What is Special functions: Definition and 20 Discussions

Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined by consensus, and thus lacks a general formal definition, but the List of mathematical functions contains functions that are commonly accepted as special.

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  1. benorin

    I Looking for Someone Strong in Analysis

    Are you strong in analysis? I need a favor please? My go to Analysis advisor (fresh_42) is unavailable for now, I was hoping one of you might help me please: This (it's the top sticky post in this sub-forum) is my insight article I've recently re-written and have shortened quite a bit. You...
  2. Tapias5000

    Integral ## \int _{ }^{ }\frac{1}{\sqrt{x^3+1}}dx ##

    I also don't understand how to get the descending factorials for this hypergeometric series, I also know that there is another way to write it with gamma functions, but in any case how am I supposed to do this? If I write it as a general term, wolfram will give me the result which leaves me...
  3. Fred Wright

    A What is the unique special function in this integral problem?

    While studying the solution to a integral problem I found online I ran across a special function I am unfamiliar with. The integral is $$ \int_0^{\infty}\frac{t^{\frac{m+1}{n}-1}}{1+t}dt=\mathcal{B}(\frac{m+1}{n},1-\frac{m+1}{n}) $$ This certainly isn't the normal beta function. What is it...
  4. MathematicalPhysicist

    Mathematica Series expansion from the red book on special functions by Richard Ask

    I want to check my calculations via mathematica. In the book I am reading there's this expansion: $$\frac{(1+\frac{1}{j})^x}{1+x/j}=1+\frac{x(x-1)}{2j^2}+\mathcal{O}(1/j^3)$$ though I get instead of the term ##\frac{x(x-1)}{2j^2}## in the rhs the term: ##-\frac{x(x+1)}{2j^2}##. So I want to...
  5. yucheng

    B Significant figures for special functions (square roots)

    I am using square roots, however, I am confused over how many significant figures (s.f.) to keep. Suppose I have ##\sqrt{3.0}##, which has 2 s.f. From three different sources, I'll put a summary in brackets: https://www.kpu.ca/sites/default/files/downloads/signfig.pdf (if 2 s.f. in the data...
  6. benorin

    Insights A Path to Fractional Integral Representations of Some Special Functions

    Introduction This bit is what new thing you can learn reading this:) As for original content, I only have hope that the method of using the sets $$C_N^n: = \left\{ { \vec x \in {\mathbb{R}^n}|{x_i} \ge 0\forall i,\sum\limits_{k = 1}^n {x_k^{2N}} < n - 1 } \right\}$$ and Dirichlet integrals to...
  7. Mr Davis 97

    I Special Functions: Complete Answers?

    I have a relatively light question about special functions. As an example, it can be shown that ##\displaystyle \int_0^{\frac{\pi}{2}} \sqrt{\sin x} ~ dx = \frac{\sqrt{\pi} ~\Gamma (\frac{3}{4})}{2 \Gamma (\frac{5}{4})}##. Generally, the expression on the right would be taken as "the answer" to...
  8. F

    I How to Understand the Primordial Power Spectrum in TASI Lectures on Inflation?

    I have been reading the TASI Lectures on Inflation by William Kinney, (https://arxiv.org/pdf/0902.1529v2.pdf). I came across the mode function eq (128) (which obeys a generalization of the Klein-Gordon equation to an expanding spacetime), as I read through until eq (163), I know that it is the...
  9. Houeto

    Is this ODE a Bernoulli Equation? Exploring Solutions with Substitution

    consider ODE : Show that the solution to this ODE is: Can someone tell what kind of ODE is it?I thought,it's on the form of Bernoulli ODE with P(x)=0.Is it possible to still solve it by using Bernoulli Methodology?I mean by substituting u=y^1-a with a=2? Thanks
  10. Emmanuel_Euler

    Integration and special functions.

    what is the relationship between special functions and integration ? why integral of some function like (sqrt(ln(x)) and (cos(1/x) and more) are entering us to special functions?? PLEASE HELP ME TO UNDERSTAND.
  11. A

    MHB Special Funcs: Evaluating $$\int \frac{1}{\sqrt{x}\ln(x)}$$

    Hi, Recently, I had stumbled across: $$\int \frac{1}{\sqrt{x}\ln(x)}$$ Let $f(x) = \frac{1}{\sqrt{x}\ln(x)}$ I noticed there is no elementary antiderivative. I want to evaluate this using special functions, but as of right now, I would like some advice as I have no clue about special...
  12. J

    Name of some special functions

    I'd like of know if the following functions have name: Y(σ), H(σ) X(σ), Y(iω), X(iω), Y(s), X(s). PS, I suppose that H(σ) must be the "exponential response"...
  13. M

    C/C++ C++ implementation of Longman's method, and special functions

    Hi all, I'm a programming newbie teaching myself C++ mainly for interest/ because I might want a real job after my physics PhD, but I have a problem in my research some code might be useful for. I have some functions defined by integrals of the form $$A(q)=\int db~ b* J_{0}(b*q)...
  14. N

    How Can I Integrate Special Functions Like x^x(ln x + 1)dx?

    This is a problem that came to me when i was doing implicit differentiation and i got curious as to how to integrate a problem like this. I was fascinated by the simplicity if an equation would have a complex integration problem. Homework Statement ∫x^x(ln x + 1)dx, Question 1 ∫x^x dx...
  15. D

    Beta - special functions - manipulation

    Homework Statement I have this incomplete Beta function question I need to solve using the Beta function. \int^{a}_{0}y^{4}\sqrt{a^{2}-y^{2}}dy Homework Equations Is there an obvious substitution which will help convert to a variant of Beta? Beta function and variants are in Beta_function...
  16. M

    Mathematica Mathematica: numerical non-evaluation of special functions

    Hi all, I've been getting Mathematica to do some integrals for me, which are typically returning sums of Meijer-G functions. When I try and obtain numerical values for these sums, some of my results have contained terms which Mathematica has refused to evaluate numerically; an example is...
  17. K

    How can I learn special functions and differential equation

    Legendre Functions, Spherical Harmonic Functions, Bessel Functions, Neumann Functions, Airy Functions, Confluent Hypergeometric Functions, Laguerre Functions, Hermitte Functions... I find this learning is so tedious, traumatic, and miserable. I find it so difficult to manage. But I have to...
  18. T

    Special Functions: Valid Gauss Formula Parameters

    Homework Statement It is known that Euler's integral representation http://img12.imageshack.us/img12/5578/euler.png is valid for Re(c)>Re(b)>0 and |z|<1. The series (Gauss Formula) [PLAIN][PLAIN]http://img830.imageshack.us/img830/2365/gaussz.png on the other hand converges for...
  19. Astronuc

    Special Functions and Polynomials

    PF Member Careful pointed to the website of Gerardus 't Hooft, Dutch physicist and winner of 1999 Nobel Prize in Physics with Martinus J.G. Veltman. 't Hooft has a very interesting and useful website, which includes the following useful pdf file about 'Special Functions and Polynomials'...
  20. J

    Special functions and application to physics

    can anyone help me in solving my doubt that what is the application of special functions and Hermite,Legenders,Laguerre function to the various branches of physics. could u please specify any link or site adress. thank you :mad:
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