# Tag Info

### Separating Gamma in two independent functions

$\newcommand\Ga\Gamma$If this factorization were true, we would get $$1=\frac{f(3)g(1)\,f(4)g(2)}{f(4)g(1)\,f(3)g(2)} =\frac{\Ga(3,1)\Ga(4,2)}{\Ga(4,1)\Ga(3,2)}=\frac{19}{16},$$ a contradiction. So, ...
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### Analytic continuation of convergent integral

Technically, your integral is not well-defined because the path goes through $z=1$; the remedy I see presently (unless you have a definition for the contour going through $z=1$), is to move the ...
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### $\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} z^{2k}$ is an elementary function

As Carlo noted, for $n$ an even integer, $S_{n,m}(z)$ is an elementary function of $z$. What about $n$ odd? When $n,m$ are both odd, I get something in terms of arcsinh, also elementary. But for $n$ ...
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### Extended binomial coefficients and the gamma function

There's nothing special about the gamma function; the failure of the limit to exist when $a$, $b$, and $n$ are negative is exactly the same as the failure of $$\lim_{(x,y,z)\to(0,0,0)} \frac{xy}{z}$$ ...
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### Integral calculus with Gamma function

You fix $\alpha$ and denote your integral to the left by $I(\beta )$. Then $I$ is convergent and analytic on the semi-plane $H=\{\beta\in{\mathbb C}\mid\Re (\beta )>0\}$. The right hand side too is ...

Let me first look at a simpler example, instead of the square root consider the inverse Laplace transform of $e^{-s}$. If you write the series expansion and invert term by term you obtain $$L^{-1}_s\... • 180k 6 votes Accepted ### Eisenstein E_2 at imaginary quadratic arguments Exactly the same Chowla--Selberg formula is valid, but you must apply it to the modified (non-holomorphic)$$E_2^*(\tau)=E_2(\tau)-3/(\pi\Im(\tau)) In other words, $E_2^*(\tau)/\eta^4(\tau)$ is an ...
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Let us calculate $f_{\epsilon}(x) - \frac{1}{\epsilon}\log(1 + x)$ using your formula for $\log(1 + x)$ as $f_1(x)$(I didn't checked it but believe that it is correct): \$f_\epsilon(x) - \frac{1}{\...