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

Famous quotes containing the words special and/or cases:

    “Myths, as compared with folk tales, are usually in a special category of seriousness: they are believed to have “really happened,” or to have some exceptional significance in explaining certain features of life, such as ritual. Again, whereas folk tales simply interchange motifs and develop variants, myths show an odd tendency to stick together and build up bigger structures. We have creation myths, fall and flood myths, metamorphose and dying-god myths.”
    —Northrop Frye (1912–1991)

    “The world men inhabit ... is rather bleak. It is a world full of doubt and confusion, where vulnerability must be hidden, not shared; where competition, not co-operation, is the order of the day; where men sacrifice the possibility of knowing their own children and sharing in their upbringing, for the sake of a job they may have chosen by chance, which may not suit them and which in many cases dominates their lives to the exclusion of much else.”
    —Anna Ford (b. 1943)