# Questions tagged [analytic-continuation]

For questions related to analytic continuation

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### Finding the image of a point by analytic continuation of regular branch of $\sqrt{π^2 + \ln^2(z)}$ in $\mathbb{C}\backslash Γ$

Denote $\varphi$ the regular branch of analytic function $\sqrt{π^2 + \ln^2(z)}$ in $\mathbb{C}\backslashΓ$ defined by $φ(1) = \pi$, and where $\Gamma$ is: With only this information, I would like to ...
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### How does this identity so elegantly combine an infinite sum in $\eta$ and an improper integral in $\Gamma$?

This is all well and good, but where did this come from? In the article on the Gamma function, Wikipedia shows most of its alternate definitions with clear proofs, yet in the article on the Dirichlet ...
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### Does my odd proof for the Abel sum for $\eta(-2)$ work?

EDIT: The correct answer to the Abel sum of $\eta(-2)$ has been given by the comments under this post. The focus of the question is now whether there is any sense to my method and my "proof" ...
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### two disjoint compact sets, holomorphic function there exists a decomposition $f=f_1+f_2$

Let $D_1$ and $D_2$ be two compact sets in $\Bbb C$, $D_1\cap D_2=\emptyset$, and $f\colon \Bbb C\setminus(D_1\cup D_2)\to\Bbb C$ be a holomorphic function. Show that there exist two holomorphic ...
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### Analytic continuation of a periodic function on the real line

Suppose $f(x+1)=f(x)$ is real analytic for $x\in \mathbb{R}\setminus \mathbb{Z}$. Moreover, it is once-differentiable at $x\in \mathbb{Z}$. In addition, suppose that on the interval $(0,1)$, we know ...
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### Extending $\sum_{n=0}^\infty s^{n^2}$ beyond its natural boundary

Let $\mathbb{D} = \{s \in \mathbb{C} : |s| < 1\}$. Let $f : \mathbb{D} \rightarrow \mathbb{C}$ where $$f(s) = \sum_{n=0}^\infty s^{n^2}$$ $f$ is analytic on $\mathbb{D}$. This is what it looks ...
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### Analytic continuation of a periodic function on the real line using a Bernoulli polynomial

Consider a function $\Gamma(z;\tau,\sigma)$ which is meromorphic in $z$ and periodic under $z\to z+1$. I am interested in a certain limit of this function, where $\tau$ and $\sigma$ take specific ...
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### Prove that an analytic function is zero

Suppose that I have an analytic function in four variables $$f = f(z_1,z_2,z_3,z_4)$$ such that I know the following facts: $$\Re [f(z_1,z_2,0,0)] = 0$$ and that: $$f(0,0,z_2,z_3) = 0$$ I was ...
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### Closed form for sums containing exponential function:

How to get closed form of following sums: $$\sum_n e^{-n/2}n^{k-1}\left(s-\frac{1}{mn}\right)^k$$ $$\sum_n \frac{e^{-n/2}}{n^2\left(s-\frac{1}{mn}\right)}$$ Here $s,k,m$ are constants and $n$ runs ...
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