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

learn more… | top users | synonyms

1
vote
1answer
124 views

Why does the Gamma function interpolate $(n-1)!$?

Why does the Gamma function interpolate $(n-1)!$ and not $n!$ instead? What is the historical reason?
2
votes
1answer
125 views

Reasoning about the gamma function using the digamma function

I am working on evaluating the following equation: $\log\Gamma(\frac{1}{2}x) - \log\Gamma(\frac{1}{3}x)$ If I'm understanding correctly, the above is an increasing function which can be demonstrated ...
2
votes
0answers
136 views

Inequalities involving regularized incomplete Gamma functions

I am new to the world of the Gamma functions and am wondering if there exist positive functions $f_1(x)>0$ and $g_1(x)>0$, and non-negative functions $f_2(x)\geq0$ and $g_2(x)\geq0$ such that ...
2
votes
2answers
171 views

Bessel functions of the first kind

How would I show that $$J_1(x)+J_3(x)=\frac 4x J_2(x)$$ Using the series definition of the Bessel Function, which is $$J_p(x)=\sum ^\infty _{n=0} \frac{(-1)^n}{\Gamma(n+1)\Gamma(n+1+p)}\left(\frac x2\...
2
votes
0answers
487 views

Approximations of the incomplete elliptic integral of the second kind

For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I know that this is a ...
1
vote
0answers
63 views

What is the closed form expression for this?

Let $r_1...r_k$ be the $k$ roots unity or solutions to the expression $x^k = 1$ What is the expression: $$\frac{1}{x^{k-1}}\frac{1}{\Gamma(-xr_1)}\frac{1}{\Gamma(-xr_2)}\ldots\frac{1}{\Gamma(-xr_k)}$...
3
votes
2answers
143 views

Bounds on $ \sum\limits_{n=0}^{\infty }{\frac{a..\left( a+n-1 \right)}{\left( a+b \right)…\left( a+b+n-1 \right)}\frac{{{z}^{n}}}{n!}}$

I have a confluent hypergeometric function as $ _{1}{{F}_{1}}\left( a,a+b,z \right)$ where $z<0$ and $a,b>0$ and integer. I am interested to find the bounds on the value it can take or an ...
5
votes
1answer
219 views

Dirichlet L-series and Gamma function question

Could someone help me, please, with this exercise? Consider a sequence of complex numbers $\{a_n\}$ such that $a_n=a_m $ iff $ n\cong m $ mod $q$ for some positive integer $q$. Define the ...
8
votes
2answers
573 views

Approximation of $\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$ [duplicate]

I am reading about the Riemann hypothesis, and the article mentioned the Li function: $$\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$$ They said that this function can be approximated: $$\...
0
votes
1answer
327 views

Nonlinear method to solve an equation with the error function in it

My question is to find a method to solve the following non-linear equation. I know it should be an iterative method, but I don't know what would be the best method to use. Any help is highly ...
0
votes
1answer
224 views

More general reflection formula for Gamma function

It is know that $\Gamma(z)\, \Gamma{(1-z)}=\pi \csc( \pi z)$. Is there any formula for $\Gamma{(a+z)}\Gamma{(a-z)}$ where $a$ is a rational number, i. e., $a=p/k$ with $p, k$ integers and $z$ is ...
27
votes
2answers
995 views

Interesting log sine integrals $\int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx= \frac{7\pi^3}{108}$

Show that $$\begin{aligned} \int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx &= \frac{7\pi^3}{108} \\ \int_0^{\pi/3}x\log^2 \left(2\sin\frac{x}{2} \right)dx &= \frac{17\pi^4}{6480}\...
2
votes
2answers
299 views

Calculate an infinite continued fraction

Is there a way to algebraically determine the closed form of any infinite continued fraction with a particular pattern? For example, how would you determine the value of $$b+\cfrac1{m+b+\cfrac1{2m+b+\...
0
votes
2answers
161 views

