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

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6
votes
2answers
144 views

Another integral related to Fresnel integrals

How would we prove this result by real methods ? $$\int_0^{\infty } \frac{\sin \left(\pi x^2\right)}{x+2} \, dx=\frac{1}{4} \left(\pi-2 \pi C\left(2 \sqrt{2}\right)-2 \pi S\left(2 ...
0
votes
1answer
120 views

Improper integral $\int_{0}^{\infty}\frac{x^n}{x^{m+n+1}} \ dx=\frac{n! {(m-1)}!}{(m+n)!}.$

How can I prove that $$\int_{0}^{\infty}\frac{x^n}{x^{m+n+1}} \ dx=\frac{n! {(m-1)}!}{(m+n)!}\quad ?$$ I tried to do induction on $n$ and on $m$, separately, but I could only do the base case ($n=1$ ...
4
votes
1answer
113 views

Solving integral with spherical bessel functions

I would like to find if possible a solution (closed form) for the following integral: $$\frac{1}{2 \pi}\cdot\int\limits_0^{2\pi}\exp\bigg[-ia(\cos x+\sin x)\bigg]\,j_{0}(b\cos x)\,j_{0}(b\sin ...
25
votes
1answer
643 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 ...
6
votes
1answer
204 views

How to calculate this complementary Bessel function?

I am trying to calculate this complementary Bessel function $$\Psi(a,b,\gamma)=\int_0^\infty\Phi({a\over \sqrt{u}}+b\sqrt{u}){u^{\gamma-1}e^{-u}\over \Gamma(\gamma)}du$$ where $\Phi$ is the standard ...
2
votes
1answer
79 views

When are the binomial coefficients equal to a generalization involving the Gamma function?

Let $\Gamma$ be the Gamma function and abbreviate $x!:=\Gamma(x+1)$, $x>-1$. For $\alpha>0$ let us generalize the binomial coefficients in the following way: ...
3
votes
1answer
105 views

Hypergeometric function values and the Baxter constant

While I was working on this question by @Vladimir Reshetnikov, I've found the following relations between Gaussian hypergeometric function values and the Baxter constant: ...
0
votes
0answers
91 views

Riemann zeta function, functional equation, what completes this analogy?

What completes this analogy? This: $$\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s)\;\;\;\;\;\;\;\;\;\;(1)$$ is to: $$\chi(s)=\pi ^{-\frac{s}{2}} \Gamma ...
4
votes
2answers
63 views

Closed form of a series (dilogarithm)

We are all aware of the dilogarithm function (Spence's function): $$\sum_{n=1}^{\infty} \frac{x^n}{n^2}, \;\; x \in (-\infty, 1]$$ Also it is known that: $$\sum_{n=1}^{\infty} \frac{\cos n x}{n^2}= ...
6
votes
3answers
119 views

Closed-form of $\operatorname{Li}_2\left(1 \pm i\sqrt{3}\right)$

I've found the following identity while I was going through a quite difficult path. $$ \Re\operatorname{Li}_2\left(1 \pm i\sqrt{3}\right) = \frac{\pi^2}{24} -\frac{1}{2}\ln^2 2 - ...
2
votes
1answer
333 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 ...
3
votes
1answer
166 views

The ratio of jacobi theta functions

Let $q=e^{2\pi i\tau}$. If $\theta_2$ and $\theta_3$ are jacobi theta functions , is it true that the ratio of the two functions can be expressed as a continued fraction of the form $$ ...
1
vote
1answer
25 views

Question Regarding a Second Order Ordinary Differential Equation

I was wondering if the solution to the following differential equation belongs to a class of special functions. If not, is it exactly solvable? \begin{equation} ...
1
vote
1answer
14 views

Various forms of the Confluent Heun Equation

The Confluent Heun equation is expressed in various forms. It's non-symmetrical canonical form is: \begin{equation} ...
2
votes
2answers
162 views

A theta function around its natural boundary

Let $q = e^{2\pi i\tau}$, if $$\psi(q^2)=\sum_{n=0}^{\infty} q^{n(n+1)}$$ is one of ramanujan theta functions,is it possible to evaluate the limit $$\lim_{q\rightarrow 1} (1-q){\psi^2(q^2)}$$ In fact ...
3
votes
1answer
39 views

Prove or disprove that Weierstrass $\wp$ function is holonomic

Recall that a holonomic function $f$ (say over $\mathbb C$) is one that is a solution to a differential equation of the form: $$p_0(z)f(z)+p_1(z)f'(z)+p_2(z)f''(z)+\dots+p_k(z)f^{(k)}(z) = 0$$ where ...
-3
votes
0answers
28 views

clarification domain and range of functions [closed]

Want a clarification on the domains and ranges of addition, subtraction, multiplication, division and enhancement of functions, if possible with an example with roots and absolute value for a ...
4
votes
2answers
67 views

Does the elliptic function $\operatorname{cn}\left(\frac{2}{3}K\left(\frac{1}{2}\right)\big|\frac{1}{2}\right)$ have a closed form?

