Special Functions

This chapter describes the AMPL bindings for the GSL special function library. 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 and Zeta functions.

• Airy Functions and Derivatives
  • Airy Functions
  • Zeros of Airy Functions
  • Zeros of Derivatives of Airy Functions
• Bessel Functions
  • Regular Cylindrical Bessel Functions
  • Irregular Cylindrical Bessel Functions
  • Regular Modified Cylindrical Bessel Functions
  • Irregular Modified Cylindrical Bessel Functions
  • Regular Spherical Bessel Functions
  • Irregular Spherical Bessel Functions
  • Regular Modified Spherical Bessel Functions
  • Irregular Modified Spherical Bessel Functions
  • Regular Bessel Function - Fractional Order
  • Irregular Bessel Function - Fractional Order
  • Regular Modified Bessel Functions - Fractional Order
  • Irregular Modified Bessel Functions - Fractional Order
  • Zeros of Regular Bessel Functions
• Clausen Function
• Coulomb Functions
  • Normalized Hydrogenic Bound States
  • Coulomb Wave Function Normalization Constant
• Coupling Coefficients
• Dawson Function
• Debye Functions
• Dilogarithm
• Elliptic Integrals
  • Definition of Legendre Forms
  • Definition of Carlson Forms
  • Legendre Form of Complete Elliptic Integrals
  • Legendre Form of Incomplete Elliptic Integrals
  • Carlson Forms
• Error Functions
  • Probability functions
• Exponential Integrals
  • Exponential Integral
  • Ei(x)
  • Hyperbolic Integrals
  • Ei_3(x)
  • Trigonometric Integrals
  • Arctangent Integral
• Fermi-Dirac Function
  • Complete Fermi-Dirac Integrals
  • Incomplete Fermi-Dirac Integrals
• Gamma and Beta Functions
  • Gamma Functions
  • Pochhammer Symbol
  • Incomplete Gamma Functions
  • Beta Functions
  • Incomplete Beta Function
• Gegenbauer Functions
• Hypergeometric Functions
• Laguerre Functions
• Lambert W Functions
• Legendre Functions and Spherical Harmonics
  • Legendre Polynomials
  • Associated Legendre Polynomials and Spherical Harmonics
  • Conical Functions
  • Radial Functions for Hyperbolic Space
• Logarithm and Related Functions
• Mathieu Functions
  • Angular Mathieu Functions
  • Radial Mathieu Functions
• Power Function
• Psi (Digamma) Function
  • Digamma Function
  • Trigamma Function
  • Polygamma Function
• Synchrotron Functions
• Transport Functions
• Zeta Functions
  • Riemann Zeta Function
  • Riemann Zeta Function Minus One
  • Hurwitz Zeta Function
  • Eta Function
• References and Further Reading

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Elementary Functions

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Airy Functions and Derivatives