Hyperfactorial curiosity

I've been familiarizing myself with the hyperfactorial, and I'm simply curious if it has an extension/analogue into the world of rational numbers, irrational numbers, and complex numbers like the ...
4
votes
1answer
176 views

visualizing functions invariant (or almost) under modular transformation

In the spirit of Möbius Transformations Revealed, I'd like to make a pair of movies depicting how Klein's absolute invariant $j(\tau)$ and the Dedekind eta function $\eta(\tau)$ transform when $\tau\...
3
votes
2answers
227 views

Polylogarithm - derivative with respect to order

Does anybody know where I could find the expression for $$\frac{\partial}{\partial s}\mathrm{Li}_s(z)\bigg|_{s=0}$$ or something similar?
4
votes
1answer
508 views

Euler's infinite product for the sine function and differential equation relation

Euler's infinite product for the sine function $$\displaystyle \sin( x) = x \prod_{k=1}^\infty \left( 1 - \frac{x^2}{\pi^2k^2} \right)$$ http://en.wikipedia.org/wiki/Basel_problem We know that $\...
3
votes
1answer
270 views

looking for upper bound on quantity with “erf”

Math people: I am looking for an upper bound on $$g(x,t) = \frac{-\operatorname{erf}(t)+2\operatorname{erf}(\frac{1}{2}t(x+1))-\operatorname{erf}(xt)}{(x-1)^2},$$ where $x > 1$ and $t > 0$. ...
2
votes
2answers
670 views

Definite integral involving modified bessel function of the first kind

I would like to solve the following integral that is a variation of this one (Integral involving Modified Bessel Function of the First Kind). Namely, I have: $$\frac{1}{\sqrt{2\pi w^2}}\int_{-\infty}...
8
votes
3answers
392 views

How to find integral of $\int_0^\infty \frac{\ln ^2z} {1+z^2}\mathrm{d}z$?

How do I find the value of $$\int_{0}^{\infty} \frac{(\ln z)^2}{1+z^2}\mathrm{d}z$$ without using contour integration, - using the usual special functions, e.g., zeta/gamma/beta/etc. Thank you,
2
votes
1answer
440 views

Limit involving a hypergeometric function

I am new to hypergeometric function and am interested in evaluating the following limit: $$L(m,n,r)=\lim_{x\rightarrow 0^+} x^m\times {}_2F_1\left(-m,-n,-(m+n);1-\frac{r}{x}\right)$$ where $n$ and $...
15
votes
2answers
967 views

Interesting Integral $\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx$

I am asking this question out of curiosity. $$\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx = \frac{ \left(2\cos \frac{n}{2} \right)^{\alpha +\beta-2}}{\Gamma(\alpha+\...
7
votes
1answer
327 views

Missing term in series expansion

I asked a similar question before, but now I can formulate it more concretely. I am trying to perform an expansion of the function $$f(x) = \sum_{n=1}^{\infty} \frac{K_2(nx)}{n^2 x^2},$$ for $x \ll 1$....
8
votes
1answer
224 views

the solution for an integral including exponential integral function

I have the following integral $$\int_c^\infty{x^{a-1} e^{\ p \ x} \ \mathrm{Ei}(-p\ x) \ \mathrm{d}x}.$$ I'd like you to help me to evaluate it or giving me a hint to proceed.
2
votes
1answer
94 views

Growth of $\Gamma(n+1,n)$ and $\operatorname{E}_{-n}(n)$

Quite often when I ask W|A to solve something it gives me an answer in terms of $\Gamma(n+1,n)$ or exponential integral $\operatorname{E}_{-n}(n)$. Looking up the definition of the incomplete gamma ...
1
vote
1answer
207 views

Dilogarithm Identities

Is there a cleaner way to write: $$ f(x) = \operatorname{Li}_2(i x) - \operatorname{Li}_2(-i x) $$ in terms of simpler functions? I don't know enough about dilogarithms, and the basic identities I see ...
3
votes
2answers
309 views

Integration over a combination of confluent hypergeometric, power, and exponential functions

