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

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1
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1answer
201 views

Power series expansion of the lemniscate function

We consider the following function. $$u(x) = \int_{0}^{x} \frac{dt}{\sqrt{1 - t^4}}$$ $u(x)$ is defined on $[-1, 1]$. Since $u'(x) = \frac{1}{\sqrt{1 - x^4}} > 0$ on $(-1, 1)$, $u(x)$ is strctly ...
7
votes
2answers
470 views

Deriving the addition formula for the lemniscate functions from a total differential equation

The lemniscate of Bernoulli $C$ is a plane curve defined as follows. Let $a > 0$ be a real number. Let $F_1 = (a, 0)$ and $F_2 = (-a, 0)$ be two points of $\mathbb{R}^2$. Let $C = \{P \in \mathbb{...
3
votes
1answer
200 views

Help Understanding Spectral Method for solving Differential Equations

I've posted a more detailed version of this question here : SE-ComputationalSci but I'm really struggling with a simpler and related question. Lets say one wants to solve (I made this equation up, ...
5
votes
2answers
261 views

Detailed proof of $\zeta(s)-1/(s-1)$ extends holomorphically to $\Re(s)>0$

I'm trying to understand the proof of PNT by Don Zagier. But his proof is too simplified so I can't understand it. I got stumped at step II: $\zeta(s)-1/(s-1)$ extends holomorphically to $\Re(s)>0$....
6
votes
3answers
1k views

Derivatives of the Riemann zeta function at $s=0$

It's a curious fact that for $n>0$, $\zeta^{(n)}(0)\approx -n!$. Apostol gave a table for $\frac{\zeta^{(n)}(0)}{n!}$, among other results on $\zeta^{(n)}(0)$ . the sequence : $$\delta_{n}=\left | ...
0
votes
1answer
671 views

Problem with ratios of integrals

I have the following integral from a paper I'm reading: $$f(z)=\frac{\displaystyle\int_0^{\pi/2}\,\tan \alpha\, J_0(z \sin\alpha)\, d\alpha}{\displaystyle \int_0^{\pi/2}\tan\alpha\,d\alpha}$$ ...
5
votes
1answer
734 views

Hypergeometric functions & integral

I'm having difficulty re-deriving a result a calculation from a paper. The integral is $$\int_0^{2\pi} \int_0^{2\pi} \frac{\sinh\eta}{(\cosh\eta-\cos\theta)^2}\left(1-c\sinh^2\eta\sin\phi\right)^\...
23
votes
4answers
1k views

Evaluation of $\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)$?

I would like to evaluate the sum $$ \sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right) $$ Where $\operatorname{Si}$ is the sine integral, defined as: $$\operatorname{Si}(x) := ...
4
votes
3answers
147 views

Does there exist a nicer form for $\beta(x + a, y + b) / \beta(a, b)$?

I have the expression $$\displaystyle\frac{\beta(x + a, y + b)}{\beta(a, b)}$$ where $\beta(a_1,a_2) = \displaystyle\frac{\Gamma(a_1)\Gamma(a_2)}{\Gamma(a_1+a_2)}$. I have a feeling this should ...
3
votes
3answers
255 views

Trying to prove that $\lim_{n\rightarrow\infty}(\frac{\Gamma '(n+1)}{n!} -\log(n))=0$

In my attempt to prove that $\Gamma'(1)=-\gamma$, I've reduced the problem to proving that $\lim_{n\rightarrow\infty}(\frac{\Gamma '(n+1)}{n!} -\log(n))=0$. Where $\gamma$ is the Euler-Mascheroni ...
5
votes
1answer
357 views

Topology of Branch Cuts and Elliptic Integrals

In reading these notes (elliptic curves starting from elliptic integrals) I came across a couple claims about the topology of some complex surfaces. On page 4, they discuss the integral $$\phi(x) = \...
3
votes
1answer
98 views

Calculating the divisor, known to be small, of two Stirling approximations of the logarithmic Gamma function without overflows

Earlier, I asked a question on MathOverflow regarding how one might analytically approximate a function of the form: $f(n) = \prod_{i=1}^{n-1} (1-ai)$ for $a \ge 0$, $(ai) < 1$, and $n > 10^5$ ...
3
votes
2answers
177 views