Given the complete elliptic integral of the first kind $K(k)$ for the modulus $k$, can the elliptic function $$\text{cn}\left(\frac{2}{3}K\left(\frac{1}{2}\right)\bigg|\frac{1}{2}\right)$$ be ...
4
votes
2answers
152 views

Help with C is Euler's constant and $\Gamma(0)=\infty$ in paper

I am referring to a paper by S. Nadarajah & S. Kotz. The notation is simple enough to understand, however i having trouble with $C$ is Euler’s constant and $\Gamma(0)=\infty$ by equation (2.3) I ...
4
votes
1answer
61 views

closed-form of an integral similar to Bessel function

The integral form of the $n$-th modified Bessel function of the first kind is $$ I_n(z)=\frac{1}{\pi}\int_0^{\pi}e^{z\cos\theta}\cos(n\theta)\;d\theta. $$ However, I found an integral $$ ...
3
votes
1answer
56 views

Is the functional equation for $\zeta (s) \left(1-\frac{1}{3^{s-1}}\right)$ known?

It says in wikipedia that Hardy gave a simple proof of the functional equation for: $$\eta(s)=\zeta (s) \left(1-\frac{1}{2^{s-1}}\right)$$ and that it is: $$\eta(-s) = 2 ...
4
votes
3answers
142 views

Closed-form of $\int_0^1 \operatorname{Li}_3\left(1-x^2\right) dx$

By using dilogarithm functional equations we can show that $$ \int_0^1 \operatorname{Li}_2\left(1-x^2\right)\,dx = \frac{\pi^2}{2}-4, $$ where $\operatorname{Li}_2$ is the dilogarithm function. Could ...
2
votes
1answer
27 views

Confusion with Heaviside Step Function and Ramp Function.

I want to know how to represent the following graph with Heaviside step functions and ramp functions. My guess is that this is represented as $r(t) -r(t-2) +2u(t-2)$, where $r(t)$ is the unit ramp ...
0
votes
0answers
35 views

Can this relation be made into a functional equation?

I am trying to find the functional equation for this: $$\zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ Therefore I let: $$x_1=\left(1-\frac{1}{n^{s-1}}\right)$$ which I substitute with ...
5
votes
1answer
197 views

Integral $\int_0^1 \ln(x)^n \operatorname{Ei}(x) \, dx$

I've conjectured the following identity for $n\geq0$ integers: $$ \int_0^1 \ln(x)^n \operatorname{Ei}(x) \, dx = (-1)^{n+1}n! \cdot \left(-\operatorname{Ei}(1)+\sum_{k=1}^{n+1} ...
2
votes
2answers
43 views

Inequality with decay of modified Bessel functions of the second kind

I think that the following inequality holds for all $x > 0$ and all $\nu$ above some constant that is somewhere around 0.2: $$ K_\nu(2 x) \le \frac{2^{2 - 2 \nu}}{\Gamma(\nu)} x^\nu K_\nu^2(x) $$ ...
5
votes
2answers
608 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 ...
1
vote
1answer
41 views

Show that the expression holds for any $x\gt 1$

Let $f(u)$ be a continuous function and, for any real number $u$, let $[u]$ denote the greatest integer less than or equal to $u$. Show that for any $x \gt 1$, $$\int \limits_1 ^x ...
1
vote
2answers
51 views

Ellptic\Jacobi theta function and its residue integral

The Ellptic\Jacobi theta function is given by \begin{align} \theta_1(\tau|z)&=\theta_1(q,y)=-iq^{\frac{1}{8}}y^{\frac{1}{2}}\prod_{k=1}^{\infty}(1-q^k)(1-yq^k)(1-y^{-1}q^{k-1}) \\ &= ...
1
vote
0answers
19 views

Generalized Hyperbolic and Circular Functions

I have recently posted a couple of questions in regards to Generalized Hyperbolic and Circular Functions and I was hoping to find a couple more papers available on the particular subject. The papers ...
1
vote
3answers
54 views

Exponential integral representation

According to exponential integral eqn. (8) $\; E_{1}(x) \;$ can be represented by: $$ E_1(x)= - \gamma - \ln(x) - \sum _{n=1}^{\infty } \frac{(-1)^n x^n}{n n!} $$ where $\gamma$ is the ...
3
votes
1answer
130 views

