Text Example Data	1	nag_specfun_cosh Hyperbolic cosine, cosh $x$
s11aac Example Text Example Data	1	nag_specfun_arctanh Inverse hyperbolic tangent, arctanh $x$
s11abc Example Text Example Data	1	nag_specfun_arcsinh Inverse hyperbolic sine, arcsinh $x$
s11acc Example Text Example Data	1	nag_specfun_arccosh Inverse hyperbolic cosine, arccosh $x$
s13aac Example Text Example Data Example Plot	1	nag_specfun_integral_exp Exponential integral $E_{1} (x)$
s13acc Example Text Example Data Example Plot	1	nag_specfun_integral_cos Cosine integral $Ci (x)$
s13adc Example Text Example Data Example Plot	1	nag_specfun_integral_sin Sine integral $Si (x)$
s14aac Example Text Example Data Example Plot	1	nag_specfun_gamma Gamma function $Γ (x)$
s14abc Example Text Example Data Example Plot	1	nag_specfun_gamma_log_real Log gamma function $\ln (Γ (x))$
s14acc Example Text Example Data Example Plot	7	nag_specfun_polygamma $ψ (x) - \ln x$
s14adc Example Text Example Data	7	nag_specfun_polygamma_deriv Scaled derivatives of $ψ (x)$
s14aec Example Text Example Data	6	nag_specfun_psi_deriv_real Derivative of the psi function $ψ (x)$
s14afc Example Text Example Data	6	nag_specfun_psi_deriv_complex Derivative of the psi function $ψ (z)$
s14agc Example Text Example Data	7	nag_specfun_gamma_log_complex Logarithm of the gamma function $\ln Γ (z)$ , complex argument
s14ahc Example Text Example Data Example Plot	9	nag_specfun_gamma_log_scaled_real Scaled log gamma function $\ln G (x)$ , where $G (x) = Γ (x + 1) / {(x / e)}^{x}$
s14anc Example Text Example Data	27	nag_specfun_gamma_vector Gamma function, vectorized $Γ (x)$
s14apc Example Text Example Data	27	nag_specfun_gamma_log_real_vector Log gamma function, vectorized $\ln (Γ (x))$
s14bac Example Text Example Data Example Plot	1	nag_specfun_gamma_incomplete Incomplete gamma functions $P (a, x)$ and $Q (a, x)$
s14bnc Example Text Example Data	27	nag_specfun_gamma_incomplete_vector Incomplete gamma functions, vectorized $P (a, x)$ and $Q (a, x)$
s14cbc Example Text Example Data	23	nag_specfun_beta_log_real Logarithm of the beta function $\ln B (a, b)$
s14ccc Example Text Example Data	23	nag_specfun_beta_incomplete Regularized incomplete beta function $I_{x} (a, b)$ and its complement $1 - I_{x}$
s14cpc Example Text Example Data	27	nag_specfun_beta_log_real_vector Logarithm of the beta function, vectorized $\ln B (a, b)$
s14cqc Example Text Example Data	27	nag_specfun_beta_incomplete_vector Regularized incomplete beta function, vectorized $I_{x} (a, b)$ and its complement $1 - I_{x}$
s15abc Example Text Example Data	1	nag_specfun_cdf_normal Cumulative Normal distribution function $P (x)$
s15acc Example Text Example Data	1	nag_specfun_compcdf_normal Complement of cumulative Normal distribution function $Q (x)$
s15adc Example Text Example Data	1	nag_specfun_erfc_real Complement of error function $erfc (x)$
s15aec Example Text Example Data	1	nag_specfun_erf_real Error function $\erf (x)$
s15afc Example Text Example Data	7	nag_specfun_dawson Dawson's integral
s15agc Example Text Example Data	9	nag_specfun_erfcx_real Scaled complement of error function, $erfcx (x)$
s15apc Example Text Example Data	27	nag_specfun_cdf_normal_vector Cumulative Normal distribution function, vectorized $P (x)$
