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

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0
votes
2answers
87 views

Evaluating an integral involving $(a-x)^ne^{-1/x}/x^2$

Let $a>0$ be a small parameter and consider a fixed integer $n\geq 0$. Is it true that $$ \int_0^a \frac{(a-x)^n e^{-1/x}}{x^2}\ dx=n! a^{2n}e^{-1/a}(1+O(a)) $$ as $a\to 0$? I have verified this ...
4
votes
2answers
96 views

Evaluating a certain integral which generalizes the ${_3F_2}$ hypergeometric function

Euler gave the following well-known integral representations for the Gauss hypergeometric function ${_2F_1}$ and the generalized hypergeometric function ${_3F_2}$: for ...
2
votes
2answers
68 views

Integration of Associated Legendre Polynomial

I am interested in the following integral $$I=\int_{-1}^1P_\ell^2(x)P_n(x)\mathrm{d}x,$$ where $P_n(x)$ is Legendre Polynomial of $n$th order, and $P_\ell^2$ is Associated Legendre Polynomial. Any one ...
9
votes
0answers
169 views

$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$

I have recently met with this integral: $$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$$ I want to evaluate it in a closed form, if possible. 1st functional equation: $\displaystyle f(a)=\ln^2 2-f\left ( ...
2
votes
1answer
61 views

Calculating $I=\int_{-1}^1{\dfrac{1}{\sqrt{1-x}}P_n(x)} \, dx$ where $P_n$ is a Legendre Polynomial.

Calculating $$I=\int_{-1}^1{\dfrac{1}{\sqrt{1-x}}P_n(x)} \, dx$$ Where $P_n$ is a Legendre Polynomial. My progress: For any integral of the form: $$\int_{-1}^1{f(x)P_n(x)} \, dx$$ Usinng Rodrigues ...
28
votes
1answer
722 views

Weber-type integral

In connection with this answer, I came across the following integral: $$\int_{0}^{\infty} \frac{du}{u} \: \,e^{-t u^2} \frac{J_0(u) Y_0(r u)-J_0(r u) Y_0(u)}{J_0^2(u)+Y_0^2(u)}$$ where $r \gt 1$. I ...
0
votes
0answers
50 views

Prove that this expression involving $_2 F_1$ and Gamma functions is identically zero

While attempting an answer to the question Evaluation of $\displaystyle \int_{1}^{3}\left[\sqrt{1+(x-1)^3}+(x^2-1)^{\frac{1}{3}}\right]dx$, after a few manipulations I came across the following ...
1
vote
1answer
56 views

Meijer G-function limit for $z\rightarrow\infty$

I am trying to understand if the integral $$f(R)=\int_a^R\frac{K_1(r)dr}{r}$$ has a finite limit for $R\rightarrow\infty$. With Wolfram Mathematica I got the following primitive: $$\frac{1}{4} ...
0
votes
2answers
1k views

Find intersection of linear and logarithmic lines

I have equations for two lines, one of which is linear and the other is logarithmic, ie: $$y = m_1 x + c_1$$ $$y = m_2 \cdot \ln(x) + c_2$$ ..and I need to find out where (if at all) these lines ...
1
vote
0answers
60 views

How To Prove The following equation?

The equation arised in the paper:Exact and asympototic representations of the sound field in a stratified ocean.That is the equation(3.12) for solving the problem $$\Delta ...
2
votes
1answer
70 views

A limit involving the Hurwitz zeta function

I want to show that $$ \lim_{s \to 1} \left( \zeta(s,a) - \frac{1}{s-1} \right) = - \psi(a)$$ where $\zeta(s,a)$ is the Hurwitz zeta function and $\psi(a)$ is the digamma function. The only ...
0
votes
1answer
79 views

What is the following expression equal to?

