Calculus How To

Meijer G-function: Simple Definition

Share on

Types of Functions >

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

Where Γ = the gamma function.

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

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,


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:
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:

Stephanie Glen. "Meijer G-function: Simple Definition" From Calculus for the rest of us!

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!

Leave a Reply

Your email address will not be published. Required fields are marked *