Confluent Hypergeometric Function - Special Cases

Special Cases

Functions that can be expressed as special cases of the confluent hypergeometric function include:

  • Bateman's function
  • Bessel functions and many related functions such as Airy functions, Kelvin functions, Hankel functions.

For example, the special case the function reduces to a Bessel function:

\begin{align}\, _1F_1(a,2a,x)&= e^{\frac x 2}\, _0F_1 (; a+\tfrac{1}{2}; \tfrac{1}{16}x^2) \\ &= e^{\frac x 2} \left(\tfrac{1}{4}x\right)^{\tfrac{1}{2}-a} \Gamma\left(a+\tfrac{1}{2}\right) I_{a-\frac 1 2}\left(\tfrac{1}{2}x\right).\end{align}

This identity is sometimes also referred to as Kummer's second transformation. Similarly

where is related to Bessel polynomial for integer .

  • The error function can be expressed as
\mathrm{erf}(x)= \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt= \frac{2x}{\sqrt{\pi}}\,_1F_1\left(\frac{1}{2},\frac{3}{2},-x^2\right).
  • Coulomb wave function
  • Cunningham functions
  • Elementary functions such as sin, cos, exp. For example,
  • Exponential integral and related functions such as the sine integral, logarithmic integral
  • Hermite polynomials
  • Incomplete gamma function
  • Laguerre polynomials
  • Parabolic cylinder function (or Weber function)
  • Poisson–Charlier function
  • Toronto functions
  • Whittaker functions Mκ,μ(z), Wκ,μ(z) are solutions of Whittaker's equation that can be expressed in terms of Kummer functions M and U by
  • The general p-th raw moment (p not necessarily an integer) can be expressed as
(the function's second branch cut can be chosen by multiplying with ).

Read more about this topic:  Confluent Hypergeometric Function

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