s15aqc Example Text Example Data	27	nag_specfun_compcdf_normal_vector Complement of cumulative Normal distribution function, vectorized $Q (x)$
s15arc Example Text Example Data	27	nag_specfun_erfc_real_vector Complement of error function, vectorized $erfc (x)$
s15asc Example Text Example Data	27	nag_specfun_erf_real_vector Error function, vectorized $\erf (x)$
s15atc Example Text Example Data	27	nag_specfun_dawson_vector Dawson's integral, vectorized
s15auc Example Text Example Data	27	nag_specfun_erfcx_real_vector Scaled complement of error function, vectorized $erfcx (x)$
s15ddc Example Text Example Data	7	nag_specfun_erfc_complex Scaled complex complement of error function, $\exp (- z^{2}) erfc (- i z)$
s15drc Example Text Example Data	27	nag_specfun_erfc_complex_vector Scaled complex complement of error function, vectorized $\exp (- z^{2}) erfc (- i z)$
s17acc Example Text Example Data Example Plot	1	nag_specfun_bessel_y0_real Bessel function $Y_{0} (x)$
s17adc Example Text Example Data Example Plot	1	nag_specfun_bessel_y1_real Bessel function $Y_{1} (x)$
s17aec Example Text Example Data Example Plot	1	nag_specfun_bessel_j0_real Bessel function $J_{0} (x)$
s17afc Example Text Example Data Example Plot	1	nag_specfun_bessel_j1_real Bessel function $J_{1} (x)$
s17agc Example Text Example Data Example Plot	1	nag_specfun_airy_ai_real Airy function $Ai (x)$
s17ahc Example Text Example Data Example Plot	1	nag_specfun_airy_bi_real Airy function $Bi (x)$
s17ajc Example Text Example Data Example Plot	1	nag_specfun_airy_ai_deriv Airy function ${Ai}^{'} (x)$
s17akc Example Text Example Data Example Plot	1	nag_specfun_airy_bi_deriv Airy function ${Bi}^{'} (x)$
s17alc Example Text Example Data	6	nag_specfun_bessel_zeros Zeros of Bessel functions $J_{α} (x)$ , $J_{α}^{'} (x)$ , $Y_{α} (x)$ or $Y_{α}^{'} (x)$
s17aqc Example Text Example Data	23	nag_specfun_bessel_y0_real_vector Bessel function vectorized $Y_{0} (x)$
s17arc Example Text Example Data	23	nag_specfun_bessel_y1_real_vector Bessel function vectorized $Y_{1} (x)$
s17asc Example Text Example Data	23	nag_specfun_bessel_j0_real_vector Bessel function vectorized $J_{0} (x)$
s17atc Example Text Example Data	23	nag_specfun_bessel_j1_real_vector Bessel function vectorized $J_{1} (x)$
s17auc Example Text Example Data	23	nag_specfun_airy_ai_real_vector Airy function vectorized $Ai (x)$
s17avc Example Text Example Data	23	nag_specfun_airy_bi_real_vector Airy function vectorized $Bi (x)$
s17awc Example Text Example Data	23	nag_specfun_airy_ai_deriv_vector Derivatives of the Airy function, vectorized ${Ai}^{'} (x)$
s17axc Example Text Example Data	23	nag_specfun_airy_bi_deriv_vector Derivatives of the Airy function, vectorized ${Bi}^{'} (x)$
s17dcc Example Text Example Data	7	nag_specfun_bessel_y_complex Bessel functions $Y_{ν + a} (z)$ , real $a \geq 0$ , complex $z$ , $ν = 0, 1, 2, \dots$
s17dec Example Text Example Data	7	nag_specfun_bessel_j_complex Bessel functions $J_{ν + a} (z)$ , real $a \geq 0$ , complex $z$ , $ν = 0, 1, 2, \dots$
s17dgc Example Text Example Data	7	nag_specfun_airy_ai_complex Airy functions $Ai (z)$ and ${Ai}^{'} (z)$ , complex $z$
s17dhc Example Text Example Data	7	nag_specfun_airy_bi_complex Airy functions $Bi (z)$ and ${Bi}^{'} (z)$ , complex $z$
s17dlc Example Text Example Data	7	nag_specfun_hankel_complex Hankel functions $H_{ν + a}^{(j)} (z)$ , $j = 1, 2$ , real $a \geq 0$ , complex $z$ , $ν = 0, 1, 2, \dots$
