# Questions tagged [analytic-continuation]

For questions related to analytic continuation

236 questions
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### On every simply connected domain, there exists a holomorphic function with no analytic continuation.

I am working on a question that requires me to prove that on every simply connected open subset of $\mathbb{C}$, there exists a holomorphic function that cannot be extended to a holomorphic function ...
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### Intuition behind $\zeta(-1)$ = $\frac{-1}{12}$ [duplicate]

When I first watched numberphile's 1+2+3+... = $\frac{-1}{12}$ I thought the sum actually equalled $\frac{-1}{12}$ without really understanding it. Recently I read some wolframalpha pages and watched ...
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### Fourier transform of meromorphic function

Suppose that I have a function $f(z)$ which is meromorphic on the entire complex plane, meaning holomorphic everywhere except for a discrete set of poles. I then take a vertical slice of this ...
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### Analytic continuation of harmonic series

Is there an accepted analytic continuation of $\sum_{n=1}^m \frac{1}{n}$? Even a continuation to positive reals would be of interested, though negative and complex arguments would also be interesting. ...
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### Evaluate the integral $\int_0^\infty x^{t-1}e^{-\beta x}dx$

I want to evaluate the following integral $$\int_0^\infty x^{t-1}e^{-\beta x}dx$$ where $\beta$ is a complex number. Now, if $\beta$ was real, we could just set $y = \beta x$ and we will reduce to ...
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### Asymptotic expansion of $Li^{-1}$ and zeros of $F(s)$ and $G(s)$

If you downvote please leave some constructive feedback. I would like to compare and visualize/gain insight about the zeros of two functions, $F(s)$ and $G(s).$ $\pi(m)$ is the prime counting ...
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### Analytic continuation of $\sum (z/a)^n$

I'm having trouble continuing this function beyond its convergence radius, $R=a$. $$f(z)=\sum (z/a)^n$$ Given the context (a textbook in complex analysis) I suspect it should have a simple closed-...
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### Is the Lambert W function analytic? If not everywhere then on what set is it analytic?

I would appreciate if someone can help me answer the following questions. Although I read several papers and documents on the Lambert W function, I could not assess on what set is this function (or ...
Let $\rho$ be the conformal mapping from the interior of the triangle with vertices $-1,i\sqrt{3},1$ onto the upper half-plane. Show that $\rho$ has an elliptic extension. The hypothesis makes sense ...