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Questions tagged [special-functions]

Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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1answer
47 views

In Mathematics is there a discrete logarithm function?

I find it difficult to understand this part in this book. Because, as far as I know, there is no unique function or formula for discrete logarithms. I cann't understand what this formula does. Is ...
3
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0answers
89 views

∫ sin(sin(x)) dx as a “suitably generalized Kampé de Fériet hypergeometric function of two variables”

I've reviewed the general theory at How can you prove that a function has no closed form integral? and the links therein. But they do not give me sufficient traction to show the impossibility of ...
4
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3answers
56 views

Prove $\text{Li}_2(e^{-2 i x})+\text{Li}_2(e^{2 i x})=\frac{1}{3} (6 x^2-6 \pi x+\pi ^2)$ when $0<x<\pi$

This is an identity I deduced when playing with the initial-boundary value problem of heat conduction equation asked here. It's easy to verify numerically with Mathematica: ...
2
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1answer
43 views

Special functions for non-elementry antiderivatives.

A lot of functions don't have elementary antiderivatives. Quite often I observed new special functions were defined that many of them are written in form of these special functions. Example: the ...
1
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2answers
28 views

Prove or Disprove: Summation of two functions (at least one discontinuous) supports IVP, if both of them support IVP.

Let, f and g be two functions on R, support IVP and at least one them is discontinuous. Then prove or disprove ( with example) whether f+g also supports IVP. If f and g, both are continuous, then it ...
3
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1answer
78 views

Did I solve the integral of $\ln (\sec x + \tan x)$ correctly?

I attempted this integral, and I got a wildly different answer from wolframalpha, so I couldn't get a proper confirmation whether or not the answer I got is correct. To simplify this integral, first ...
5
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4answers
86 views

Asymptotic approximation regarding the Gamma function $\Gamma$.

On the wikipedia page for Gamma function I saw an interesting formula $$ \lim_{n\to \infty} \frac{\Gamma(n+\alpha)}{\Gamma(n)n^\alpha} = 1 $$ for all $\alpha\in\Bbb C$. I couldn't find the source of ...
0
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0answers
31 views

Existence of smooth function which has compact support

Assume we have two sets $V,W$, for which $V \subset \subset W \subset \subset \Omega \subset \mathbb{R}^n$ holds. Now we want to find a smooth function $\psi$, such that \begin{equation*} \begin{...
0
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0answers
21 views

What hypergeometric transformation rules might I apply to try to simplify a certain expression?

I have (https://mathematica.stackexchange.com/questions/189538/sum-a-certain-hypergeometric-function-based-expression-pertaining-to-an-integrat) a Mathematica expression involving the following six (...
1
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0answers
61 views

using zeta function in the regularization of functional trace and determinant

I'm looking for books or lectures concerning Zeta function regularization. In particular, I'm interested in using zeta function in the regularization of functional trace and determinant . To be more ...
2
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0answers
39 views

Integral involving a Gaussian hypergeometric function and a rational function

Let $x_0,x \in (0,1/10)$ and define: \begin{equation} g(x):= F_{2,1}\left[\frac{1}{13},\frac{1}{17},\frac{1}{5}; 100 x^2 \right] \end{equation} Then the following identity holds true: \begin{eqnarray} ...
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0answers
13 views

Identities involving the Gaussian hypergeometric function

By applying the algorithm from Solving linear ordinary, 2nd order differential equations via global integral bases. to the five parameter family of ODEs defined in my first answer to Gauge ...
0
votes
1answer
42 views

Is $f(x;a)=\sum_{n=0}^{\infty}\cfrac{n!(1-x)^n}{(1-a)(2-a)\cdots(n-a)}$ some kind of special function?

In deriving the solution to a differential equation I arrived at the following series expression: $f(x;a)=\sum_{n=0}^{\infty}\cfrac{n!(1-x)^n}{(1-a)(2-a)\cdots(n-a)};\quad 0\le x \le 1$ where $a$ is ...
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0answers
29 views

Does there exist a function such that derivatives along X and Y axis are given but the derivative is not necessarily continuous?

