S Chapter Contents : NAG Library CL Interface Manual, Mark 30

Function	Mark of Introduction	Purpose
s01bac	7	nag_specfun_log_shifted $\ln (1 + x)$
s10aac	1	nag_specfun_tanh Hyperbolic tangent, tanh $x$
s10abc	1	nag_specfun_sinh Hyperbolic sine, sinh $x$
s10acc	1	nag_specfun_cosh Hyperbolic cosine, cosh $x$
s11aac	1	nag_specfun_arctanh Inverse hyperbolic tangent, arctanh $x$
s11abc	1	nag_specfun_arcsinh Inverse hyperbolic sine, arcsinh $x$
s11acc	1	nag_specfun_arccosh Inverse hyperbolic cosine, arccosh $x$
s13aac	1	nag_specfun_integral_exp Exponential integral $E_{1} (x)$
s13acc	1	nag_specfun_integral_cos Cosine integral $Ci (x)$
s13adc	1	nag_specfun_integral_sin Sine integral $Si (x)$
s14aac	1	nag_specfun_gamma Gamma function $Γ (x)$
s14abc	1	nag_specfun_gamma_log_real Log gamma function $\ln (Γ (x))$
s14acc	7	nag_specfun_polygamma $ψ (x) - \ln x$
s14adc	7	nag_specfun_polygamma_deriv Scaled derivatives of $ψ (x)$
s14aec	6	nag_specfun_psi_deriv_real Derivative of the psi function $ψ (x)$
s14afc	6	nag_specfun_psi_deriv_complex Derivative of the psi function $ψ (z)$
s14agc	7	nag_specfun_gamma_log_complex Logarithm of the gamma function $\ln Γ (z)$ , complex argument
s14ahc	9	nag_specfun_gamma_log_scaled_real Scaled log gamma function $\ln G (x)$ , where $G (x) = Γ (x + 1) / {(x / e)}^{x}$
s14anc	27	nag_specfun_gamma_vector Gamma function, vectorized $Γ (x)$
s14apc	27	nag_specfun_gamma_log_real_vector Log gamma function, vectorized $\ln (Γ (x))$
s14bac	1	nag_specfun_gamma_incomplete Incomplete gamma functions $P (a, x)$ and $Q (a, x)$
s14bnc	27	nag_specfun_gamma_incomplete_vector Incomplete gamma functions, vectorized $P (a, x)$ and $Q (a, x)$
s14cbc	23	nag_specfun_beta_log_real Logarithm of the beta function $\ln B (a, b)$
s14ccc	23	nag_specfun_beta_incomplete Regularized incomplete beta function $I_{x} (a, b)$ and its complement $1 - I_{x}$
s14cpc	27	nag_specfun_beta_log_real_vector Logarithm of the beta function, vectorized $\ln B (a, b)$
s14cqc	27	nag_specfun_beta_incomplete_vector Regularized incomplete beta function, vectorized $I_{x} (a, b)$ and its complement $1 - I_{x}$
s15abc	1	nag_specfun_cdf_normal Cumulative Normal distribution function $P (x)$
s15acc	1	nag_specfun_compcdf_normal Complement of cumulative Normal distribution function $Q (x)$
s15adc	1	nag_specfun_erfc_real Complement of error function $erfc (x)$
s15aec	1	nag_specfun_erf_real Error function $\erf (x)$
s15afc	7	nag_specfun_dawson Dawson's integral
s15agc	9	nag_specfun_erfcx_real Scaled complement of error function, $erfcx (x)$
s15apc	27	nag_specfun_cdf_normal_vector Cumulative Normal distribution function, vectorized $P (x)$
s15aqc	27	nag_specfun_compcdf_normal_vector Complement of cumulative Normal distribution function, vectorized $Q (x)$
s15arc	27	nag_specfun_erfc_real_vector Complement of error function, vectorized $erfc (x)$
s15asc	27	nag_specfun_erf_real_vector Error function, vectorized $\erf (x)$
s15atc	27	nag_specfun_dawson_vector Dawson's integral, vectorized
s15auc	27	nag_specfun_erfcx_real_vector Scaled complement of error function, vectorized $erfcx (x)$