s17gac Example Text Example Data Example Plot	26.1	nag_specfun_struve_h0 Struve function of order $0$ , $H_{0} (x)$
s17gbc Example Text Example Data Example Plot	26.1	nag_specfun_struve_h1 Struve function of order $1$ , $H_{1} (x)$
s18acc Example Text Example Data Example Plot	1	nag_specfun_bessel_k0_real Modified Bessel function $K_{0} (x)$
s18adc Example Text Example Data Example Plot	1	nag_specfun_bessel_k1_real Modified Bessel function $K_{1} (x)$
s18aec Example Text Example Data Example Plot	1	nag_specfun_bessel_i0_real Modified Bessel function $I_{0} (x)$
s18afc Example Text Example Data Example Plot	1	nag_specfun_bessel_i1_real Modified Bessel function $I_{1} (x)$
s18aqc Example Text Example Data	23	nag_specfun_bessel_k0_real_vector Modified Bessel function vectorized $K_{0} (x)$
s18arc Example Text Example Data	23	nag_specfun_bessel_k1_real_vector Modified Bessel function vectorized $K_{1} (x)$
s18asc Example Text Example Data	23	nag_specfun_bessel_i0_real_vector Modified Bessel function vectorized $I_{0} (x)$
s18atc Example Text Example Data	23	nag_specfun_bessel_i1_real_vector Modified Bessel function vectorized $I_{1} (x)$
s18ccc Example Text Example Data	2	nag_specfun_bessel_k0_scaled Scaled modified Bessel function $e^{x} K_{0} (x)$
s18cdc Example Text Example Data	2	nag_specfun_bessel_k1_scaled Scaled modified Bessel function $e^{x} K_{1} (x)$
s18cec Example Text Example Data	2	nag_specfun_bessel_i0_scaled Scaled modified Bessel function $e^{- \|x\|} I_{0} (x)$
s18cfc Example Text Example Data	2	nag_specfun_bessel_i1_scaled Scaled modified Bessel function $e^{- \|x\|} I_{1} (x)$
s18cqc Example Text Example Data	23	nag_specfun_bessel_k0_scaled_vector Scaled modified Bessel function vectorized $e^{x} K_{0} (x)$
s18crc Example Text Example Data	23	nag_specfun_bessel_k1_scaled_vector Scaled modified Bessel function vectorized $e^{x} K_{1} (x)$
s18csc Example Text Example Data	23	nag_specfun_bessel_i0_scaled_vector Scaled modified Bessel function vectorized $e^{- \|x\|} I_{0} (x)$
s18ctc Example Text Example Data	23	nag_specfun_bessel_i1_scaled_vector Scaled modified Bessel function vectorized $e^{- \|x\|} I_{1} (x)$
s18dcc Example Text Example Data	7	nag_specfun_bessel_k_complex Modified Bessel functions $K_{ν + a} (z)$ , real $a \geq 0$ , complex $z$ , $ν = 0, 1, 2, \dots$
s18dec Example Text Example Data	7	nag_specfun_bessel_i_complex Modified Bessel functions $I_{ν + a} (z)$ , real $a \geq 0$ , complex $z$ , $ν = 0, 1, 2, \dots$
s18ecc Example Text Example Data	6	nag_bessel_i_nu_scaled Scaled modified Bessel function $e^{- x} I_{ν / 4} (x)$
s18edc Example Text Example Data	6	nag_bessel_k_nu_scaled Scaled modified Bessel function $e^{x} K_{ν / 4} (x)$
s18eec Example Text Example Data	6	nag_bessel_i_nu Modified Bessel function $I_{ν / 4} (x)$
s18efc Example Text Example Data	6	nag_bessel_k_nu Modified Bessel function $K_{ν / 4} (x)$
s18egc Example Text Example Data	6	nag_bessel_k_alpha Modified Bessel functions $K_{α + n} (x)$ for real $x > 0$ , selected values of $α \geq 0$ and $n = 0, 1, \dots, N$
s18ehc Example Text Example Data	6	nag_bessel_k_alpha_scaled Scaled modified Bessel functions $e^{x} K_{α + n} (x)$ for real $x > 0$ , selected values of $α \geq 0$ and $n = 0, 1, \dots, N$