Does there exist a function such that derivatives along X and Y axis are given but the derivative is not necessarily continuous ?
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0answers
13 views

How one can define inverse polylog function?

I am not a matematician therefore word "define" can be wrong. I know that polylog function can be defined as series for argument $|z|<1$ and analytically continued for $|z|\geq 1$. My question: ...
0
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1answer
41 views

How to find integration of function, in form of hypergeometric function, given below?

I would like to prove the left side to right hand side which is in form of hypergeometeric function. Looking for your hints, suggestions and solultions. $$ \alpha_{1} \int_{0}^{1} (1-z)^{\alpha_{1}+\...
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0answers
28 views

Solving linear ordinary, 2nd order differential equations via global integral bases.

Consider a linear, homogenous 2nd order ODE: \begin{equation} L\left[y(x)\right] = \left[\frac{d^2}{d x^2} + a_1(x) \frac{d}{d x} + a_0(x)\right] y(x)=0 \end{equation} In https://arxiv.org/pdf/1606....
1
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0answers
21 views

special functions defined as a solution to a trig equation with a variable parameter

Warning: This is kind of an open-ended question. Often I have seen people interested in solving equations like $f(t)=g(t)$, where $f(t)$ is made up of trig functions and $g(t)$ is not. I was ...
3
votes
1answer
61 views

Solution to the parabolic cylinder equation

In the Gradshteyn & Ryzhik (7th ed.) the differential equation (9.255) leading to parabolic cylinder functions is $$\frac{d^2u}{dz^2}+(p+\frac{1}{2}-\frac{z^2}{4})u=0.$$ The solutions are $u=D_p(z)...
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0answers
25 views

Show that if $|f(x)| \leq \phi(x) + \psi (x)$, there exist $g,h$ such that $f(x) = g(x)+h(x),\, |g(x)| \leq \phi (x), \, \, \, |h(x)| \leq \psi(x)$ [duplicate]

Given $\phi$ and $\psi$ two seminorms in a vector space X, and a functional $f:X \rightarrow \mathbb{K}$, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$, such that $|f(x)| \leq \phi(x) + \psi (x) \, \...
1
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0answers
59 views

Functions whose input is the same as the output?

Given the Dedekind eta function $\eta(\tau)$ and complex number $\tau$. I came across these family of functions, $${f_2(\tau)= \frac{i}{\sqrt{2}}\frac{\,_2F_1\left(\tfrac14,\tfrac34,1,\,1-\alpha_2\...
3
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2answers
77 views

Show that a complex function differentiable in all $\mathbb C$ is constant if it is bounded.

Given $X$ a complex normed space, and $f: \mathbb{C} \rightarrow X$ a function that is differentiable at all the points of $\mathbb{C}$. Show that if $f$ is bounded, then $f$ is constant. Any idea ...
1
vote
1answer
69 views

Bernoulli Polynomial BesselJ expansion

I have been reading a paper on classes of polynomials and it gives the following series: $$J_{\nu }(x)=\sum _{n=0}^{\infty } \frac{\left(x^{\nu } B_n\left(x^2\right)\right) \left(\frac{(-1)^{n+2} 2^{-\...
0
votes
1answer
57 views

what is integral of this function?

Let $f(x)$ be an arbitrary continuous function, $n\in \mathbb{N}$ and $$g(x) = \frac{1}{1+n\cdot f(x)^2}$$ then what is anti-derivative of this: $$ \int \left(\frac{d}{dx}g(x)\right)\cdot\tanh\left(n\...
2
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0answers
41 views

learning to solve inhomogeneous versions of Legendre, Hermite, Laguerre etc.

Does anyone know of any good links (or even books though preferably not as there's a chance I won't be able to acquire them ) which go through worked examples of solving inhomogeneous versions of the ...
1
vote
1answer
55 views

How to solve the differential equation $(1-x^2)y''-2xy'=\sum_{n=1}^{\infty}P_n(x)$

So I was reading back over my course problem sheet for differential equations ( I'm studying for exams right now) and I came across this question: Given that $$\tfrac{2}{(5-4x)^{1/2}}=\sum_{n=0}^{\...
2
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1answer
132 views

What is anti-derivative of this function? [closed]

Let $f(x)$ be an arbitrary continuous function, $n\in \mathbb{N}$ and $$g(x) = \frac{1}{1+n\cdot f(x)^2}$$ then what is anti-derivative of this: $$ \int \left(\frac{d}{dx}g(x)\right)\cdot\tanh\left(n\...
0
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1answer
30 views

Complex zeroes of Error Function and Parabolic Cylinder Function.

