Special Functions: Simple Definition

Types of Functions >

Special functions are well-known, classical functions. Many are named after the person who first studied it. For example, the Bessel Function is named after the astronomer Friedrich Bessel (1784-1846). Most of these functions have been used for centuries, although several dozen have been formulated in recent years.

Paul Turan (as cited in Andrews et. al, 1999) once said that a more suitable name for these types of functions is “useful functions.” You can also think of them as commonly used functions, or elementary functions—ones that are building blocks for other function types.

What Functions are “Special”?

In order for a function to be defined as “special,” it also has to be useful in a practical sense, especially in finding solutions for differential equations. For example, Bessel functions, which deal with circular or spherical symmetry, are useful for studying heat flow. In order to grasp the use of these functions, you’ll usually have to study multivariable calculus and differential equations as prerequisites.

Common special functions (click the bold item for more information):

• Bessel functions: solutions to differential equations and popular in problems involving circular or cylindrical symmetry or wave propagation.
• Beta functions: definite integrals, related to the gamma function.
• Chi Function: a special case of the Lerch transcendent; Resembles the Dirichlet series for the polylogarithm.
• Error Function: Important in the study of errors. It
• Gamma functions: the multivariate form is used extensively in many sub-fields of multivariate analysis, including Bayesian Psychometric Modeling and Signal Processing,
• Legendre functions, which are solutions to Legendre equations— used extensively in physics.
• Trigonometric functions: functions of angles,
• Hypergeometric functions: building blocks for many other functions, including closed form solutions of linear differential equations with polynomial coefficients.

Perhaps surprisingly, special functions do not include those you’ll come across in the first semester or two of calculus, like power functions, logarithm function, and exponential functions. Although they are certainly “classical” functions that have been around for centuries, they aren’t very useful for solving physical problems. For that, you need more complex functions. Some authors do include the trig functions (cosine function, sine function, etc.) in the definition of special functions as a group (as opposed to individual functions, which are deemed as too simplistic on their own).

There are many more. In fact, dozens of books have been written about special functions (you can find two in the references below). For definitions of specific function types, see: Types of Functions. You can even take an entire university course on them.

What is the Chi Function?

The Chi Function (also called Legendre’s Chi Function) is a special function defined with the following formulas:


0 < a ≤ 1, Re ν > 1,

And



|z| ≤ 1, Re ν > 1 with ν = 2, 3, 4, …
Where “re” stands for Real.

The second formula is usually called the Legendre chi function of order n. The function’s Taylor series is also a Dirichlet series. It is a special case of the Lerch transcendent.

Relation to Polylogarithm

The chi function resembles the Dirichlet series for the polylogarithm. It is related to the polylogarithm function for integral ν = 2, 3 by:

History of the Chi Function

Adrien-Marie Legendre studied the chi function in 1811 (in Exercices de calcul integral) (1811) and used the letter phi(φ) instead of chi(χ) to denote it. The use of χ dates to Edward’s 1889 text A Treatise on the Integral Calculus.

References

Abramowitz, M. & Stegun, C. (Eds.). Sine and Cosine Integrals. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 9th edition.
Andrews, G. et al. (1999). Special Functions. Cambridge University Press.
Bell, W. (2013). Special Functions for Scientists and Engineers. Courier Corporation.
Cvijovic, D. and Klinowski, J. (1995). Closed-Form Summation of Some Trigonometric Series. Math. Comput. 64, 205-210.
Cvijovic, D. and Klinowski, J. (1999). Values of the Legendre Chi and Hurwitz Zeta Functions at Rational Arguments. Mathematics of Computation. Volume 68, Number 228, Pages 1623–1630. Retrieved November 21, 2019 from: https://www.ams.org/journals/mcom/1999-68-228/S0025-5718-99-01091-1/S0025-5718-99-01091-1.pdf
Cvijovic, D. (2006). Integral representations of the Legendre chi. J. Math. Anal. Appl. 332 (2007) 1056–1062
Lewin, L. (1981). Polylogarithms and Associated Functions. Elsevier Science Ltd.
Narcowish, F. (2005). Notes on…
Retrieved November 27, 2019 from: https://www.math.tamu.edu/~francis.narcowich/psfiles/special_fun.pdf

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