What is the following expression equal to? $$z^{\alpha } \left(\, _2F_1\left(1,-\alpha ;1-\alpha ;\frac{1}{z}\right)+\, _2F_1(1,\alpha ;\alpha +1;z)-1\right)$$ The derivative of it with respect to z ...
3
votes
2answers
389 views

Solve equation with lower gamma function: $A \gamma(2;x/B)=x$ for $x$

I need to find an expression for $x$ given: $A \gamma(2;x/B)=x$ where $\gamma(a,x)=\int\limits_0^x t^{a-1} e^{-t} \mathrm{d}t$ is the lower incomplete gamma function. $A$ and $B$ are real, positive ...
2
votes
1answer
109 views

Re-Expressing the Digamma

I was reading some articles on the digamma function, and I was wondering if anyone knows how to express the digamma function $\psi^{(0)}(n)$ in terms of a trigonometric function or a logarithmic ...
8
votes
4answers
283 views

Another beautiful arctan integral $\int_{1/2}^1 \frac{\arctan\left(\frac{1-x^2}{7 x^2+10x+7}\right)}{1-x^2} \, dx$

Do you think we can express the closed form of the integral below in a very nice and short way? As you already know, your opinions weighs much to me, so I need them! Calculate in closed-form ...
1
vote
1answer
29 views

gegenbauer polynomial

Usually, Gegenbuaer polynomial is denoted by $C^{(\lambda )}_{n}(x)$ with $\lambda >-1/2$. My question: is it possible to generalize Gegenbuaer polynomial for $Re(\lambda)>-1/2, \lambda \in ...
5
votes
2answers
614 views

Question Relating Gamma Function to Riemann Zeta function evaluated at integers

I was just reading a paper of Ramanujan entitled "On question 330 of Professor Sanjana" when I got confused regarding a proposition which I am unable to answer. The proposition is: $ \displaystyle ...
3
votes
0answers
45 views

Airy transform of gaussian on half-line: $\int_{0}^\infty dx\, e^{-x^2}\text{Ai}(y-x)$

Background. The Airy transform of $f$ is defined as $$\int_{-\infty}^\infty dx\, f(x)\,\text{Ai}(y-x)\;.$$ $\text{Ai}$ denotes Airy function, $$\text{Ai}(x)=\frac{1}{\pi}\int_{-\infty}^\infty ...
2
votes
1answer
115 views

Extract imaginary part of $\text{Li}_3\left(\frac{2}{3}-i \frac{2\sqrt{2}}{3}\right)$ in closed form

We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g. $\mathrm{Re}[\text{Li}_2(i)]=-\frac{\pi^2}{48}$ Is there a closed form (free of polylogs and ...
1
vote
0answers
22 views

Estimate lower incomplete gamma function $\left|\gamma(n,z)-\gamma(n,-z)\right|$.

From Wikipedia, the upper incomplete gamma function is defined as: $$\Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t ,\,\!$$ whereas the lower incomplete gamma function is defined as: ...
13
votes
3answers
207 views

Conjecture $\sum_{m=1}^\infty\frac{y_{n+1,m}y_{n,k}}{[y_{n+1,m}-y_{n,k}]^3}\overset{?}=\frac{n+1}{8}$, where $y_{n,k}=(\text{BesselJZero[n,k]})^2$

While solving a quantum mechanics problem using perturbation theory I encountered the following sum $$ S_{0,1}=\sum_{m=1}^\infty\frac{y_{1,m}y_{0,1}}{[y_{1,m}-y_{0,1}]^3}, $$ where ...
1
vote
0answers
45 views

What is the general theory of solving polynomial equations “beyond radicals”?

For example, using Bring radicals or elliptic functions to solve quintic equations. Wikipedia says that similar methods can be used for higher degree polynomials, but I'm struggling on finding ...
7
votes
0answers
108 views

Weierstrass product expression for Klein's j-invariant

The first sentence of @ccorn's answer to a previous question of mine was: “Because of the modular symmetries of $j(\tau)$, the zeros of $j(\tau)$ are precisely the ...
7
votes
0answers
248 views

How to solve the general sextic equation with Kampé de Fériet functions?

It is frequently stated, for example on Wolfram Mathworld, that the general sextic equation $$x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x^1 + a_0 = 0$$ can be solved in terms of Kampé de ...
2
votes
0answers
106 views

Limit with the Appell F1 function

While attempting to solve this problem I ran into a nasty limit. Mathematica claims that the indefinite integral ...
0
votes
0answers
14 views

Wigner 3j for Interchanged $m$

I am given with two wigner j coefficients $$\begin{pmatrix} \ell_1&\ell_2&\ell_3\\ m& 0& -m\end{pmatrix},$$ and $$\begin{pmatrix} \ell_1&\ell_2&\ell_3\\ -m& 0& ...
8
votes
1answer
224 views

Closed-forms of real parts of special value dilogarithm identities from inverse tangent integral function

The inverse tangent integral is defined as $$\operatorname{Ti}_2(x)=\Im\operatorname{Li}_2\left(ix\right)$$ Because this we have some special value identitiy. Let $c_1 = \operatorname{Li}_2(i)$, ...
0
votes
0answers
18 views

Reference for Fractional calculus and Differential Operators

I`ve been struggling with Fractional calculus and differential operators while studying special functions, and got to the conclusion that I need some references for them. So I ask for as many ...
2
votes
1answer
40 views

Does the Borel-transform of the Lerch-Transcendent have a name/simple expression?

The Lerch-transcendent as given in Mathworld is $$ \Phi(z,s,a)= \sum_{k=0}^\infty {z^k\over (a+k)^s}$$ I'm fiddling with series of the form $$ f_n(z)=\sum_{k=0}^\infty {z^k\over (1+k)^n} $$ and their ...
0
votes
0answers
19 views

How to compute this integrale $\int_{\mathbb R^3} e^{-i\left<x,y\right>} e^{-a\| x\|} \| x\|^{\frac{5}{2} } dx$

I would like to calculate the following integral $$I(a,y)=\int_{\mathbb R^3} e^{-i\left<x,y\right>} e^{-a\| x\|} \| x\|^{\frac{5}{2}} dx, \quad a>0, y\in \mathbb R^3 .$$ Here's what I did: In ...
0
votes
0answers
28 views

The inverse Laplace transform of $\Gamma\left(\zeta\right) \, W_{\zeta,\mu}(z) $

Someone has a reference that addresses an integral of the followns type $$I = \frac{1}{2i\pi} \int_{\sigma-i\infty}^{\sigma+i\infty} e^{t\zeta} \, \Gamma\left(\zeta\right) \, W_{\zeta,\mu}(z) \, ...
1
vote
0answers
28 views

Integration on complex spheres and Gamma function

I'm studying special functions, especially Jacobi functions, related to the rank one groups ($U(1, n; \mathbb{F})$ where $\mathbb{F}$ is $\mathbb{C}$ or $\mathbb{H}$, the skew-field of quaternions), ...
0
votes
0answers
32 views

Integral in terms of special function

I'm trying to express the integral given below in terms of special functions. $$\int_{1}^{\infty}\left(-1\right)^{n}x^{-1-a}\left[\ln\left(e^{\left(x-1\right)/b}-1\right)\right]^{n}\mathrm{d}x$$ I ...
15
votes
2answers
482 views

Integral $\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}$

$$ I:=\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}. $$ Thank you. The Gamma function is given by $\Gamma(n)=(n-1)!$ and its integral representation is $$ \Gamma(x)=\int_0^\infty t^{x-1} ...
2
votes
0answers
57 views

What is the series expansion of reciprocal of theta function $\frac{1}{\theta(z;q)}$?

"The" theta function is an ambiguous concept, but one definition I have found is: $$ \theta(z;q) = (z;q)_\infty(q/z;q)_\infty = \frac{1}{(q;q)_\infty}\sum_{k \in \mathbb{Z}}z^k q^{\binom{k}{2}} ...
0
votes
1answer
41 views

Does this limit involving the Dirichlet eta function and the Riemann zeta function make sense?

Let $p_n$ the sequence of prime numbers (and you will consider below, too, the sequence $\frac{1}{n}$ with $n>1$). And if it isn't wrong for $0<\Re s<1$ the known equation between Dirichlet ...
2
votes
1answer
36 views

Is the Lambert W function multivalued everywhere?

