All Questions
Tagged with closed-form gamma-function
109
questions
1
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1
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45
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Expressing the coefficients of $(1-x)^{1/4}$ using factorials
From the fact that $1\times3\times5\times\ldots\times(2n-1)=\frac{(2n)!}{2^nn!}$, we can show that
$$
(1-x)^{1/2}=\sum_{n=0}^\infty \frac{(2n-2)!}{(n-1)!n!2^{2n+1}}x^n.
$$
However, can I do the same ...
- 7,534
4
votes
1
answer
171
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closed form of the following integral :$\int_{0}^{\infty}- \sqrt{x}+ \sqrt{x}\coth (x) dx$?
I have tried to evaluate this:$\int_{0}^{\infty}- \sqrt{x}+ \sqrt{x}\coth (x)$ using the the following formula
$$2 \Gamma(a) \zeta(a) \left(1-\frac{1}{2^{a}} \right) = \int_{0}^{\infty}\Big( \frac{x^{...
- 7,195
1
vote
0
answers
37
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Closed form for an integral involving an incomplete Gamma function?
I am trying to find a closed form for this integral:
$$\int_{0}^{\infty}\int_{0}^{\infty}e^{-d_{p,s}^v\,x-d_{s,p}^v\, y+d_{p,p}^v\,\frac{\xi_1\,\sigma^2\,xy}{P_p\,\xi_2\,xy+\sigma^2}}\mathrm dy\...
- 11
3
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2
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397
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How to Evaluate $\sum_{n=0}^{\infty}\frac{(-1)^n(4n+1)(2n)!^3}{2^{6n}n!^6}$
I want to Evaluate
$\sum_{n=0}^{\infty}\frac{(-1)^n(4n+1)(2n)!^3}{2^{6n}n!^6}.$
I tried from arcsin(x) series and got $\frac{1-z^4}{(1+z^4)^{\frac{2}{3}}}= 1-5(\frac{1}{2})z^4+9(\frac{(1)(3)}{(2)(4)})...
- 347
1
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0
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68
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Closed form for product over Gamma function
Is there a "closed form" (with which I mean an expression not involving an indexed sum or product) for any of these four products?
$$\prod_{k=1}^{n} \Gamma(\frac{x}{k*2+1})$$
$$\prod_{k=1}^{n} \Gamma(...
3
votes
2
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152
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Integration problem related to Gamma function: $ \int_{0}^{\infty} u^{\alpha + b - 1} \exp\left(-ub + u^{\alpha}c\right)du $
During my work on some statistics problem, I stumbled across the following integral:
$$ \int_{0}^{\infty} u^{\alpha + b - 1} \exp\left(-ub + u^{\alpha}c\right)du,\qquad \alpha, b, c>0 $$
I tried ...
- 131
1
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0
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Yet another bizarre identity involving hypergoemetric functions and gamma functions.
Let $d=4$, $T\ge d$ and $p\ge 0$ be integers.
By solving Spectral densities of finite dimensional sample covariance matrices we stumbled on a following identity.
\begin{eqnarray}
&&\sum\...
- 11.5k
3
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0
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84
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Closed form for infinite series $\sum_{n=1}^\infty \prod_{j=1}^n \left[1-(\tfrac{j}{n}u+v)^2\right]^{-1} x^n$
I've been struggling to find a closed form for the following series that depends on $u$ and $v$ as parameters:
$$
f(u,v,x) = \sum_{n=1}^\infty c_n x^n
$$
with
$$
c_n = \frac{1}{\prod_{j=1}^n \left[...
- 207
0
votes
0
answers
60
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closed form of series involving gamma function
In one of my calculation, I came across the series:
$$\sum_{n=0}^\infty\frac{z^{n}}{Γ(n\alpha)}$$
$0<\alpha<1$.
- 11
2
votes
1
answer
130
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On a closed form for $\int_{-\infty}^\infty\frac{dx}{\left(1+x^2\right)^p}$ [duplicate]
Consider the following function of a real variable $p$ , defined for $p>\frac{1}{2}$:
$$I(p) = \int_{-\infty}^\infty\frac{dx}{\left(1+x^2\right)^p}$$
Playing around in Wolfram Alpha, I have ...
- 3,309
1
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1
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143
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Series including the Gamma funtion
In one of my calculation, I came across the series: $$\sum_{n=0}^{\infty} \frac{x^n}{\Gamma(n+\alpha)}.$$
When $\alpha=1,$ this equals $e^x$ and when $\alpha=\frac{1}{2},$ this equals $\frac{1}{2\sqrt{...
- 16.8k
3
votes
1
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284
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Definite Gamma function integral
Curiosity Question
It's very well known that
$$\int_a^{a+1} \ln\Gamma(x)\ dx = \frac{1}{2}\ln(2\pi) - a - \ln(a) - (a+1)\ln\Gamma(a) + (a+1)\ln\Gamma(1+a)$$
Clearly provided that $\Gamma(a)\geq ...
- 26.3k
3
votes
2
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177
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Evaluate $\int_0^{\pi/2}\frac{1-\sqrt[18]{\cos u}}{1-\cos u}du$, in terms of particular values of special functions and constants
I wondered about this question when I've considered similar integrals with different integrand as an analogous of a formula for harmonic number.
As example that I know that can be calculated using ...
user243301
2
votes
1
answer
76
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How to compute: $I_x =\int_0^1 t^x e^{-\frac{1}{1-t^2}} dt~~~x>0$?
$$I_x =\int_0^1 t^x e^{-\frac{1}{1-t^2}} dt~~~x>0$$
I first tried to see if I could solve the integer case.
$$I_n =\int_0^1 t^ne^{-\frac{1}{1-t^2}} dt $$
I have tried to find a possible ...
- 24.2k
1
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1
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What is the closed form of : $\int_{0}^\infty\alpha(\frac{x}{\beta})^{\alpha}e^{-(\frac{x}{\beta})^\alpha}dx$.
can anyone suggest how to solve this intergral?
$$\int_{0}^\infty\alpha\Big(\frac{x}{\beta}\Big)^{\alpha}e^{-\left(\frac{x}{\beta}\right)^\alpha}dx,~~\alpha>0,~~\beta>0$$.
I'm not familiar ...
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