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

learn more… | top users | synonyms

4
votes
2answers
823 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 ...
5
votes
1answer
184 views

An infinite series of a product of three logarithms

I was told this very interesting question today, and despite my efforts I did not manage to get very far. Evaluate $$\sum_{n=1}^\infty \log \left(1+\frac{1}{n}\right)\log ...
3
votes
2answers
257 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
114 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} ...
15
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
168 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
684 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 $\Gamma(k+1)=k\Gamma(k)$ ...
9
votes
0answers
223 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
236 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
187 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
250 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. ...
0
votes
1answer
175 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 $n<2?$ ...
4
votes
3answers
1k 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
301 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
226 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
721 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
178 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 ...
10
votes
4answers
650 views

Solving $(t^2+1)(y''-2y+1)=e^t$ with the initial conditions: $y(0)=y'(0)=1$

Since it is important to me I would like to award a user who would kindly explain me what are my mistakes and what is the correct way to solve the whole problem with 500 points. I'd really like your ...
8
votes
3answers
374 views

An infinite product for $\left(\frac{\eta(13\tau)}{\eta(\tau)}\right)^2$?

Given the Dedekind eta function, $$\eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1-q^n)$$ where $q = \exp(2\pi i\tau)$. Consider the following "family", $\begin{align} ...
8
votes
3answers
454 views

Evaluation of $\sum_{n=1}^\infty \frac{1}{\Gamma (n+s)}$

I want to try and evaluate this interesting sum: $$\sum_{n=1}^\infty \frac{1}{\Gamma (n+s)}$$ where $0 \le s < 1$ WolframAlpha evaluates this sum to be $$\sum_{n=1}^\infty \frac{1}{\Gamma ...
0
votes
1answer
98 views

Finding a general coefficient in the multiplication of the two series

Help me please to find a general coefficient $a_j$ of the following series $$ ...
20
votes
1answer
519 views

Does $\left(n^2 \sin n\right)$ have a convergent subsequence?

I'm wrestling with the following: Question: For what values of $\alpha > 0$ does the sequence $\left(n^\alpha \sin n\right)$ have a convergent subsequence? (The special case $\alpha = 2$ in ...
5
votes
2answers
1k views

Are Complex Substitutions Legal in Integration?

This question has been irritating me for awhile so I thought I'd ask here. Are complex substitutions in integration okay? Can the following substitution used to evaluate the Fresnel integrals: ...
4
votes
2answers
244 views

To find closed form of $\int_0^{\frac{\pi}{2}} e^{-x\tan t+\alpha t} \;dt $

Let $x\geq 0$, then $$\int_0^{\frac{\pi}{2}} e^{-x\tan t+\alpha t} \;dt = U_{\alpha} (x) $$ $$-\int_0^{\frac{\pi}{2}} \tan t \ e^{-x\tan t+\alpha t} \;dt = \frac{d (U_{\alpha} (x) )}{dx} $$ ...
7
votes
2answers
667 views

Approximate $\int_a^b \frac{1}{\sqrt{2 \pi \sigma^2}}e^{-(x-\mu)^2/2 \sigma^2}\log(1+e^{-x}) \ \ dx $

I am trying to find an approximation to $$ I = \int_a^b \frac{1}{\sqrt{2 \pi \sigma^2}}e^{-(x-\mu)^2/2 \sigma^2}\log(1+e^{-x}) \ \ dx. $$ My attempt is as follows: $$ \begin{align} I &= \int_a^b ...
2
votes
0answers
67 views

Is this formula for $ \sum_{n} (n^{2}+z^{2})^{-s} $ correct?

I would like to know if this formula is true: $$\sum_{n=1}^{\infty}\frac{1}{(z^{2}+n^{2})^s}=\frac{1}{\Gamma(s)} \sum_{n=0}^{\infty}\Gamma(s+n)\zeta(2s+2n)\frac{ (-z^2)^n}{n!}.$$ I have used the ...
5
votes
2answers
156 views

Riemann zeta sums and harmonic numbers

Given the nth harmonic number of order s, $$H_n(s) =\sum_{m=1}^n \frac{1}{m^s}$$ It can be empirically observed that, for $s > 2$, then, $$\sum_{n=1}^\infty\Big[\zeta(s)-H_n(s)\Big] = ...
6
votes
1answer
286 views

On the Dirichlet beta function sum $\sum_{k=2}^\infty\Big[1-\beta(k) \Big]$

Given the Dirichlet beta function, $$\beta(k) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^k}$$ (The cases k = 2 is Catalan's constant.) It seems, $$\sum_{k=2}^\infty\Big[1-\beta(k) \Big] = ...
1
vote
0answers
85 views

integral with bessel function represented as a series [duplicate]

Possible Duplicate: prove equality with integral and series This integral was my homework question with $p=2$ and $n=1$. I am wondering if one can get the general formula for p, or at least ...
2
votes
1answer
320 views

Why is $\pi$ the Limit of the Absolute Value of the Prime $\zeta$ Function?