Bounding an expression involving digamma function

Let $\psi$ be the digamma function. I have a conjecture that $$\frac ax > \log(x) - \psi(x)$$ holds for all $x > 0$ if (and only if) $a \ge 1$. I do not know how to prove it. Please help.
12
votes
2answers
575 views

Evaluating the elliptic integral $\int_{-\pi}^\pi\frac{dx}{\sqrt{(t-2\cos x)^2-4}}$

I have the following integral, $$I(t)=\int_{-\pi}^\pi\frac{dx}{\sqrt{(t-2\cos x)^2-4}},$$ where $t>4$ is a real parameter. I know from messing around numerically and playing with Mathematica that ...
0
votes
1answer
129 views

Derivative of HeunC function

Given the HeunC function: $$ \operatorname{HeunC}\left( \frac{a^2}{2} \sqrt{2k+3},-1/2,-1+\frac{a^2}{2},-\frac{a^2}{8}(-1 +a^2 k), \frac{1}{2}-\frac{a^2}{4}, -\frac{x^2}{a^2} \right) $$ where $a$ is ...
0
votes
1answer
123 views

Evaluating $\int_0^1 \! C(x) \, \mathrm dx$ through integration by parts

$$ \int_0^1 \! C(x) \, \mathrm{d} x. $$ where $C(x) = \int_0^x \cos(t^2) \, \mathrm{d} t$. I am really not quite sure how to go about this one, especially given that it needs to be calculated ...
3
votes
1answer
284 views

Concerning the lower incomplete gamma function

$\gamma$ is the lower incomplete gamma function. Is $\gamma(1, x) \ge \gamma(k, kx)$ when $k \in Z^+$, $x \in (0,1)$?
2
votes
1answer
235 views

To find the closed form of $ f^{-1}(x)$ if $3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$

$$3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$$ where $\alpha=e^{\frac{2\pi i}{3} }$ I would like to find a closed form of $ f^{-1}(x)$ $$f(x)=\sum \limits_{k=0}^\infty \frac{x^{3k}}{(3k)!}$$ We can ...
1
vote
1answer
198 views

Evaluate an integral of the Airy function $\operatorname{Bi}(x)$

How do I evaluate this interesting integral with the Airy function: $$\int_0^x \operatorname{Bi}(u)^2 du$$ More generally, how do I evaluate $$\int_0^x \operatorname{Bi}(u)^n du$$
2
votes
1answer
325 views

Proving an identity involving the derivative of the Laguerre polynomials with respect to $n$

I've recently come across the following equality in a paper: suppose one defines an analytic function $L(n,x)$ which is equal to the $n$th Laguerre polynomial for $n\in\{0,1,\ldots\}$, and let* $L^{(1,...
3
votes
2answers
128 views

Compute $\lim_{x\to\infty} \frac{{(x!)}^{\frac{1}{x}-1} (x\Gamma(x+1) \psi^{(0)}(x+1)-x! \log(x!))}{x^2}$

What's the strategy one may use when facing a limit like this one? I think it's more important to know the possible ways to go than the answer itself. It's a problem that came to my mind again when I ...
2
votes
1answer
134 views

Calculate $I(\alpha, x,y)=\int\limits_0^1 {{v^{\alpha - 1}}{{(1 - vx)}^{\alpha - 1}}{e^{vy}}dv,\,\,\,0 < \alpha ,x,y < 1}.$

I want to calculate this integral with singularity: $$I(\alpha, x,y)=\int\limits_0^1 {{v^{\alpha - 1}}{{(1 - vx)}^{\alpha - 1}}{e^{vy}}dv,\,\,\,0 < \alpha ,x,y < 1}. $$ I hope to obtain a ...
3
votes
1answer
294 views

Poisson summation formula (in general)

Define Poisson kernel as $$ P_r ( \theta) := \frac{1}{2\pi} \frac{1-r^2}{1- 2r \cos \theta + r^2} $$ Then I want to prove the Poisson summation formula which is $$ P_r (2\pi x) = \sum_{n=-\infty}^\...
7
votes
1answer
1k views

About the asymptotic formula of Bessel function

For $ \nu \in \Bbb R$, I want to prove the well-known formula $$ J_\nu (x) \sim \sqrt{\frac{2}{\pi x}} \cos \left( x - \frac{2 \nu +1}{4} \pi \right) + O \left( \frac{1}{x^{3/2}} \right) \;\;\;\;(x \...
2
votes
1answer
76 views

A certain family of continuous functions on $[0,1]^2$ the closure of which linear span is $\tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$

First of all I must apologize for the vague title and am open to suggestions. This is not a Homework Assignment but something I once again encountered while reading a very compactly written paper. $\...
5
votes
2answers
399 views

A question of the norm calculation of Hermite function.