s18ejc Example Text Example Data	6	nag_bessel_i_alpha Modified Bessel functions $I_{α + n - 1} (x)$ or $I_{α - n + 1} (x)$ for real $x \neq 0$ , non-negative $α < 1$ and $n = 1, 2, \dots, \|N\| + 1$
s18ekc Example Text Example Data	6	nag_bessel_j_alpha Bessel functions $J_{α + n - 1} (x)$ or $J_{α - n + 1} (x)$ for real $x \neq 0$ , non-negative $α < 1$ and $n = 1, 2, \dots, \|N\| + 1$
s18gac Example Text Example Data Example Plot	26.1	nag_specfun_struve_l0 Modified Struve function of order $0$ , $L_{0} (x)$
s18gbc Example Text Example Data Example Plot	26.1	nag_specfun_struve_l1 Modified Struve function of order $1$ , $L_{1} (x)$
s18gcc Example Text Example Data Example Plot	26.1	nag_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
s18gdc Example Text Example Data Example Plot	26.1	nag_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
s18gkc Example Text Example Data	7	nag_specfun_bessel_j_seq_complex Bessel function of the 1st kind $J_{α \pm n} (z)$
s19aac Example Text Example Data	1	nag_specfun_kelvin_ber Kelvin function $ber x$
s19abc Example Text Example Data	1	nag_specfun_kelvin_bei Kelvin function $bei x$
s19acc Example Text Example Data	1	nag_specfun_kelvin_ker Kelvin function $\ker x$
s19adc Example Text Example Data	1	nag_specfun_kelvin_kei Kelvin function $kei x$
s19anc Example Text Example Data	23	nag_specfun_kelvin_ber_vector Kelvin function vectorized $ber x$
s19apc Example Text Example Data	23	nag_specfun_kelvin_bei_vector Kelvin function vectorized $bei x$
s19aqc Example Text Example Data	23	nag_specfun_kelvin_ker_vector Kelvin function vectorized $\ker x$
s19arc Example Text Example Data	23	nag_specfun_kelvin_kei_vector Kelvin function vectorized $kei x$
s20acc Example Text Example Data Example Plot	1	nag_specfun_fresnel_s Fresnel integral $S (x)$
s20adc Example Text Example Data Example Plot	1	nag_specfun_fresnel_c Fresnel integral $C (x)$
s20aqc Example Text Example Data	23	nag_specfun_fresnel_s_vector Fresnel integral vectorized $S (x)$
s20arc Example Text Example Data	23	nag_specfun_fresnel_c_vector Fresnel integral vectorized $C (x)$
s21bac Example Text	1	nag_specfun_ellipint_symm_1_degen Degenerate symmetrised elliptic integral of 1st kind $R_{C} (x, y)$
s21bbc Example Text	1	nag_specfun_ellipint_symm_1 Symmetrised elliptic integral of 1st kind $R_{F} (x, y, z)$
s21bcc Example Text	1	nag_specfun_ellipint_symm_2 Symmetrised elliptic integral of 2nd kind $R_{D} (x, y, z)$
s21bdc Example Text	1	nag_specfun_ellipint_symm_3 Symmetrised elliptic integral of 3rd kind $R_{J} (x, y, z, r)$
s21bec Example Text Example Plot	9	nag_specfun_ellipint_legendre_1 Elliptic integral of 1st kind, Legendre form, $F (ϕ ∣ m)$
s21bfc Example Text Example Plot	9	nag_specfun_ellipint_legendre_2 Elliptic integral of 2nd kind, Legendre form, $E (ϕ ∣ m)$
s21bgc Example Text	9	nag_specfun_ellipint_legendre_3 Elliptic integral of 3rd kind, Legendre form, $Π (n; ϕ ∣ m)$
s21bhc Example Text	9	nag_specfun_ellipint_complete_1 Complete elliptic integral of 1st kind, Legendre form, $K (m)$
s21bjc Example Text	9	nag_specfun_ellipint_complete_2 Complete elliptic integral of 2nd kind, Legendre form, $E (m)$
s21cac Example Text Example Data	7	nag_specfun_jacellip_real Jacobian elliptic functions sn, cn and dn of real argument
s21cbc Example Text Example Data	6	nag_specfun_jacellip_complex Jacobian elliptic functions sn, cn and dn of complex argument