s15ddc	7	nag_specfun_erfc_complex Scaled complex complement of error function, $\exp (- z^{2}) erfc (- i z)$
s15drc	27	nag_specfun_erfc_complex_vector Scaled complex complement of error function, vectorized $\exp (- z^{2}) erfc (- i z)$
s17acc	1	nag_specfun_bessel_y0_real Bessel function $Y_{0} (x)$
s17adc	1	nag_specfun_bessel_y1_real Bessel function $Y_{1} (x)$
s17aec	1	nag_specfun_bessel_j0_real Bessel function $J_{0} (x)$
s17afc	1	nag_specfun_bessel_j1_real Bessel function $J_{1} (x)$
s17agc	1	nag_specfun_airy_ai_real Airy function $Ai (x)$
s17ahc	1	nag_specfun_airy_bi_real Airy function $Bi (x)$
s17ajc	1	nag_specfun_airy_ai_deriv Airy function ${Ai}^{'} (x)$
s17akc	1	nag_specfun_airy_bi_deriv Airy function ${Bi}^{'} (x)$
s17alc	6	nag_specfun_bessel_zeros Zeros of Bessel functions $J_{α} (x)$ , $J_{α}^{'} (x)$ , $Y_{α} (x)$ or $Y_{α}^{'} (x)$
s17aqc	23	nag_specfun_bessel_y0_real_vector Bessel function vectorized $Y_{0} (x)$
s17arc	23	nag_specfun_bessel_y1_real_vector Bessel function vectorized $Y_{1} (x)$
s17asc	23	nag_specfun_bessel_j0_real_vector Bessel function vectorized $J_{0} (x)$
s17atc	23	nag_specfun_bessel_j1_real_vector Bessel function vectorized $J_{1} (x)$
s17auc	23	nag_specfun_airy_ai_real_vector Airy function vectorized $Ai (x)$
s17avc	23	nag_specfun_airy_bi_real_vector Airy function vectorized $Bi (x)$
s17awc	23	nag_specfun_airy_ai_deriv_vector Derivatives of the Airy function, vectorized ${Ai}^{'} (x)$
s17axc	23	nag_specfun_airy_bi_deriv_vector Derivatives of the Airy function, vectorized ${Bi}^{'} (x)$
s17dcc	7	nag_specfun_bessel_y_complex Bessel functions $Y_{ν + a} (z)$ , real $a \geq 0$ , complex $z$ , $ν = 0, 1, 2, \dots$
s17dec	7	nag_specfun_bessel_j_complex Bessel functions $J_{ν + a} (z)$ , real $a \geq 0$ , complex $z$ , $ν = 0, 1, 2, \dots$
s17dgc	7	nag_specfun_airy_ai_complex Airy functions $Ai (z)$ and ${Ai}^{'} (z)$ , complex $z$
s17dhc	7	nag_specfun_airy_bi_complex Airy functions $Bi (z)$ and ${Bi}^{'} (z)$ , complex $z$
s17dlc	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	26.1	nag_specfun_struve_h0 Struve function of order $0$ , $H_{0} (x)$
s17gbc	26.1	nag_specfun_struve_h1 Struve function of order $1$ , $H_{1} (x)$
s18acc	1	nag_specfun_bessel_k0_real Modified Bessel function $K_{0} (x)$
s18adc	1	nag_specfun_bessel_k1_real Modified Bessel function $K_{1} (x)$
s18aec	1	nag_specfun_bessel_i0_real Modified Bessel function $I_{0} (x)$
s18afc	1	nag_specfun_bessel_i1_real Modified Bessel function $I_{1} (x)$
s18aqc	23	nag_specfun_bessel_k0_real_vector Modified Bessel function vectorized $K_{0} (x)$
s18arc	23	nag_specfun_bessel_k1_real_vector Modified Bessel function vectorized $K_{1} (x)$
s18asc	23	nag_specfun_bessel_i0_real_vector Modified Bessel function vectorized $I_{0} (x)$
s18atc	23	nag_specfun_bessel_i1_real_vector Modified Bessel function vectorized $I_{1} (x)$
s18ccc	2	nag_specfun_bessel_k0_scaled Scaled modified Bessel function $e^{x} K_{0} (x)$
s18cdc	2	nag_specfun_bessel_k1_scaled Scaled modified Bessel function $e^{x} K_{1} (x)$
s18cec	2	nag_specfun_bessel_i0_scaled Scaled modified Bessel function $e^{- \| x \|} I_{0} (x)$
s18cfc	2	nag_specfun_bessel_i1_scaled Scaled modified Bessel function $e^{- \| x \|} I_{1} (x)$
s18cqc	23	nag_specfun_bessel_k0_scaled_vector Scaled modified Bessel function vectorized $e^{x} K_{0} (x)$
s18crc	23	nag_specfun_bessel_k1_scaled_vector Scaled modified Bessel function vectorized $e^{x} K_{1} (x)$
s18csc	23	nag_specfun_bessel_i0_scaled_vector Scaled modified Bessel function vectorized $e^{- \| x \|} I_{0} (x)$
s18ctc	23	nag_specfun_bessel_i1_scaled_vector Scaled modified Bessel function vectorized $e^{- \| x \|} I_{1} (x)$
s18dcc	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	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	6	nag_bessel_i_nu_scaled Scaled modified Bessel function $e^{- x} I_{ν / 4} (x)$
s18edc	6	nag_bessel_k_nu_scaled Scaled modified Bessel function $e^{x} K_{ν / 4} (x)$
s18eec	6	nag_bessel_i_nu Modified Bessel function $I_{ν / 4} (x)$
s18efc	6	nag_bessel_k_nu Modified Bessel function $K_{ν / 4} (x)$
s18egc	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	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	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	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	26.1	nag_specfun_struve_l0 Modified Struve function of order $0$ , $L_{0} (x)$
s18gbc	26.1	nag_specfun_struve_l1 Modified Struve function of order $1$ , $L_{1} (x)$
s18gcc	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	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	7	nag_specfun_bessel_j_seq_complex Bessel function of the 1st kind $J_{α \pm n} (z)$
s19aac	1	nag_specfun_kelvin_ber Kelvin function $ber x$
s19abc	1	nag_specfun_kelvin_bei Kelvin function $bei x$
s19acc	1	nag_specfun_kelvin_ker Kelvin function $\ker x$
s19adc	1	nag_specfun_kelvin_kei Kelvin function $kei x$
s19anc	23	nag_specfun_kelvin_ber_vector Kelvin function vectorized $ber x$
s19apc	23	nag_specfun_kelvin_bei_vector Kelvin function vectorized $bei x$
s19aqc	23	nag_specfun_kelvin_ker_vector Kelvin function vectorized $\ker x$
s19arc	23	nag_specfun_kelvin_kei_vector Kelvin function vectorized $kei x$
s20acc	1	nag_specfun_fresnel_s Fresnel integral $S (x)$
s20adc	1	nag_specfun_fresnel_c Fresnel integral $C (x)$
s20aqc	23	nag_specfun_fresnel_s_vector Fresnel integral vectorized $S (x)$
s20arc	23	nag_specfun_fresnel_c_vector Fresnel integral vectorized $C (x)$
s21bac	1	nag_specfun_ellipint_symm_1_degen Degenerate symmetrised elliptic integral of 1st kind $R_{C} (x, y)$
s21bbc	1	nag_specfun_ellipint_symm_1 Symmetrised elliptic integral of 1st kind $R_{F} (x, y, z)$
s21bcc	1	nag_specfun_ellipint_symm_2 Symmetrised elliptic integral of 2nd kind $R_{D} (x, y, z)$
s21bdc	1	nag_specfun_ellipint_symm_3 Symmetrised elliptic integral of 3rd kind $R_{J} (x, y, z, r)$
s21bec	9	nag_specfun_ellipint_legendre_1 Elliptic integral of 1st kind, Legendre form, $F (ϕ ∣ m)$
s21bfc	9	nag_specfun_ellipint_legendre_2 Elliptic integral of 2nd kind, Legendre form, $E (ϕ ∣ m)$
s21bgc	9	nag_specfun_ellipint_legendre_3 Elliptic integral of 3rd kind, Legendre form, $Π (n; ϕ ∣ m)$
