Special functions Definition and 20 Threads

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...

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...

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...

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...

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...

[url="https://www.physicsforums.com/insights/the-orin-fractional-calculus/"]Continue reading...

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...

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...

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

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.

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...

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"...

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)...

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...

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...

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...

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...

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...

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'...

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: