Special functions

14 Special functions

The special function approximations are defined in the header pnl/pnl_specfun.h.

Most of these functions rely on the Cephes library which uses its own error mechanism which can be activated or deactivated using the two following functions

void pnl_deactivate_mtherr ()
Description Deactivate Cephes error mechanism
void pnl_activate_mtherr ()
Description Activate Cephes error mechanism

14.1 Real Bessel functions

double pnl_bessel_i (double v, double x)
Description Modified Bessel function of the first kind of order v.
double pnl_bessel_i_scaled (double v, double x)
Description Modified Bessel function of the first kind of order v divided by e^|x|.
double pnl_bessel_rati (double v, double x)
Description Ratio of modified Bessel functions of the first kind : I_v+1(x)∕I_v(x).
double pnl_bessel_j (double v, double x)
Description Bessel function of the first kind of order v.
double pnl_bessel_j_scaled (double v, double x)
Description Bessel function of the first kind of order v. Same function as pnl_bessel_j.
double pnl_bessel_y (double v, double x)
Description Modified Bessel function of the second kind of order v.
double pnl_bessel_y_scaled (double v, double x)
Description Modified Bessel function of the second kind of order v. Same function as pnl_bessel_y.
double pnl_bessel_k (double v, double x)
Description Bessel function of the third kind of order v.
double pnl_bessel_k_scaled (double v, double x)
Description Bessel function of the third kind of order v multiplied by e^x.
dcomplex pnl_bessel_h1 (double v, double x)
Description Hankel function of the first kind of order v.
dcomplex pnl_bessel_h1_scaled (double v, double x)
Description Hankel function of the first kind of order v and divided by e^Ix.
dcomplex pnl_bessel_h2 (double v, double x)
Description Hankel function of the second kind of order v.
dcomplex pnl_bessel_h2_scaled (double v, double x)
Description Hankel function of the second kind of order v and multiplied by e^Ix.

14.2 Complex Bessel functions

dcomplex pnl_complex_bessel_i (double v, dcomplex z)
Description Complex Modified Bessel function of the first kind of order v.
dcomplex pnl_complex_bessel_i_scaled (double v, dcomplex z)
Description Complex Modified Bessel function of the first kind of order v divided by e^|Creal(z)|.
dcomplex pnl_complex_bessel_rati (double v, dcomplex x)
Description Ratio of complex modified Bessel functions of the first kind : I_v+1(x)∕I_v(x).
dcomplex pnl_complex_bessel_j (double v, dcomplex z)
Description Complex Bessel function of the first kind of order v.
dcomplex pnl_complex_bessel_j_scaled (double v, dcomplex z)
Description Complex Bessel function of the first kind of order v divided by e^|Cimag(z)|.
dcomplex pnl_complex_bessel_y (double v, dcomplex z)
Description Complex Modified Bessel function of the second kind of order v.
dcomplex pnl_complex_bessel_y_scaled (double v, dcomplex z)
Description Complex Modified Bessel function of the second kind of order v divided by e^|Cimag(z)|.
dcomplex pnl_complex_bessel_k (double v, dcomplex z)
Description Complex Bessel function of the third kind of order v.
dcomplex pnl_complex_bessel_k_scaled (double v, dcomplex z)
Description Complex Bessel function of the third kind of order v multiplied by e^z.
dcomplex pnl_complex_bessel_h1 (double v, dcomplex z)
Description Complex Hankel function of the first kind of order v.
dcomplex pnl_complex_bessel_h1_scaled (double v, dcomplex z)
Description Complex Hankel function of the first kind of order v and divided by e^Iz.
dcomplex pnl_complex_bessel_h2 (double v, dcomplex z)
Description Complex Hankel function of the second kind of order v.
dcomplex pnl_complex_bessel_h2_scaled (double v, dcomplex z)
Description Complex Hankel function of the second kind of order v and multiplied by e^Iz.

14.3 Error functions

double pnl_sf_erf (double x)
Description Compute the error function ∫ ₀^x e-t² dt.
dcomplex pnl_sf_complex_erf (dcomplex z)
Description Same as pnl_sf_erf for complex arguments.
double pnl_sf_erfc (double x)
Description Compute the complementary error function 1. - erf(x).
dcomplex pnl_sf_complex_erfc (dcomplex x)
Description Same as pnl_sf_erfc for complex arguments.
double pnl_sf_erfcx (double x)
Description Compute the scaled complementary error function of x, defined by ex² erfc(x).
dcomplex pnl_sf_complex_erfcx (dcomplex z)
Description Same as pnl_sf_erfcx for complex arguments. Note that erfcx(-i x) = w(x).
dcomplex pnl_sf_w (dcomplex z)
Description Compute e-z² erfc(-iz).
double pnl_sf_w_im (double x)
Description Compute 2Dawson(x)∕
double pnl_sf_erfi (double x)
Description Compute -i erf(i z)
dcomplex pnl_sf_complex_erfi (dcomplex z)
Description Same as pnl_sf_erfi for complex arguments.
double pnl_sf_dawson (double x)
Description Compute ∕2 e-x² erfi(x).
dcomplex pnl_sf_complex_dawson (dcomplex z)
Description Same as pnl_sf_dawson for complex arguments.
double pnl_sf_log_erf (double x)
Description Compute log pnl_sf_erf(x)
double pnl_sf_log_erfc (double x)
Description Compute log pnl_sf_erfc(x)

