What is a special function? - Mathematics Stack Exchange
In particular, elementary functions are also considered as special functions." Also, looking at the Wikipedia list (linked above) the indicator function, step functions, the absolute value function and the sign are special functions -- this sounds very wrong to me. So what is a special function and what should be under the (special-function) tag?
What are special functions for? - Mathematics Stack Exchange
That often occuring, simply stated relations, functions, series, basic for most of our operations (and may be: even of our thinking, mentally modeling) may be called "elementary functions". And functions, which are needed in less general and only specific context or depth, and possibly need much explanation, may then be called "special functions" .
Reference Book on Special Functions - Mathematics Stack Exchange
2014年6月24日 · Now I'm studying the topic that uses the special functions frequently, so I find myself in need for some good reference book on the properties and equalities of the special functions. The optimal one might be the book with dictionary-like or handbook type lists of the properties of the special functions with sufficient amount of proofs.
ordinary differential equations - Why Special Functions are called ...
2019年7月15日 · "Special" is to be seen as being opposed to "general", in the sense of "select". "General functions" refers to arbitrary functions - in the case of real number functions defined on $\mathbb{R}$ or a subset thereof, arbitrary functions comprise an enormous set with size $\beth_2$ (equiv. the size of the power set of the real numbers, or the number of subsets of …
Newest 'special-functions' Questions - Mathematics Stack Exchange
Questions tagged [special-functions] Ask Question This tag is for questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).
Book recommendation on special functions - Mathematics Stack …
2016年2月4日 · I am currently studying real analysis from rudin and really like the chapter on special functions. But Rudin does not give much knowledge about those topics. Reading the references I found book by Emil artin on The Gamma function and really liked it. Are there more books like Emil artin's gamma function on special functions.
Solving Special Function Equations Using Lie Symmetries
The Lie group and representation theory approach to special functions, and how they solve the ODEs arising in physics is absolutely amazing. I've given an example of its power below on Bessel's equation. Kaufman's article describes algebraic methods for dealing with Hermite, Legendre & associated Legendre functions. Can we take the other ...
special functions - An explanation of spherical harmonics ...
2012年4月17日 · Consider Laplace's equation in three dimensional space, $$\nabla^2 V({\bf r}) = 0.$$ Such a function is called harmonic. Harmonic functions describe a multitude of physical objects, typically called potentials. There are gravitational, electric, and …
special functions - Closed-form solution to the transcendental …
2023年2月28日 · Expanding $\P_{2n+1}(y)$ as a series and then as a binomial series will again give incomplete beta functions from the remaining integral. There should be a simpler solution than an infinite series and 2 finite series maybe through complex integrals or with Teixeira’s theorem etc, but I have little experience with contour integrals.
Summation and integral representations of special functions
2020年5月20日 · Your question : << Is it reasonable to seek out possible routes for getting special functions defined by such representations such that their properties can be found... ? >>. This is a big question which is far to be fully solved. In fact this question cannot be raised without to specify the list of special functions to be considered in the search.