I am trying to work out this integral. If there is no closed form, can you think of any approximations to it? $$\int_0^T e^{a (T-x)} (T-x)^{1+m+n} x^k \, _1F_1\Big(1+n;2+m+n;a (x-T) \Big) \, dx$$ ...
10
votes
2answers
203 views

Infinite Series :$ \sum_{n=0}^\infty \frac{\Gamma \left(n+\frac{1}{2} \right)\psi \left(n+\frac{1}{2} \right)}{n! \left(n+\frac{3}{2}\right)^2}$

Prove that: $$\sum_{n=0}^\infty \frac{\Gamma \left(n+\frac{1}{2} \right)\psi \left(n+\frac{1}{2} \right)}{n! \left(n+\frac{3}{2}\right)^2} = \frac{-\pi^{\frac{3}{2}}}{12}\left( \pi^2+6\gamma(1-2\log ...
2
votes
0answers
89 views

Series expansion of a series

I would like to perform an asymptotic expansion of the function $$f(x) = \sum_{n=1}^{\infty}\frac{1}{(nx)^2}K_2(n x),$$ where $K_2(x)$ is the modified Bessel function of the second kind, around $x=0$. ...
11
votes
1answer
725 views

Why $p$-adically interpolate?

I'm studying $p$-adic analysis now and particularly $p$-adic interpolation; for example, constructions like $p$-adic $L$-functions (Kubota-Leopoldt style). I'm having some difficulty though, and I'd ...
5
votes
1answer
196 views

Prove that $\int_0^{\pi/2} \cos^{p+q-2}(\theta) \cos((p-q)\theta)d\theta = \frac{\pi}{(p+q-1)2^{p+q-1}B(p,q)}$

Does anybody know how to prove this identity? $$\int_0^{\pi/2} \cos^{p+q-2}(\theta) \cos((p-q)\theta)d\theta = \frac{\pi}{(p+q-1)2^{p+q-1}B(p,q)}\quad p+q>1,q<1$$ $B(x,y)$ denotes Beta ...
2
votes
1answer
498 views

Verification of integral over $\exp(\cos x + \sin x)$

I found the following integral in a paper I was reading: \begin{equation} \frac{1}{2\pi} \int\limits_{-\pi}^{\pi} \exp\left(a \cos x + b \sin x\right) dx = I_0\left(\sqrt{a^2+b^2}\right), \end{...
2
votes
2answers
115 views

How to solve $x \geq \frac{y}{z-\ln{x}}$ for positive variables?

How can you solve $x \geq \frac{y}{z-\ln{x}}$ for $x$ when the variables are real positive values? I am only really interested in the case where the values are large and $z > \ln x$. How ...
0
votes
2answers
185 views

Bessel function confirmation

I'm trying to see that $J_0(x)$ is indeed a solution for the Bessel equation $x^2y''+xy'+x^2y=0$, so: $$J_0(x)=\sum_{k=0}^\infty \frac{(-1)^kx^{2k}}{(k!)^22^{2k}}$$ Pluging it in the equation and and ...
2
votes
1answer
114 views

Constant term of recursively defined polynomials related to the Lambert W function

The Lambert $W$ function has the property that $$ W'(x) = \frac{W(x)}{x[1+W(x)]}, $$ and using this one can show that its Taylor expansion about $x=a$ has the form $$ W(x) = W(a) + \sum_{n=1}^{\...
1
vote
0answers
70 views

The cdf of a beta variable, evaluated at the mean

Consider a Beta random variable $X$ with shape parameters $k/2$ and $(d-k)/2$, where the parameters $k, d$ are integers that satisfy $0 < k < d$. What is the best possible upper bound for the ...
11
votes
3answers
387 views

Integral $\int\limits_0^\infty \prod\limits_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx$

Does anybody know how to prove this identity? $$\int_0^\infty \prod_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx=\frac{\sqrt{\pi}}{2}\frac{\Gamma \left(a+\frac{1}{2}\right)\...
8
votes
2answers
221 views

