# Questions tagged [analytic-continuation]

For questions related to analytic continuation

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### Deriving the logarithm of negative numbers

is this approach correct? $$\log_a b = \frac{\ln b}{\ln a} \tag{1}$$ but from the famous formula $e^{i\pi} = -1$ we can extend the domain of the natural logarithm and have $i\pi = \ln (-1)$ therefore ...
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### Description of complete analytic function for $\sqrt{4z-\sqrt{z}}$.

The problem is to describe all branches and all the curves of analytic continuation of $\sqrt{4z-\sqrt{z}}$. I started with representing function in a way $\sqrt{4w^3-w}\circ\sqrt{z}$ So, there ...
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### Question on analytic continuation of the principal branch of logarithms on $B(1, 1)$

(Excercise $6$ ,Conway ,page $217$ ) Let $D_0=B(1, 1)$ and $f_0$ be the restriction to $D_0$ of the principal branch of logarithm. For $n\in \Bbb{Z}$ let $\gamma(t) =e^{2\pi i nt}$ for $0\le t\le 1$ ....
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### How to guarantee uniqueness of analytic continuation of the zeta function

The Riemann zeta function has one analytic continuation to $\Re(s)>0$ given by: $$\zeta(s)=\frac{s}{s-1}-s\int_{1}^{\infty}\frac{\{x\}}{x^{s+1}}dx \space \space (1)$$ It also has anther analytic ...
1 vote
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### Evaluate a complex integral. Does it agree with a previous result?

Let $$\Phi(s)=\sum_{n=1}^\infty e^{-n^{s}}$$ converging iff $s>0$ is real. So far the best method for an analytic continuation of $\Phi(s)$ is to use the Cahen-Mellin integral outlined below (see ...
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### Analytic continuation from real axis?

I was given the following question. Use the theorem on analytic continuation discussed in class, perform an analytic continuation for each of the following functions from the real axis to the ...
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### How to find the analytic continuation of this function

Lets say that I have the following function $$f(x,\gamma)=\frac{x^{\gamma-2}}{(\gamma-2)!}+\frac{x^{\gamma-4}}{(\gamma-4)!}+...$$ $$f(x,\gamma)=x^\gamma\sum_{k=1}^\infty \frac{x^{-2k}}{(\gamma-2k)!}$$ ...
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### Does this function come up anywhere in mathematics? $f(s)=\sum_{n=0}^\infty\frac{(-1)^n}{n!}\zeta(-ns)$

I'm wondering if anyone knows if this function comes up anywhere in mathematics: $$f(s)=\sum_{n=0}^\infty\frac{(-1)^n}{n!}\zeta(-ns)$$ where $\zeta(s)$ is the Riemann Zeta function. I'm asking because ...
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