0
votes
0answers
20 views

zeros of Incomplet Gamma function

for which values of complex variable $z$ let us getting the zeros of incomplet gamma function ($\Gamma(0.5,z)$) ? I would be interest for any replies or any comments
2
votes
0answers
39 views

Why does the Riemann Xi function $(\xi(s))$ have order of growth 1

Why does $s(s-1)\xi(s)$, have order of growth 1? In other words, why is it that $\forall \epsilon > 0 $ $\exists A_{\epsilon},B_{\epsilon} \in \mathbb R_+$ so that $\forall s \in \mathbb C$, ...
0
votes
2answers
46 views

Values of Incomplete gamma function

Look this claim : Does $\Gamma(0.5,-x^2)= i\alpha$, for $x$ large real number? i=unity imaginary part $\alpha$ is real number I would like someone to prove me this if it's a true claim
3
votes
2answers
141 views

Proof of my conjecture on closed form of $\int _{0}^{\infty}\frac{x^{a-1}e^{-mbx}}{1-e^{-bx}}$

Let $a$, $b\in \Bbb R^+$ and $m \in \Bbb N$ then My conjectural closed form is $$\int _{0}^{\infty}\frac{x^{a-1}e^{-mbx}}{1-e^{-bx}}\,{\rm d}x = ...
12
votes
0answers
469 views

The log gamma integral $\int_{0}^{z} \log \Gamma (x) \ \mathrm dx$

One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \ \mathrm dx $ is in terms of the Barnes G function. $$ \int_{0}^{z} \ln \Gamma(x) \ \mathrm dx = \frac{z}{2} \log (2 \pi) + ...
3
votes
1answer
438 views

Real and imaginary part of Gamma function

Is there a way to separate the real and imgainary part of the gamma function $$\Gamma (a+ib)$$ I thought of using the formula $$\zeta(z) \Gamma(z) = \int^{\infty}_0\frac{t^{z-1}}{e^t-1}\, dt$$ ...
3
votes
1answer
106 views

Calculating an almost Gamma integral

How would you proof that $$I:=\int_{0}^{\infty}\frac{z^{x-1}}{e^{z}+1}dz=\left(1-2^{1-x}\right)\Gamma(x)\zeta(x)$$ I can rewrite the integral as ...
12
votes
1answer
289 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, ...
0
votes
0answers
139 views

Determining well-definedness for functions

How does one determine well-definedness in analytical continuation for $\Gamma(s)\zeta(s)$ function? Firstly: $$\Gamma(s)\zeta(s) = \int_0^\infty dt \frac{t^{s-1}}{e^t - 1},\quad Re(s) > 1$$ ...
3
votes
2answers
381 views

Is it problem of Mathematica or my own?

The following is a plot comparing Exp[Derivative[1,0][Zeta][0,x]+1/2Log[2 Pi]] and Gamma[x]: In theory the blue and the red ...
4
votes
2answers
577 views

Question Relating Gamma Function to Riemann Zeta function evaluated at integers

I was just reading a paper of Ramanujan entitled " On question 330 of Professor Sanjana" when i got stuck up with a Proposition which i am unable to answer. The proposition is if $ \displaystyle ...