# 6.43 S Approximations of Special Functions

## s - Approximations of Special Functions

Routine Name Mark of Introduction Purpose
s01bac 7 nag_shifted_log
ln (1+x)
s10aac 1 nag_tanh
Hyperbolic tangent, tanh x
s10abc 1 nag_sinh
Hyperbolic sine, sinh x
s10acc 1 nag_cosh
Hyperbolic cosine, cosh x
s11aac 1 nag_arctanh
Inverse hyperbolic tangent, arctanh x
s11abc 1 nag_arcsinh
Inverse hyperbolic sine, arcsinh x
s11acc 1 nag_arccosh
Inverse hyperbolic cosine, arccosh x
s13aac 1 nag_exp_integral
Exponential integral E1 (x)
s13acc 1 nag_cos_integral
Cosine integral Ci(x)
Sine integral Si(x)
s14aac 1 nag_gamma
Gamma function Γ (x)
s14abc 1 nag_log_gamma
Log Gamma function ln(Γ (x))
s14acc 7 nag_polygamma_fun
ψ (x) - ln x
Scaled derivatives of ψ (x)
s14aec 6 nag_real_polygamma
Derivative of the psi function ψ (x)
s14afc 6 nag_complex_polygamma
Derivative of the psi function ψ (z)
s14agc 7 nag_complex_log_gamma
Logarithm of the Gamma function lnΓ (z)
s14ahc 9 nag_scaled_log_gamma
Scaled log gamma function lnG(x), where G(x) = γ(x + 1) / (x / e)
s14bac 1 nag_incomplete_gamma
Incomplete Gamma functions P(a,x) and Q(a,x)
s14cbc 23 nag_log_beta
Logarithm of the beta function lnB(a,b)
s14ccc 23 nag_incomplete_beta
Incomplete beta function Ix(a,b) and its complement 1-Ix
s15abc 1 nag_cumul_normal
Cumulative Normal distribution function P(x)
s15acc 1 nag_cumul_normal_complem
Complement of cumulative Normal distribution function Q(x)
Complement of error function erfc(x)
s15aec 1 nag_erf
Error function erf(x)
s15afc 7 nag_dawson
Dawson's integral
s15agc 9 nag_erfcx
Scaled complement of error function, erfcx(x)
s15ddc 7 nag_complex_erfc
Scaled complex complement of error function, exp(-z2) erfc(-iz)
s17acc 1 nag_bessel_y0
Bessel function Y0 (x)
Bessel function Y1 (x)
s17aec 1 nag_bessel_j0
Bessel function J0 (x)
s17afc 1 nag_bessel_j1
Bessel function J1 (x)
s17agc 1 nag_airy_ai
Airy function Ai(x)
s17ahc 1 nag_airy_bi
Airy function Bi(x)
s17ajc 1 nag_airy_ai_deriv
Airy function Ai'(x)
s17akc 1 nag_airy_bi_deriv
Airy function Bi'(x)
s17alc 6 nag_bessel_zeros
Zeros of Bessel functions Jα(x), J'α(x), Yα(x) or Y'α(x)
s17aqc 23 nag_bessel_y0_vector
Bessel function vectorized Y0 (x)
s17arc 23 nag_bessel_y1_vector
Bessel function vectorized Y1 (x)
s17asc 23 nag_bessel_j0_vector
Bessel function vectorized J0 (x)
s17atc 23 nag_bessel_j1_vector
Bessel function vectorized J1 (x)
s17auc 23 nag_airy_ai_vector
Airy function vectorized Ai(x)
s17avc 23 nag_airy_bi_vector
Airy function vectorized Bi(x)
s17awc 23 nag_airy_ai_deriv_vector
Derivatives of the Airy function, vectorized Ai'(x)
s17axc 23 nag_airy_bi_deriv_vector
Derivatives of the Airy function, vectorized Bi'(x)
s17dcc 7 nag_complex_bessel_y
Bessel functions Yν+a(z), real a ≥ 0, complex z, ν =0,1, 2,...
s17dec 7 nag_complex_bessel_j
Bessel functions J ν +a(z), real a ≥ 0, complex z, ν =0,1, 2,...