Motivation: I was looking at the approximation of the truncated Prime $\zeta$ function $$ P_x(s)=\sum_{p\leq x}p^{-s}= \mathrm{li}(x^{1-s}) + O \left(\cdot \right) $$ (to be found here with or ...
2
votes
1answer
116 views

To simplify $f_a(x)= \int_{-a}^{+a} e^ {-\frac{x}{t^2-a^2}}\;dt$

Let $x\leq0$, then $$ f_a(x)= \int_{-a}^{+a} e^ {-\frac{x}{t^2-a^2}}\;dt$$ $$ f'(x)= -\int_{-a}^{+a} \frac{1}{t^2-a^2} e^ {-\frac{x}{t^2-a^2}}dt$$ $$ f'(x)= -\int_{-a}^{+a} ...
1
vote
0answers
127 views

How to find a function with the following properties?

I want to find a function $f(s,x)$ such that $f(s,x)$ is analytic for any $s \in Z^+ $, $f(s,x)=B_s(x)$, where $B_s(x)$ are the Bernoulli polynomials $f(a, x)$ is elementary against $x$ at any ...
8
votes
2answers
2k views

Euler's product formula for $\sin(\pi z)$ and the gamma function

I want to derive Euler's infinite product formula $$\displaystyle \sin(\pi z) = \pi z \prod_{k=1}^\infty \left( 1 - \frac{z^2}{k^2} \right)$$ by using Euler's reflection equation ...
3
votes
0answers
203 views

Common zeros of associated Legendre functions

Suppose that $x_{0}$ is a zero of the associated Legendre function $P_{n}^{m}(x)$ (the degree $n$ is a positive integer while the order $m$ is an integer in the range from $0$ to $n$). If there exist ...
3
votes
1answer
134 views

Hypergeometric functions inequality

Let $_2F_1(a,b;c,z)$ be the (Gauss) hypergeometric function, and $m$ and $n$ positive integers. From a simple plot it looks like $_2F_1(m+n,1,m+1,\frac{m}{m+n})>\frac{m}{n} ...
5
votes
1answer
226 views

prove equality with integral and series

I am stuck on one question with integral. Help me please to show that with $n=1$ the following is true $$ ...
15
votes
4answers
502 views

The function $f(x) = \int_0^\infty \frac{x^t}{\Gamma(t+1)} \, dt$

Does anyone know if this function has a name? I came up with it by looking at the power series for $e^z$, changing the summation to an integral, and substituting the gamma function for the factorial ...
15
votes
1answer
352 views

On the zeta sum $\sum_{n=1}^\infty[\zeta(5n)-1]$ and others

For p = 2, we have, $\begin{align}&\sum_{n=1}^\infty[\zeta(pn)-1] = \frac{3}{4}\end{align}$ It seems there is a general form for odd p. For example, for p = 5, define $z_5 = e^{\pi i/5}$. Then, ...
11
votes
1answer
220 views

A particular case of Truesdell's unified theory of special functions

I'm reading through Clifford Truesdell's "An essay toward a unified theory of special functions", Princeton Univ. Press, 1948. All his exposition is based on the functional equation ...
4
votes
2answers
797 views

Integral of product of exponential function and two complementary error functions (erfc)

I found the following integral evaluation very interesting to me: Integral of product of two error functions (erf) and I hoped that I could use that result to evaluate the following integral: $$ ...
0
votes
1answer
95 views

Condition for frame of $L_2$

Let $f$ be continuous, real valued and compactly supported with exactly one maximum function in $L_2$. Form the functions $$ f_{m,k}=f^m(x-2^k) $$ Under which conditions $\{f_{m,k}\}$ would be a ...
1
vote
0answers
655 views

solving Bessel function equation by hand

I have a Bessel function of the first kind given by the equation $$J_\alpha (\beta) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha +1)} \left(\frac{\beta}{2}\right)^{2m+\alpha}$$ I am trying to ...
10
votes
3answers
317 views

Nicer expression for the following differential operator

I have the following sequence of differential operators: $$D_n = \underbrace{t \partial_t t \partial_t \dots t \partial_t}_{\text{$n$ times}}.$$ Is there any expression involving a sum of "normal" ...
7
votes
0answers
155 views

Evaluting $ \int_0^{\infty}\frac{v}{\sqrt{v + c}}e^{-\frac{y^2}{2(v + c)} - \frac{(u-v)^2}{u^2v}}dv$

While working on mixture (variance) of normal distribution and keep running into these two integrals $$ \int_0^{\infty}\dfrac{v}{\sqrt{v + c}}e^{-\dfrac{y^2}{2(v + c)} - \dfrac{(u-v)^2}{u^2v}}dv,$$ ...