The library includes routines for calculating the values of Airy functions, Bessel functions, Clausen functions, Coulomb wave functions, Coupling coefficients, the Dawson function, Debye functions, Dilogarithms, Elliptic integrals, Jacobi elliptic functions, Error functions, Exponential integrals, Fermi-Dirac functions, Gamma functions, Gegenbauer functions, Hypergeometric functions, Laguerre functions, Legendre functions and Spherical Harmonics, the Psi (Digamma) Function, Synchrotron functions, Transport functions, Trigonometric functions and Zeta functions.
The Airy functions Ai(x) and Bi(x) are defined by the integral representations,
For further information see Abramowitz & Stegun, Section 10.4.
Return the n-th zero of the respectives functions.
Airy functions Ai and Bi.
These routines compute the regular cylindrical Bessel function of n-th order, J0(x)
Bessel functions J0(red), J1(green), J2(blue)
These routines compute the irregular cylindrical Bessel function of n-th order, Y0(x)
Bessel functions Y0(red), Y1(green), Y2(blue)
The Clausen function is defined by the following integral,
It is related to the dilogarithm by .
The Dawson integral is defined by
A table of Dawson’s integral can be found in Abramowitz & Stegun, Table 7.5.
The generalized Laguerre polynomials are defined in terms of confluent hypergeometric functions as
and are sometimes referred to as the associated Laguerre polynomials. They are related to the plain Laguerre polynomials by
and
For more information see Abramowitz & Stegun, Chapter 22.
Legendre polynomials
The Debye functions Dn(x) are defined by the following integral,
For further information see Abramowitz & Stegun, Section 27.1.
The complete Fermi-Dirac integral Fn(x) is given by,
Note that the Fermi-Dirac integral is sometimes defined without the normalisation factor in other texts.
These routines compute the dilogarithm for a real argument. In Lewin’s notation this is Li2(x), the real part of the dilogarithm of a real x. It is defined by the integral representation
Note that Im(Li2(x)) = 0 for x <= 1, and -π log(x) for x > 1.
Note that Abramowitz & Stegun refer to the Spence integral S(x)= Li2(1-x) as the dilogarithm rather than Li2(x).
The error function erf(x)
Compute the exponential integral En(x) of order n,
Computes the exponential integral Ei(x),
where PV denotes the principal value of the integral.
Compute the integral
where γE is the Euler constant.
Compute the combinatorial factor
These routines compute the Gauss hypergeometric function 2F1(a,b,c,x) = F(a,b,c,x) for .
If the arguments (a,b,c,x) are too close to a singularity then the function can return the error code `GSL_EMAXITER’ when the series approximation converges too slowly. This occurs in the region of x=1, c - a - b = m for integer m.