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Gauss Hypergeometric Function: Simple Definition

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What is the Gauss Hypergeometric Function?

The Gauss hypergeometric function (sometimes called the Euler-Gauss hypergeometric function or “the” hypergeometric equation) is a special function defined by the series:
Gauss Hypergeometric Function
Where:

  • F (a, b; c; z) is the hypergeometric function.
  • a, b, and c are parameters (reals),
  • z is a variable in the complex plane.

As the function is defined by a series, it is sometimes called the hypergeometric series. A similar function, the confluent hypergeometric function, is denoted with F (a; b; z).

The function can also be represented by integrals:
integral representation

Calculation

Calculation for anything but the simplest of hypergeometric functions is difficult, especially when all the parameters are complex. This is, in part, because of issues with cancellation and round off error.


What is the Gauss Hypergeometric Function Used For?

The function is an analytic function that generates more complex combinatorial numbers, which generalize the binomial series. The function provides information about the relationships between combinatorial numbers and their growth.

Many special functions can be expressed in terms of the hypergeometric. Closed form solutions of linear differential equations with polynomial coefficients can many times be defined in terms of Gauss’s hypergeometric function.

The Gauss Hypergeometric Function function arises in many practical areas, like:

  • Binary stars (where two stars orbiting around a common center of mass),
  • Mathematical finance (an interdisciplinary study of financial markets),
  • Photon scattering from atoms,
  • Networks,
  • Non-Newtonian fluids (a fluid with variable viscosity is depending on applied stress or force).

References

Aomoto, K. & Kita, M. (2011). Theory of Hypergeometric Functions. Springer Science and Business Media.
Erdelyi, A. Ed. (1955). Higher Transcendental Functions. McGraw-Hill.
Pearson, J. et al. (2017) Numerical methods for the computation of the confluent and Gauss hypergeometric functions. Numer Algor (2017) 74:821–866
Seaborn. Hypergeometric Functions and Their Applications. Retrieved November 26, 2019 from: https://books.google.com/books?id=HJXkBwAAQBAJ

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Stephanie Glen. "Gauss Hypergeometric Function: Simple Definition" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/gauss-hypergeometric-function/
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