Does there exist EXACT zeros of Error Function (Erfc(z)) and Parabolic Cylinder Function ($D_v(z)$)(http://functions.wolfram.com/HypergeometricFunctions/ParabolicCylinderD/). Here (https://dlmf.nist....
0
votes
1answer
40 views

Sign of the Hankel representation of the Gamma function

I have a question about the Hankel path representation of the Gamma function. The path of integration is displayed on the imege. The branch cut is taken as the negative real axis. $$ \Gamma(z) = \...
1
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0answers
19 views

bounds for hypergeometric 2F2 function

Looking for lower- and upper- bounds for the following case of the hypergeometric ${}_pF_q$ function: $$ {}_2F_{2}(1,k;\;k+1/2+b,k+1/2-b;\;z), $$ where $k\in\mathbb{N}$, $b\in[0,1/2)$, and $z\ge0$. ...
0
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1answer
19 views

Indefinite Sum Extension of a Finite Sum Equality

The other night I was considering the way in which we can split a finite sum of any arithmetic function into two finite sums, one for it's odd and another for even index terms : $$\sum _{k=1}^{n} \...
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0answers
56 views

An integral leads to complementary error functions

I am reading a paper Albrecher, Constantinescu and Loisel "2011Explicit ruin formulas for models with dependence among risks" and getting stuck at one integral (Example 2.4): $$\int _{\frac{\lambda }{...
13
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2answers
179 views

Functions with $\mathrm s(x)^n+ \mathrm c(x)^n \equiv 1$

I came across a very nice STEP question - Q8, STEP 1, 2018. It assumed the existence of function $\mathrm s(x)$ and $\mathrm c(x)$ with the properties that $\mathrm s(0)=0$, $\mathrm c(0)=1$, $\...
3
votes
2answers
133 views

Calculate $\int_{-\infty}^\infty \frac{e^{-x^2}}{x^2+a^2}\ dx$. [duplicate]

Let $$F(a)=\int_{-\infty}^\infty \frac{e^{-x^2}}{x^2+a^2}\ dx, \quad a>0.$$ Is it possible to relate $F(a)$ to some known (special) functions?
0
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1answer
17 views

A proving question based on greatest integer function.

If $$[x]=[\frac{x}{2}]+[\frac{x+1}{2}] $$b , where [°] denotes the greatest integer function and $n$ be a positive integer, then show that $$[\frac{n+1}{2}]+[\frac{n+2}{4}]+[\frac{n+4}{8}]+[\frac{n+8}...
2
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1answer
46 views

Is the hypergeometric function $_2F_1\left[\frac{1}{2},n+\frac{1}{2};n+1;z\right]$ expressible in terms of more elementary functions?

Is this special case of hypergeometric function expressible in terms of more elementary functions : $$_2F_1\left[\frac{1}{2},n+\frac{1}{2};n+1;z\right]$$ It will also be helpful for me to know, if ...
2
votes
1answer
67 views

How to derive the closed form of this gamma quotient?

Sorry for this very localized question. I am aware that there are myriads of algebraic gamma quotients of this kind around, but in fact I have never seen proofs involving more than the well-known ...
4
votes
1answer
81 views

Compound Binomial-Exponential: Closed form for the PDF?

