The **Meijer G-Function** is a very general family of functions, usually defined by the following complex line integral:

Where Γ = the gamma function.

Each G-function is a linear combination of special functions. These include as special cases:

- Higher transcendental hypergeometric functions,
- All four Bessel functions and family members (e.g. Struve Functions).

In fact, most special functions and products of special functions are either G-functions or can be represented by products of G-functions with elementary functions.

The G-function can also be extended to reproduce a wide variety of smooth functions, including exponential functions, trigonometric functions and many others.

## Uses for the Meijer G-function

The Meijer G-function has greater flexibility for choices of parameters in comparison to other special functions, which makes it a popular choice for finding solutions to a wide variety of problems. In addition, some problematic integrals—which could previously only be represented numerically—can be evaluated in closed form with use of the G-function.

The Meijer G-Function has many practical applications to many areas, where the function can represent solutions for a wide variety of problems. Fields that use the Meijer G-function extensively include:

- Generalized birth and death processes,
- Mathematical physics (hydrodynamics, potential theory, theory of elasticity etc.),
- Mathematical statistics,
- Optimization theory,
- Queuing theory,
- Sinusoidal signals,
- Theoretical physics,

## References

Baricz et. al (2018). Series of Bessel and Kummer-Type Functions. Springer.

Beals, R. (2013). Meijer G-Functions: A Gentle Introduction. Notices of the American Mathematical Society. Retrieved December 1, 2019 from: https://www.academia.edu/26546005/Meijer_G-Functions_A_Gentle_Introduction

Kiryakova, V. (1993). Generalized Fractional Calculus and Applications. CRC Press.

Toth, V. (2007). Maple and Meijer’s G-function: a numerical instability and a cure. Retrieved December 1, 2019 from: vttoth.com/CMS/index.php/technical-notes/67

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