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
1answer
254 views

Upper bound of function including Pochhammer symbol

How can I find the upper bound of $$\left\vert\frac{(c+1/2+\lambda)_{n}}{\lambda^{n}}\right\vert,\quad\text{where}\quad(c+1/2+\lambda)_{n}=\frac{\Gamma(c+1/2+\lambda+n)}{\Gamma(c+1/2+\lambda)}$$ and ...
3
votes
1answer
57 views

on the sum $ \sum _{n=1}^{\infty}J_{0} (2\pi nx) $

given the zeroeth order Bessel function.. is then possible to compute the sum $ \sum _{n=1}^{\infty}J_{0} (2\pi nx) $ for every 'x' positive real number ?
-1
votes
2answers
565 views

what are the properties of gamma function? [closed]

In mathematics, the gamma function (represented by the capital Greek letter $\Gamma$) is an extension of the factorial function, example: $\Gamma(x)$, $\Gamma(ix)$. What are the physical properties ...
3
votes
1answer
374 views

Chebyshev polynomial properties [duplicate]

Possible Duplicate: Chebyshev polynomial question I am trying to prove a property of Chebyshev polynomials. Given the polynomials $T_n(x), n = 0, 1, \ldots$ which are recursively defined ...
4
votes
1answer
342 views

Oscillation frequencies in an ODE

Given the following ODE: $$\ddot{x}(t)+\sin(\omega t)x(t)=0$$ its solution can be expressed in terms of the Mathieu functions. Plotting this solutions and assuming known the initial conditions it can ...
0
votes
1answer
4k views

Calculation of bessel function versus matlab solution

I am looking to calculate the Bessel function of the first kind $J_o(\beta)$. I am using the formula (referenced from wikipedia) to accomplish this. $$J_\alpha (\beta) = ...
4
votes
1answer
96 views

Show $\lim_{x\to0}\frac{\Gamma(x)}{\psi(x)}=-1$

How to show that $$ \lim_{x\to0}\frac{\Gamma(x)}{\psi(x)}=-1 $$ where $\psi(x)$ is the digamma function.
3
votes
0answers
296 views

Properties of the lemniscate functions as meromorphic functions on $\mathbb{C}$

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 ...
1
vote
1answer
199 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
452 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 ...
3
votes
1answer
197 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
260 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 ...
6
votes
3answers
972 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
655 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
698 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} ...
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
146 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
250 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
351 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
97 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
175 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
567 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
127 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
121 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
278 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
233 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
197 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
315 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* ...
3
votes
2answers
127 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
290 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) = ...
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
376 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
265 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 ...
4
votes
2answers
839 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 ...
4
votes
1answer
197 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 ...
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
116 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} ...
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 ...
7
votes
3answers
690 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 $\Gamma(k+1)=k\Gamma(k)$ ...
10
votes
0answers
226 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
304 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 ...
1
vote
0answers
245 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
189 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
252 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 ...
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. ...