Define the Hermite function $H_n (x)$ by $$H_n (x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} $$ then prove that $$ \int_{\mathbb R} |H_n (x) |^2 e^{-x^2} dx = 2^n n! \sqrt{\pi}$$
1
vote
1answer
273 views

About the Legendre differential equation

Consider the Legendre differential equation $$ (1-x^2) y'' - 2xy' + n(n+1)y = 0 $$ Then its solution is given by $$ y = c_1 P_n (x) + \text{an infinite series} $$ In fact $y = c_1 P_n (x) + c_2 Q_n (x)...
4
votes
2answers
854 views

Laplace transform of a product of Modified Bessel Functions

Working with a scalar field in 2 dimensions I've come to the following integral, from which I can extract the proper ultraviolet behavior ($a \ll 1$) of the theory: $\int_0^\infty e^{-(4+a^2)x}\left[...
4
votes
1answer
205 views

An infinite series of a product of three logarithms

I was told this interesting question today, but I haven't managed to get very far: Evaluate $$\sum_{n=1}^\infty \log \left(1+\frac{1}{n}\right)\log \left(1+\frac{1}{2n}\right)\log \left(1+\frac{1}{...
3
votes
2answers
261 views

Proving that special functions do not have closed-form expression

When dealing with special functions, like Erf, one should encounter the following statement This function cannot be expressed in terms of classical functions This seems pretty true, but I was ...
1
vote
2answers
117 views

Summing Lerch Transcendents

The Lerch transcendent is given by $$ \Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s}. $$ While computing $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \sum_{p=1}^{\infty}\frac{(-1)...
17
votes
1answer
2k views

Evaluation of $\sum_{x=0}^\infty e^{-x^2}$

Most of us are aware of the classic Gaussian Integral $$\int_0^\infty e^{-x^2}\, dx=\frac{\sqrt{\pi}}{2}$$ I would be interested in evaluating the similar sum $$\sum_{x=0}^\infty e^{-x^2}$$ Now, ...
3
votes
2answers
169 views

Does anyone recognize this function?

I am looking for a function $f(n)$ that satisfies the following two conditions at the same time $$ \frac{f(n-1)}{f(n)}=(-1)^n\quad ,\quad \frac{f(n+1)}{f(n)}=(+1)^n\equiv 1,\quad \forall n\in\mathbb{...
7
votes
3answers
694 views

$e^x(\ln x-c) =\sum \limits_{k=0}^\infty \frac{ x^{k} \Gamma'(k+1)}{ (k!)^2}$ Is it correct result?

$e^x=\sum \limits_{k=0}^\infty \frac{x^k}{k!}$ We can write $e^x=\sum \limits_{k=0}^\infty \frac{x^k}{ \Gamma(k+1)}$ Where $\Gamma(x)$ is Gamma function $\Gamma(k+1)=k\Gamma(k)$ $\frac{\Gamma(k+...
10
votes
0answers
229 views

Different notions of q-numbers

It seems that most of the literature dealing with q-analogs defines q-numbers according to $$[n]_q\equiv \frac{q^n-1}{q-1}.$$ Even Mathematica uses this definition: with the built-in function QGamma ...
4
votes
1answer
305 views

To express $f(x,z)=\sum \limits_{n=0}^\infty \frac{e^{-\alpha n^2 x+\beta n z}}{n!}$ as known functions

$\alpha,\beta >0$ $$f(x,z)=\sum \limits_{n=0}^\infty \frac{e^{-\alpha n^2 x+\beta n z}}{n!}$$ $$\frac{\partial{f(x,z)}}{\partial z}=\beta \sum \limits_{n=1}^\infty \frac{e^{-\alpha n^2 x+\beta n z}...
1
vote
0answers
257 views

How to derive to inverse z transform of $\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$ from Laguerre differential equation?