s17dgc 7 nag_complex_airy_ai
Airy functions Ai(z) and Ai'(z) , complex z
s17dhc 7 nag_complex_airy_bi
Airy functions Bi(z) and Bi'(z) , complex z
s17dlc 7 nag_complex_hankel
Hankel functions Hν+a(j)(z), j=1,2, real a ≥ 0, complex z, ν =0,1,2,...
s18acc 1 nag_bessel_k0
Modified Bessel function K0 (x)
Modified Bessel function K1 (x)
s18aec 1 nag_bessel_i0
Modified Bessel function I0 (x)
s18afc 1 nag_bessel_i1
Modified Bessel function I1(x)
s18aqc 23 nag_bessel_k0_vector
Modified Bessel function vectorized K0 (x)
s18arc 23 nag_bessel_k1_vector
Modified Bessel function vectorized K1 (x)
s18asc 23 nag_bessel_i0_vector
Modified Bessel function vectorized I0 (x)
s18atc 23 nag_bessel_i1_vector
Modified Bessel function vectorized I1 (x)
s18ccc 2 nag_bessel_k0_scaled
Scaled modified Bessel function exK0(x)
s18cdc 2 nag_bessel_k1_scaled
Scaled modified Bessel function exK1(x)
s18cec 2 nag_bessel_i0_scaled
Scaled modified Bessel function e-¦x¦I0(x)
s18cfc 2 nag_bessel_i1_scaled
Scaled modified Bessel function e-¦x¦I1(x)
s18cqc 23 nag_bessel_k0_scaled_vector
Scaled modified Bessel function vectorized exK0(x)
s18crc 23 nag_bessel_k1_scaled_vector
Scaled modified Bessel function vectorized exK1(x)
s18csc 23 nag_bessel_i0_scaled_vector
Scaled modified Bessel function vectorized e-¦x¦I0(x)
s18ctc 23 nag_bessel_i1_scaled_vector
Scaled modified Bessel function vectorized e-¦x¦I1(x)
s18dcc 7 nag_complex_bessel_k
Modified Bessel functions Kν+a(z), real a ≥ 0, complex z, ν =0,1,2,...
s18dec 7 nag_complex_bessel_i
Modified Bessel functions Iν+a(z), real a ≥ 0, complex z, ν =0,1,2,...
s18ecc 6 nag_bessel_i_nu_scaled
Scaled modified Bessel function e-x Iν/4 (x)
s18edc 6 nag_bessel_k_nu_scaled
Scaled modified Bessel function ex Kν/4 (x)
s18eec 6 nag_bessel_i_nu
Modified Bessel function Iν/4 (x)
s18efc 6 nag_bessel_k_nu
Modified Bessel function Kν/4 (x)
s18egc 6 nag_bessel_k_alpha
Modified Bessel functions Kα +n (x) for real x > 0, selected values of α ≥ 0 and n = 0,1,...,N
s18ehc 6 nag_bessel_k_alpha_scaled
Scaled modified Bessel functions Kα +n (x) for real x > 0, selected values of α ≥ 0 and n = 0,1,...,N
s18ejc 6 nag_bessel_i_alpha
Modified Bessel functions Iα +n-1 (x) or Iα -n+1 (x) for real x ≠ 0, non-negative α < 1 and n = 1,2,...,¦N¦+1
s18ekc 6 nag_bessel_j_alpha
Bessel functions Jα +n-1 (x) or Jα -n+1 (x) for real x ≠ 0, non-negative α < 1 and n = 1,2,...,¦N¦+1
s18gkc 7 nag_complex_bessel_j_seq
Bessel function of the 1st kind Jα ≠n(z)
s19aac 1 nag_kelvin_ber
Kelvin function ber x
s19abc 1 nag_kelvin_bei
Kelvin function bei x
s19acc 1 nag_kelvin_ker
Kelvin function ker x
Kelvin function kei x
s19anc 23 nag_kelvin_ber_vector
Kelvin function vectorized ber x
s19apc 23 nag_kelvin_bei_vector
Kelvin function vectorized bei x
s19aqc 23 nag_kelvin_ker_vector
Kelvin function vectorized ker x
s19arc 23 nag_kelvin_kei_vector
Kelvin function vectorized kei x
s20acc 1 nag_fresnel_s
Fresnel integral S(x)
Fresnel integral C(x)
s20aqc 23 nag_fresnel_s_vector
Fresnel integral vectorized S(x)
s20arc 23 nag_fresnel_c_vector