The double integral $\int_0^1 \int_0^1 \frac{1-x}{1-xy}(-\log(xy))^s \, dx \, dy$

$$\int_0^1 \int_0^1 \frac{1-x}{1-xy}(-\log(xy))^s \, dx \, dy=\Gamma(s+2)\left(\zeta(s+2) -\frac{1}{s+1}\right) \quad \Re(s)>-2$$ How to prove this identity?
6
votes
2answers
634 views

Turning an elliptic curve over C into a complex torus

I have been reading a lot about the Weierstrass $\wp$ function and I understand the parameterization of an elliptic curve with the elliptic function( i.e. $x=\wp(z)$ and $y=\wp^\prime(z)$). I would ...
1
vote
2answers
71 views

Roots of the equation $I_1(b x) - x I_0(b x) = 0$

I'm interested in the roots of the equation: $I_1(bx) - x I_0(bx) = 0$ Where $I_n(x)$ is the modified Bessel function of the first kind and $b$ is real positive constant. More specifically, I'm ...
3
votes
2answers
794 views

A hard log definite integral: $\int_0^{\pi/4}\ln^3\sin x\,\mathrm dx$

Show that: $$\int_0^{\pi/4}\ln ^3\sin x\text{d}x=\frac34\text{Im}\left(\text{Polylog}\left(4,i\right)\right)-3\text{Im}\left(\text{Polylog}\left(4,\frac12+\frac{i}{2}\right)\right)-\frac{23}{128}\pi^3\...
11
votes
3answers
334 views

Evaluating $\int_0^1 \frac{1}{\sqrt{\Gamma(x)}} dx$

What is the value of the following integral? $$\int_0^1 \frac{1}{\sqrt{\Gamma(x)}} \,dx$$ Here $\Gamma(x)$ is Euler's gamma function. EDIT: Can we improve the upper bound strictly smaller than $1$? ...
2
votes
0answers
100 views

A more general exponential integral

More generalized on the previous question: A improper integral with Glaisher-Kinkelin constant Show that : $\displaystyle\int_0^\infty\frac{\text{e}^{-ax}}{x^2}\left(\frac{1}{1-\text{e}^{-x}}-\frac{1}{...
2
votes
1answer
84 views

Proof of $\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$?

Can anyone show a proof of $$\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$$ where $0<a<1$, and $$\Gamma(a)=\int_0^\infty t^{a-1}e^{-t}\,dt.$$ Thank you.
5
votes
1answer
127 views

Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$

Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$ I don't know how to do this ? Note that $\gamma $ is the Euler-Mascheroni constant
4
votes
2answers
917 views

Calculate an integral involving Hermite polynomials

I have to calculate the integral $$\frac{1}{\sqrt{2^nn!}\sqrt{2^ll!}}\frac{1}{\sqrt{\pi}}\int_{-\infty}^{+\infty}H_n(x)e^{-x^2+kx}H_l(x)\;\mathrm{d}x$$ where $H_n(x)$ is the $n^{th}$ Hermite ...
0
votes
1answer
174 views

Bessel function equation

I have to prove that equation: $$\tag{1} \operatorname{J}_2(xy)\ \operatorname{Y}_0 (x) - \operatorname{J}_0(x) \operatorname{Y}_2(xy) =0$$ (where $\operatorname{J}_k$ and $\operatorname{Y}_k$ are ...
2
votes
1answer
105 views

Construct a generating function for the components of a sum

Let $j \in Z_+$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$. Find generating function $\sum_{j}a_jx^j$ so that allows to ...
5
votes
1answer
214 views

Integral representation of cosecant function

According to Wolfram website http://functions.wolfram.com/ElementaryFunctions/Csc/introductions/Csc/05/, There exists a "well-known" integral representation for the cosecant function, i.e. $$\csc(z):...
2
votes
2answers
609 views

Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes

I have to numerically calculate the following definite integral $$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$ for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...