How can I derive the inverse z-transform of: $$\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$$ If Maple is not the way, how to derive manually? With Maple code I encounter some problems ...
3
votes
1answer
113 views

Minimal $x$ for which $\phi(k) > n$ for all $k > x$

It's well-known that $$ \liminf_n\frac{\varphi(n)\log\log n}{n}=e^{-\gamma} $$ and there exists an effective version $$ \varphi(n)>\frac {n}{e^\gamma\log\log n+\frac{3}{\log\log n}} $$ valid for $...
1
vote
0answers
190 views

What is the correct differential equation for the Laguerre function?

I would like to derive the correct Laguerre function from the differential equation but the differential equations seems different from the original one. What is the correct differential equation and ...
8
votes
1answer
257 views

Hypergeometric formulas for the Rogers-Ramanujan identities?

Let $q = e^{2\pi i \tau}$. Given the j-function, $$j = j(q) = 1/q + 744 + 196884q + 21493760q^2 + \dots$$ and define, $$k = j-1728$$ Let $\tau =\sqrt{-N}$, where $N > 1$. Anybody knows how ...
1
vote
0answers
88 views

fastest way to evaluate $\arg\zeta\left(\frac{1}{2}+i\text{t}\right) $ [duplicate]

Possible Duplicate: evaluation of $ \operatorname{Arg}\zeta (1/2+is) $ ?? If we consider $$\arg\zeta\left(\frac{1}{2} + i\text{t}\right) = \text{Im }\log\zeta\left(\frac{1}{2}+i\text{t}\right)...
2
votes
1answer
1k views

Value of a scaled Bessel function for negative argument

Is the function $\hat{i}_0(x) = e^{-|x|} \sqrt{\frac{\pi}{2x}} I_{\frac{1}{2}}(x)$ positive or negative for negative $x$? $I_{\alpha}(x)$ above is a modified Bessel function. Here are my arguments. ...
0
votes
1answer
179 views

Seeking for some neat function for Hermite polynomial

Let us define $$F_n=\int f(z) |He_n(z)|^2 \, dz \, dz^*$$ is there any type of function $f$ could make that $F_n=0$ for $n\geq 2$ and $F_n>0$ for $n<2?$ $He_n(x)=2^{-\frac{n}{2}}H_n\left(\frac{...
4
votes
3answers
2k views

Estimating the Gamma function to high precision efficiently?

I know there are several approximations of the Gamma function that provide decent approximations of this function. I was wondering, how can I efficiently estimate specific values of the Gamma ...
1
vote
1answer
331 views

Operator for Laguerre polynomial

Is there any operator that could truncate Laguerre polynomial so that the polynomial is only left with the highest order term?
7
votes
1answer
231 views

Improper integral about exp appeared in Titchmarsh's book on the zeta function

May I ask how to do the following integration? $$\int_0^\infty \frac{e^{-(\pi n^{2}/x) -(\pi t^2 x)}}{\sqrt{x}} dx $$ where $t>0$, $n$ a positive integer. This came up on page 32 (image) of ...
2
votes
1answer
813 views

Closed Form of Normal Distribution

What does closed form in following sentence mean and why we need tables of c.d.f.? Normal distributions's p.d.f. cannot be integrated in closed form, and hence tables of the c.d.f. or computer ...
30
votes
2answers
1k views

Is this function decreasing on $(0,1)$?

While doing some research I got stuck trying to prove that the following function is decreasing $$f(k):= k K(k) \sinh \left(\frac{\pi}{2} \frac{K(\sqrt{1-k^2})}{K(k)}\right)$$ for $k \in (0,1)$. ...
10
votes
5answers
1k views

Calculate integrals involving gamma function

What are the usual ways to follow in order to solve the integrals given below? $$\begin{align*} I&=\int_0^1 \ln\Gamma(x)\,dx\\ J&=\int_0^1 x\ln\Gamma(x)\,dx \end{align*}$$
0
votes
1answer
191 views

Is there an infinite product for $\left(\frac{\eta(13\tau)}{\eta(\tau)}\right)^2$ analogous to the Rogers-Ramanujan identity? [closed]

Given $$ \left(\frac{\eta(5\tau)}{\eta(\tau)}\right)^{6}\;\; =\;\; \frac{r^5}{1-11r^5-r^{10}},\;\;\;\;\;\text{with}\;\;r\; =\; q^{1/5} \prod_{n=1}^\infty \frac{(1-q^{5n-1})(1-q^{5n-4})}{(1-q^{5n-2})(...