Setup: Consider the random variable $Y_N$ derived from $$Y_N = \sum_{i=0}^N X_i$$ where $N$ is a random variable with distribution $p_n = {M \choose n} p^n q^{M-n}$ (binomial i.e. $M$ Bernoulli ...
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0answers
39 views

On the Zeros of the Complex Error Function

I'm working on something that involves the error function, except with complex inputs, i.e. erf(z) with $$z\in\mathbb{C}$$. In particular, at this point, I need a formula for the zeros of this ...
14
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2answers
199 views

Closed form of an improper integral to solve the period of a dynamical system

This improper integral comes from a problem of periodic orbit. The integral evaluates one half of the period. In a special case, the integral is $$I=\int_{r_1}^{r_2}\frac{dr}{r\sqrt{\Phi^2(r,r_1)-1}}$...
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0answers
78 views

How to integrate $\int_0^{+\infty} \frac{x^2\ln x}{x^4-x^3+1}\,\mathrm{d}x$?

First I think it may use the method of contour integration,but I have no idea how to create a contour.And is there some normal method to solve it? $$\int_0^{+\infty} \frac{x^2\ln x}{x^4-x^3+1}\,\...
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0answers
42 views

Sum of hypergeometric function

I am trying to evaluate the following sum $$\sum_{n=1}^N {}_2F_1(-n,n-N,1,x) y^n $$ I notice that according to wolfram alpha, $$\sum_{n=1}^\infty {}_2F_1(-n,b,c,x) y^n = \frac{_2F_1(1,b,c,\frac{...
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1answer
35 views

Confusion regarding Kelvin functions

I am trying to implement the following equation from this paper and having some troubles in the interpretation of $bei'$ and $ber'$. I understand from the definition of Kelvin functions that for ...
6
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3answers
115 views

prove $\sum_{n=0}^{\infty}\frac{\Gamma^2(n+1)}{\Gamma(2n+2)}=\frac{2\pi}{3^{3/2}}$

I am seeking alternate proofs for $$\sum_{n\geq0}\frac{\Gamma^2(n+1)}{\Gamma(2n+2)}=\frac{2\pi}{3^{3/2}}$$ Here's mine: Recall that, for $x\in(0,2)$, $$\frac1x=\sum_{n\geq0}(1-x)^n$$ Hence we have ...
7
votes
5answers
111 views

On $\int_0^{2\pi}e^{\cos2x}\cos(\sin2x)\ \mathrm{d}x=2\pi$

Here's my attempt at an integral I found on this site. $$\int_0^{2\pi}e^{\cos2x}\cos(\sin2x)\ \mathrm{d}x=2\pi$$ I'm not asking for a proof, I just want to know where I messed up Recall that, for all ...
0
votes
1answer
49 views

Proof that β-function ∈ C^∞

I need to find correct proof that β-function is smooth on its domain. Is there some feature of such functions, I guess that we need to prove the continuity of all n-derivatives, or their partial ...
1
vote
1answer
36 views

Find a function $f_m $ that enumerates all subsets of $\{1,..,n\}$ of cardinality $m$ in ascending order

I'm Looking for a function $f_m: \{1,..,n\} \to \{(M\subseteq\{1,...,n\}: |M|=m\} $ that enumerates all subsets of $\{1,..,n\}$ of cardinality $m$ in ascending order. For example, for the set $\{1,...
2
votes
0answers
56 views

Do you recognize this infinite series? $\sum_{n=0}^\infty \frac 1{(1+an)^c} \frac{b^n}{n!} $

I came by this infinite series $$\sum_{n=0}^\infty \frac 1{(1+an)^c} \frac{b^n}{n!} $$ Is there some special function that can have this form? $c$ can be assumed to be a positive integer. While $...
1
vote
1answer
34 views

Evaluating $\int_{-\infty}^{\infty} xf(x)\delta(x-a) dx$.

Evaluate $$\int_{-\infty}^{\infty} xf(x)\delta(x-a) dx.$$ I suspect that I could create another function, let's say $g(x)=xf(x)$ and perform the integral which would just give $g(a)=af(a)$ but I'...
1
vote
1answer
59 views

Limiting value of a contour integral

Fix an integer $r \geq 1$, and a complex real $u$. (edit: I think $u$ is meant to be real) For an integer $k \geq 1$ define $$ a_k = \int_{ |z| = (2k+1)\pi } \frac{z \exp( u z)}{(e^z - 1)z^{2r+1}}dz. ...