S Chapter Contents : NAG Library Manual, Mark 25

Routine Name	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 $* (x) - \ln x$ where $* (x)$ is the psi function
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
S14BAF Example Text Example Data Example Plot	14	nagf_specfun_gamma_incomplete Incomplete gamma functions $P (a, x)$ 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 Incomplete beta function $I_{x} (a, b)$ 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, $erfcx (x)$
S15DDF Example Text Example Data	14	nagf_specfun_erfc_complex Scaled complex complement of error function, $\exp (- z^{2}) erfc (- i z)$
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 $z$ , $ν = 0, 1, 2, \dots$
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 $z$ , $ν = 0, 1, 2, \dots$
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_1f1_real Real confluent hypergeometric function ${}_{1}F_{1} (a; b; x)$
S22BBF Example Text	24	nagf_specfun_1f1_real_scaled Real confluent hypergeometric function ${}_{1}F_{1} (a; b; x)$ in scaled form
S22BEF Example Text	25	nagf_specfun_2f1_real Real Gauss hypergeometric function ${}_{2}F_{1} (a, b; c; x)$
S22BFF Example Text	25	nagf_specfun_2f1_real_scaled Real Gauss hypergeometric function ${}_{2}F_{1} (a, b; c; x)$ in scaled form.
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 Library Chapter Contents

S – Approximations of Special Functions

S Chapter Introduction

NAG Library Chapter ContentsS – Approximations of Special Functions

S Chapter Introduction

NAG Library Chapter Contents

S – Approximations of Special Functions