Fresnel integral vectorized C(x)
s21bac 1 nag_elliptic_integral_rc
Degenerate symmetrised elliptic integral of 1st kind RC(x,y)
s21bbc 1 nag_elliptic_integral_rf
Symmetrised elliptic integral of 1st kind RF(x,y,z)
s21bcc 1 nag_elliptic_integral_rd
Symmetrised elliptic integral of 2nd kind RD(x,y,z)
s21bdc 1 nag_elliptic_integral_rj
Symmetrised elliptic integral of 3rd kind RJ(x,y,z,r)
s21bec 9 nag_elliptic_integral_F
Elliptic integral of 1st kind, Legendre form, F(φ¦m)
s21bfc 9 nag_elliptic_integral_E
Elliptic integral of 2nd kind, Legendre form, E(φ¦m)
s21bgc 9 nag_elliptic_integral_pi
Elliptic integral of 3rd kind, Legendre form, Π(n ; φ¦m)
s21bhc 9 nag_elliptic_integral_complete_K
Complete elliptic integral of 1st kind, Legendre form, K(m)
s21bjc 9 nag_elliptic_integral_complete_E
Complete elliptic integral of 2nd kind, Legendre form, E(m)
s21cac 7 nag_real_jacobian_elliptic
Jacobian elliptic functions sn, cn and dn of real argument
s21cbc 6 nag_jacobian_elliptic
Jacobian elliptic functions sn, cn and dn of complex argument
s21ccc 6 nag_jacobian_theta
Jacobian theta functions with real arguments
s21dac 6 nag_elliptic_integral_f
Elliptic integrals of the second kind with complex arguments
s22aac 6 nag_legendre_p
Legendre and associated Legendre functions of the first kind with real arguments
s22bac 24 nag_specfun_1f1_real
Real confluent hypergeometric function 1F1(a;b;x)
s22bbc 24 nag_specfun_1f1_real_scaled
Real confluent hypergeometric function 1F1(a;b;x) in scaled form
s22bec 24 nag_specfun_2f1_real
Real Gauss hypergeometric function 2F1(a,b;c;x)
s22bfc 24 nag_specfun_2f1_real_scaled
Real Gauss hypergeometric function 2F1(a,b;c;x) in scaled form
s30aac 9 nag_bsm_price
Black–Scholes–Merton option pricing formula
s30abc 9 nag_bsm_greeks
Black–Scholes–Merton option pricing formula with Greeks
s30bac 9 nag_lookback_fls_price
Floating-strike lookback option pricing formula
s30bbc 9 nag_lookback_fls_greeks
Floating-strike lookback option pricing formula with Greeks
s30cac 9 nag_binary_con_price
Binary option: cash-or-nothing pricing formula
s30cbc 9 nag_binary_con_greeks
Binary option: cash-or-nothing pricing formula with Greeks
s30ccc 9 nag_binary_aon_price
Binary option: asset-or-nothing pricing formula
s30fac 9 nag_barrier_std_price
Standard barrier option pricing formula
s30jac 9 nag_jumpdiff_merton_greeks
Jump-diffusion, Merton's model, option pricing formula with Greeks
s30jbc 9 nag_jumpdiff_merton_greeks
Jump-diffusion, Merton's model, option pricing formula with Greeks
s30nac 9 nag_heston_price
Heston's model option pricing formula
s30nbc 23 nag_heston_greeks
Heston's model option pricing formula with Greeks
s30ncc 24 nag_heston_term
Heston's model option pricing with term structure
s30qcc 9 nag_amer_bs_price
American option: Bjerksund and Stensland pricing formula
s30sac 9 nag_asian_geom_price
Asian option: geometric continuous average rate pricing formula
s30sbc 9 nag_asian_geom_greeks
Asian option: geometric continuous average rate pricing formula with Greeks