Hypergeometric Function simple identity

I must prove this property but I really have no idea of how to prove it: $${}_2F_1(a,b;c;z)=(1-z)^{-a}{}_2F_1(a,c-b;c,\frac{-z}{1-z}) $$ It seems it's a 'simple' property, but I haven't been able to ...
2
votes
1answer
469 views

2-increasing functions

I'm trying to learn about copulas. Two definitions I've come across are the H-volume of a rectangle, which is defined as $V_H=H(x_2,y_2)-H(x_2,y_1)-H(x_1,y_2)+H(x_1,y_1)$. The function H, whose ...
1
vote
1answer
25 views

Caculation of involving Hermite polynomial

I have a trouble with this problem involving Hermite polynomial(probability version!). The problem is $$ \frac {(-1)^{r-1}H_{2r-1}(x)}{2^{r-1}(r-1)!x}=\sum_{s=0}^{r-1}\frac{(-1)^s}{2^ss!}H_{2s}(x) $$ ...
6
votes
1answer
143 views
0
votes
0answers
24 views

functional equation from a recurrence relation

Hoe can we get a functional equation from a recurrence relation? Lets say I have a recurrence relation $P_n(x)=a\cdot P_{n-1}(x)-b\cdot P_{n-1}(x)$. We let $\sum P_m(x) t^n=P(x,t)$ and now we have to ...
1
vote
1answer
211 views

What is the relationship between hypergeometric function and Legendre polynomials?

I have an equation $$ -\frac{1}{2}y''(x)-b~\text{sech}^2(ax)~y=-\frac{k^2}{2}y $$ and I know that it has solution in terms of Legendre polynomials: $$ y_1(x)=P_s^\epsilon(\zeta) $$ $$ ...
3
votes
1answer
77 views

Question about Meijer-G definition and identity

I'm trying to wrap my mind around computation involving the Meijer $G$ function, as defined here. (Edit: I'm actually using a somewhat mixed notation using the definitions from MathWorld and the ...
7
votes
0answers
98 views

Bear of an integral

I have a pretty ferocious integral to solve, and would be over the moon if I were able to get some sort of analytic expression / insight for it. $$ I = \int_{r}^{\infty} r_0^{-5/2} W_{-i\alpha'/2, ...
2
votes
2answers
116 views

transforming ordinary generating function into exponential generating function

I have seen a post here that says that you can convert an exponential generating function into an ordinary one with the aid of the Laplace transform. Is it possible to do the reverse transformation? ...
3
votes
1answer
419 views

Incomplete Fermi-Dirac integrals and polylogs

The complete Fermi-Dirac integrals $$ F_s(x) = \frac{1}{\Gamma(s+1)} \int\limits_{0}^{\infty} \frac{t^s}{e^{t-x}+1} \: dt $$ are related to the polylogarithms, see http://dlmf.nist.gov/25.12#iii $$ ...
16
votes
1answer
349 views

Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
1
vote
1answer
39 views

Is this a correct proof that $\tau(n)$ is eventually smaller than any $n^{\delta}$, $\delta>0$?

Is this a correct proof that $\tau(n)$ is eventually smaller than any $n^{\delta}$, $\delta>0$? I'm not even sure about the statement, let alone the proof. Let's first proof this result: $\tau(n) ...
0
votes
0answers
37 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 ...
3
votes
1answer
181 views

Expressing upper incomplete gamma function of half-integer order in terms of gamma function?

N. M. Temme, "Special Functions" (Wiley 1996) gives the following expression that expresses the upper incomplete gamma function in terms of the ordinary gamma function, for integer orders: $$ ...
2
votes
0answers
69 views

What do engineers do when they confront special integrals?

Suppose in a real life situation engineers have to calculate $\int{{2^2}^2}^xdx$ or $\int\sqrt{4-\sin^2x} dx$. The first one doesn't have an integral at all and the second one is an elliptic one. A ...
2
votes
0answers
37 views

Integral Representation of Terminating Hypergeometric function

For Hypergeometric function ${}_2F_1(a_1,a_2;b_1;z)$, if $a_1$, $a_2$ are negative integers, it will be terminated and convergence is not a problem. Under this circumstances, does anyone know the ...
1
vote
0answers
31 views

Complicated recurrence relation

I would like to know if the following recurrence relation is solvable \begin{equation} (\alpha_{1}n^{2}+\beta_{1}n+\gamma_{1})\ c(n+1)+(\alpha_{0}n^{2}+\beta_{0}n+\gamma_{0})\ ...
1
vote
1answer
25 views

Erroneous reasoning over equivalence of hankel function with logarithm

We have the following equation/solution pairs: $$(\nabla^2+k^2)G(\mathbf{x},\mathbf{x'}) = \delta(\mathbf{x}-\mathbf{x'}) \to G(\mathbf{x},\mathbf{x'}) = ...