14.4 Gamma functions

For x > 0, the Gamma Function is defined by

∫ ∞ -u x-1 Γ (x ) = e u du. 0

double pnl_sf_fact (int n)
Description Computes factorial of n Γ(n + 1).
double pnl_sf_gamma (double x)
Description Computes Γ(x),x ≥ 0
double pnl_sf_log_gamma (double x)
Description Computes log(Γ(x)),x ≥ 0
int pnl_sf_log_gamma_sgn (double x, double *y, int *sgn)
Description Computes y = log(|Γ(x)|) for x > 0 sgn contains the sign of Γ(x) (-1 or +1).
double pnl_sf_choose (int n, int k)
Description Computes the binomial coefficient = for 0 ≤k ≤n in double precision.

14.5 Digamma function

For x > 0, the digamma function ψ is defined as the logarithmic derivative of the Gamma function Γ

′ ψ (x ) =-d-logΓ (x ) = Γ-(x). dx Γ (x)

The function ψ admits the following integral representation

∫ ( ) ψ(x) = ∞ e--u- -e-xu-- . 0 u 1- e- u

double pnl_sf_psi (double x)
Description Return ψ(x).

14.6 Incomplete Gamma functions

For a ∈ℝ and x > 0, the Incomplete Gamma Function is defined by

∫ ∞ Γ (a,x ) = e-u ua-1du. x

A relation similar to the one existing for the standard Gamma function holds

- xa-e-x+-Γ (a-+-1,x) Γ (a,x) = a .

Γ(a)	= ∫ ₀^∞u^a-1 e-u du
P(a,x)	= = ∫ ₀^xu^a-1 e-u du
Q(a,x)	= 1 -P(a,x) = = ∫ _x^∞e-u u^a-1du.

double pnl_sf_gamma_inc (double a, double x)
Description Computes Γ(a,x), a ∈ℝ,x ≥ 0
void pnl_sf_gamma_inc_P (double a, double x)
Description Computes P(a,x), a > 0,x ≥ 0
void pnl_sf_gamma_inc_Q (double a, double x)
Description Computes Q(a,x), a > 0,x ≥ 0

14.7 Exponential integrals

For x > 0 and n ∈ℕ, the function E_n is defined by

∫ ∞ En (x ) = e-xu u-ndu 1

This function is linked to the Incomplete Gamma function by

∫ ∞ ∫ ∞ En (x) = e- xu(xu )-nxn-1d(xu) = xn-1 e-tt-ndt = xn-1Γ (1 - n,x) , x x

from which we can deduce

nE (x) = e- x- xE (x ). n+1 n

For n > 1, the series expansion is given by

[ 2 3 ] En(x) = xn-1Γ (1 - n)+ ---1-- + --x-- - ---x----+ --x------ ... . 1 - n 2 - n 2(3- n) 6(4 - n)

The asymptotic behaviour is given by

- x[ ] En (x ) = e-- 1 - n-+ n(n-+-1) + .... x x x2

The special case n = 1 gives

∫ ∞ e- u E1 (x) = ---du, |Arg (x )| ≥ π. x u

For any complex number x with positive real part, this can be written

∫ ∞ e-ux- E1 (x) = 1 u du, ℜ(x) ≥ 0.

By integrating the Taylor expansion of e-t ∕t, and extracting the logarithmic singularity, we can derive the following series representation for E₁(x),

∑∞ (- 1)kxk E1(x) = - γ - lnx - -------- |Arg(x)| < π. k=1 k k!

The function E₁ is linked to the exponential integral Ei

∫ x eu ∫ ∞ e-u Ei (x) = u-du = - -u--du ∀x ⁄= 0. - ∞ -x

The above definition can be used for positive values of x, but the integral has to be understood in terms of its Cauchy principal value, due to the singularity of the integrand at zero.

Ei(- x) = - E (x ), ℜ (x) ≥ 0. 1

We deduce,

∑∞ -xk- Ei (x ) = γ + lnx + k k!, x > 0. k=1

For x ∈ℝ

{ - Ei (- x) - iπ x < 0, Γ (0,x) = - Ei (- x) x > 0.

double pnl_sf_expint_En (int n, double x)
Description Computes E_n(x) for n ≥ 0,x ≥ 0, or x > 0 when n = 0 or 1.

14.8 Hypergeometric functions

double pnl_sf_hyperg_2F1 (double a, double b, double c, double x)
Description Compute the Gauss hypergeometric function 2F1(a,b,c,x) for |x| < 1 and for x < -1 when b,a,c,(b-a),(c-a),(c-b) are not integers
double pnl_sf_hyperg_1F1 (double a, double b, double x)
Description Compute the hypergeometric function 1F1(a,b,x)
double pnl_sf_hyperg_2F0 (double a, double b, double x)
Description Compute the hypergeometric function 2F0(a,b,x) for x<0 using the relation 2F0(a,b,x) = (-x)^-aU(a,1 + a -b,-).
double pnl_sf_hyperg_0F1 (double c, double x)
Description Compute the hypergeometric function 0F1(c,x)
double pnl_sf_hyperg_U (double a, double b, double x)
Description Compute the confluent hypergeometric function U(a,b,x) with x > 0

[next] [prev] [prev-tail] [front] [up]