s21bhc	9	nag_specfun_ellipint_complete_1 Complete elliptic integral of 1st kind, Legendre form, $K (m)$
s21bjc	9	nag_specfun_ellipint_complete_2 Complete elliptic integral of 2nd kind, Legendre form, $E (m)$
s21cac	7	nag_specfun_jacellip_real Jacobian elliptic functions sn, cn and dn of real argument
s21cbc	6	nag_specfun_jacellip_complex Jacobian elliptic functions sn, cn and dn of complex argument
s21ccc	6	nag_specfun_jactheta_real Jacobian theta functions with real arguments
s21dac	6	nag_specfun_ellipint_general_2 Elliptic integrals of the second kind with complex arguments
s22aac	6	nag_specfun_legendre_p Legendre and associated Legendre functions of the first kind with real arguments
s22bac	24	nag_specfun_hyperg_confl_real Real confluent hypergeometric function ${}_{1}F_{1} (a; b; x)$
s22bbc	24	nag_specfun_hyperg_confl_real_scaled Real confluent hypergeometric function ${}_{1}F_{1} (a; b; x)$ in scaled form
s22bec	24	nag_specfun_hyperg_gauss_real Real Gauss hypergeometric function ${}_{2}F_{1} (a, b; c; x)$
s22bfc	24	nag_specfun_hyperg_gauss_real_scaled Real Gauss hypergeometric function ${}_{2}F_{1} (a, b; c; x)$ in scaled form
s22cac	27	nag_specfun_mathieu_ang_periodic_real Calculates values of real periodic angular Mathieu functions
s30aac	9	nag_specfun_opt_bsm_price Black–Scholes–Merton option pricing formula
s30abc	9	nag_specfun_opt_bsm_greeks Black–Scholes–Merton option pricing formula with Greeks
s30acc	27.1	nag_specfun_opt_imp_vol Black–Scholes–Merton implied volatility
s30bac	9	nag_specfun_opt_lookback_fls_price Floating-strike lookback option pricing formula in the Black-Scholes-Merton model
s30bbc	9	nag_specfun_opt_lookback_fls_greeks Floating-strike lookback option pricing formula with Greeks in the Black-Scholes-Merton model
s30cac	9	nag_specfun_opt_binary_con_price Binary option, cash-or-nothing pricing formula
s30cbc	9	nag_specfun_opt_binary_con_greeks Binary option, cash-or-nothing pricing formula with Greeks
s30ccc	9	nag_specfun_opt_binary_aon_price Binary option, asset-or-nothing pricing formula
s30cdc	9	nag_specfun_opt_binary_aon_greeks Binary option, asset-or-nothing pricing formula with Greeks
s30fac	9	nag_specfun_opt_barrier_std_price Standard barrier option pricing formula
s30jac	9	nag_specfun_opt_jumpdiff_merton_price Jump-diffusion, Merton's model, option pricing formula
s30jbc	9	nag_specfun_opt_jumpdiff_merton_greeks Jump-diffusion, Merton's model, option pricing formula with Greeks
s30nac	9	nag_specfun_opt_heston_price Heston's model option pricing formula
s30nbc	23	nag_specfun_opt_heston_greeks Heston's model option pricing formula with Greeks
s30ncc	24	nag_specfun_opt_heston_term Heston's model option pricing with term structure
s30ndc	28.5 (Experimental)	nag_specfun_opt_heston_more_greeks Heston's model option pricing formula with Greeks, sensitivities of model parameters and negative rates
s30qcc	9	nag_specfun_opt_amer_bs_price American option, Bjerksund and Stensland pricing formula
s30sac	9	nag_specfun_opt_asian_geom_price Asian option, geometric continuous average rate pricing formula
s30sbc	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