s21ccc Example Text Example Data	6	nag_specfun_jactheta_real Jacobian theta functions with real arguments
s21dac Example Text Example Data	6	nag_specfun_ellipint_general_2 Elliptic integrals of the second kind with complex arguments
s22aac Example Text Example Data	6	nag_specfun_legendre_p Legendre and associated Legendre functions of the first kind with real arguments
s22bac Example Text	24	nag_specfun_hyperg_confl_real Real confluent hypergeometric function ${}_{1}F_{1} (a; b; x)$
s22bbc Example Text	24	nag_specfun_hyperg_confl_real_scaled Real confluent hypergeometric function ${}_{1}F_{1} (a; b; x)$ in scaled form
s22bec Example Text	24	nag_specfun_hyperg_gauss_real Real Gauss hypergeometric function ${}_{2}F_{1} (a, b; c; x)$
s22bfc Example Text	24	nag_specfun_hyperg_gauss_real_scaled Real Gauss hypergeometric function ${}_{2}F_{1} (a, b; c; x)$ in scaled form
s22cac Example Text Example Data Example Plot	27	nag_specfun_mathieu_ang_periodic_real Calculates values of real periodic angular Mathieu functions
s30aac Example Text Example Data	9	nag_specfun_opt_bsm_price Black–Scholes–Merton option pricing formula
s30abc Example Text Example Data	9	nag_specfun_opt_bsm_greeks Black–Scholes–Merton option pricing formula with Greeks
s30bac Example Text Example Data	9	nag_specfun_opt_lookback_fls_price Floating-strike lookback option pricing formula in the Black-Scholes-Merton model
s30bbc Example Text Example Data	9	nag_specfun_opt_lookback_fls_greeks Floating-strike lookback option pricing formula with Greeks in the Black-Scholes-Merton model
s30cac Example Text Example Data	9	nag_specfun_opt_binary_con_price Binary option, cash-or-nothing pricing formula
s30cbc Example Text Example Data	9	nag_specfun_opt_binary_con_greeks Binary option, cash-or-nothing pricing formula with Greeks
s30ccc Example Text Example Data	9	nag_specfun_opt_binary_aon_price Binary option, asset-or-nothing pricing formula
s30cdc Example Text Example Data	9	nag_specfun_opt_binary_aon_greeks Binary option, asset-or-nothing pricing formula with Greeks
s30fac Example Text Example Data	9	nag_specfun_opt_barrier_std_price Standard barrier option pricing formula
s30jac Example Text Example Data	9	nag_specfun_opt_jumpdiff_merton_price Jump-diffusion, Merton's model, option pricing formula
s30jbc Example Text Example Data	9	nag_specfun_opt_jumpdiff_merton_greeks Jump-diffusion, Merton's model, option pricing formula with Greeks
s30nac Example Text Example Data	9	nag_specfun_opt_heston_price Heston's model option pricing formula
s30nbc Example Text Example Data	23	nag_specfun_opt_heston_greeks Heston's model option pricing formula with Greeks
s30ncc Example Text Example Data	24	nag_specfun_opt_heston_term Heston's model option pricing with term structure
s30qcc Example Text Example Data	9	nag_specfun_opt_amer_bs_price American option, Bjerksund and Stensland pricing formula
s30sac Example Text Example Data	9	nag_specfun_opt_asian_geom_price Asian option, geometric continuous average rate pricing formula
s30sbc Example Text Example Data	9	nag_specfun_opt_asian_geom_greeks Asian option, geometric continuous average rate pricing formula with Greeks

NAG CL InterfaceS (Specfun)Approximations of Special Functions

NAG CL Interface
S (Specfun)
Approximations of Special Functions