Is the Lambert W function multivalued everywhere except at $x=0$? It is obvious that $W(0)=0\implies 0=0e^0$ because $e^u\ne0$, therefore, it is the coefficient that determines such, and the only ...
2
votes
1answer
102 views

Finding $f(x)$ such that $\int_{a}^{b}f(x)dx=\sum_{k=a}^{b}f(k)$

Does there exist any method to find the function $f(x)$ which satisfies $$\int_{a}^{b}f(x)dx=\sum_{k=a}^{b}f(k)$$ For example $$\int_{- ...
1
vote
1answer
54 views

Express $\int_{\sin nx}^{\sin(n+1)x}\sin t^2dt$ in terms of $x$ and $n$

Please help me to express $$\int_{\sin nx}^{\sin(n+1)x}\sin t^2\,dt$$ in terms of $x$ and $n$. If it is not possible please help to establish bounds on the integral again in terms of $x$ and $n$. The ...
0
votes
0answers
45 views

How to prove the following function is independent of z?

I series expanded the following expression in Mathematica and the result is independent of z: $$(1-z)^{-a }\left(\, _2F_1(1,-a ;1-a ;1-z)+\, _2F_1\left(1,a ;a+1;\frac{1}{1-z}\right)-1\right)-(1-z)^{a ...
1
vote
1answer
50 views

Relation between hypergeometric functions?

Is there any relations between the following hypergeometric functions? $$\ _2F_1(1,-a,1-a,\frac{1}{1-z})$$ $$\ _2F_1(1,-a,1-a,{1-z})$$ $$\ _2F_1(1,a,1+a,\frac{1}{1-z})$$ $$\ _2F_1(1,a,1+a,{1-z})$$
22
votes
1answer
281 views

Definite integral of arcsine over square-root of quadratic

For $a,b\in\mathbb{R}\land0<a\le1\land0\le b$, define $\mathcal{I}{\left(a,b\right)}$ by the integral ...
0
votes
1answer
35 views

How to prove this HyperGeometric function identity?

After using FullSimplify in Mathematica, I got the left hand side of the following equation: $$(a-1)(z-1)\ _2F_1(1,1,1-a,\frac{1}{z})+a\ _2F_1(1,1,2-a,\frac{1}{z})-az+z=0$$ I series expanded it and ...
3
votes
2answers
131 views

The trigonometric solution to the solvable DeMoivre quintic?

Using the relations for the Rogers-Ramanujan cfrac described in this post, $$\frac{1}{r}-r = x$$ $$\frac{1}{r^5}-r^5 = y$$ and eliminating $r$ yields, $$x^5+5x^3+5x = y$$ This is the case $a=1$ ...
2
votes
2answers
402 views

What is the geometric, physical or other meaning of the tetration?

What is the geometric, physical or other meaning of the tetration or more high hyperoperations? Is it exists in general or it has only math concept?
1
vote
1answer
31 views

Show that $f(qz) = qz^2 f(z)$ where $f(z) = [\theta(z) \theta(1/z)]^{-1}$

Let $\theta(z) = (z;q)(q/z; q)$ where $(q;z) = \prod_{i=0}^\infty (1 - zq^i) $. Then let $f(z)$ be defined by: $$ f(z) = \frac{1}{\theta(z) \theta(1/z)} $$ Show that $\boxed{\color{#0033FF}{f(qz) = ...
2
votes
1answer
53 views

Which theorems in the Gamma Function are important? [closed]

I'm interested in the Special Functions especially the Gamma Function. I decided to write a Bachelor Thesis about it but I do not know what kind of theorem(s) in the Gamma Function that is (are) very ...
31
votes
1answer
489 views

Are elementary and generalized hypergeometric functions sufficient to express all algebraic numbers?

Are (integers) plus (elementary functions) plus (generalized hypergeometric functions) sufficient to represent any algebraic number? For example, the real algebraic number $\alpha\in(-1,0)$ ...
0
votes
0answers
30 views

Express $\cos^2\theta\cos\phi\sin\phi$ in Spherical Harmonics

I am looking for a form of $$\cos^2\theta\cos\phi\sin\phi=\sum_{lm}c_{lm}Y_l^{m}(\theta,\phi),$$ where $Y_{lm}$ is the spherical harmonics. The idea I believe would be to find ...
1
vote
1answer
44 views

Monomials in terms of Legendre polynomials

Is there a closed-form expression for a monomial $x^m$ in terms of a sum of Legendre polynomials $P_n(x)$? $$ x^m = \sum_n a_n P_n(x) $$ How can I determine the coefficients $a_n$ in general? ...