# Questions tagged [analytic-continuation]

For questions related to analytic continuation

236 questions
1answer
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### Analytic continuation of a holomorphic function bounded on neighbourhood of 'small' compact subset $K\subset\mathbb{C}$

I have two questions regarding the following exercise. First, is my solution (see below) correct? Second, how would the solution go that the writer of the book had in mind (i.e. using the hint)? ...
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### Analytic continuation and complex integration of one variable of a multivariate function

Consider a function $f:(x_0,x_1,...,x_n)\in\mathbb{R}^n\rightarrow f(x_0,...,x_n)\in\mathbb{R}$. $f$ is a ratio of polynomials in $x_0,...,x_n$ which only has simple poles in the variable $x_0$, whose ...
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### Clarification of the proof of Euler's identity regarding the generating function for partitions.

In reference to this question which I asked here couple of days back but didn't get any answer I am posting this question to clarify whether we can able to extend Euler's identity regarding the ...
0answers
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### Deriving Analytic Continuation of Riemann Zeta Function [duplicate]

I'm trying to derive the formula for analytic continuation of riemann zeta function but I can't find a way. How do we find results like Zeta(-1)= -1/12 ?
2answers
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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 ...
0answers
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### Analytic continuation of circular orbits in restricted three body problem

I don't understand a Birkhoff's construction for circular restricted three body problem. "The x,y-plane is rotating. Consider the orbits belonging to the given value of C(Jacobi's constant) which ...
2answers
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0answers
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### Meromorphic continuation of the multifactorial

The double factorial $z!!$, like the normal factorial function, can be extended to the complex plane using $$z!! = 2^{z/2} \left(\frac{\pi}{2}\right)^{(\cos\pi z-1)/4} \left(\frac{z}{2}\right)!$$ ...
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### Geometric view of analytic continuation

Can the shift of center of convergence for power series from point to point in a path of overlapping circles, in analytic continuation, be interpreted as a translation in any way? https://upload....
0answers
23 views

### Correct way to analytically continue a multi-dimensional integral

Consider a multi-dimensional integral \begin{equation} \int dx_1 \int dx_2 ... \int dx_n f(x_1,...,x_n) . \end{equation} where $f$ has simple poles in each of the variables $x_1,...,x_n$. Is it ...
1answer
32 views

### Extend $f(z)=\frac{1}{z^n +z^{n-1}+…+z^2 + z^{-n}}+\frac{c}{z-1}$

find $c$ such that $f(z)=\frac{1}{z^n +z^{n-1}+...+z^2 + z^{-n}}+\frac{c}{z-1}$ can be extended to be analytic at $z=1$ , when $n\in \mathbb{N}$ when $n$ is fixed. The given function I write it ...
0answers
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### For what real part of $s$ as a function of $q$ is the Euler-Maclaurin formula a valid analytic continuation of the Riemann zeta function?

The familiar formula for the Riemann zeta function: $$\zeta(s)=\lim\limits_{k \rightarrow \infty} \left( \sum\limits_{n=1}^{n=k} \frac{1}{n^s}\right) \mbox{ is true for } \Re(s)>1$$ adding one ...
0answers
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### Relationship between cauchy principal value and integrability of a singular point

Under what conditions does an integral have a cauchy principal value and how is it related to an integral having an integrable singularity? E.g $$p.v \int_{-\delta}^{\delta} \frac{dz}{z} = 0.$$ If I ...
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30 views

### Does analytic continuation give actual values

If analytic continuation gives the wrong answer sometimes, at least under typical/basic reasoning, like when it assigns a divergent series a finite value, then why do we trust it to give us values ...
0answers
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### Quaternion algebra using analytic continuation

As for complex variables, do we use analytic continuation to find things like $sin(j)$, $i^k$, and so on? Is there another method or do these expressions even have values at all.
1answer
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### Analytic continuation of $\sin(z)$ [closed]

Why $$\sin{ (z)} =\frac{e^{iz}-e^{-iz}}{2i}$$ the only analytic function, is equal to $\sin{(x)}$ for $z=x \in \mathbb{R}$?
1answer
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### Analytic contiunation

this is more of a broader question. Say I have an analytic complex function $f$ that is defined on the open unit circle, but I know that the limit of $f$ when $|z|$ approaches 1 is 0. Can you define ...
1answer
51 views

### Simple case of analytic continuation

I have yet to formally study complex analysis, yet the topic of analytic continuation, specifically with respect to the Riemann Zeta function, is fascinating. My question is, what are some ...
1answer
42 views

### Analytic continuation of $\sum _{n=0}^{\infty} \frac{z^{2^n}} {2^n}$ beyond the unit disc

The function : $\sum _{n=0}^{\infty} \frac{z^{2^n}} {2^n}$ convergs to holomorphic function $f$ on $D_1(0)$ and is continious on $\overline{D_1(0)}$. I need to prove that f can't be exteneded to any ...
1answer
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1answer
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### how to prove that a function vanishing at an interval is identically zero?

Let $\phi(s):=\int_{0}^{\infty}\exp(-st)g(t)dt$ for $g\in L_1(0,\infty)$. Assume that $\phi(s)=0$ for $s\in[0,\frac{1}{2})$. How to prove that $\phi(s)=0$ for every $s\in[0,\infty)$.
0answers
35 views

### Contour Deformation in the Laplace Inversion Formula

Following Szpankowski - Average Case Analysis of Algorithms on Sequences the exponential generating function of $g(n)$ which is thought to be analytic in $n$ is defined as  G(z)=\sum_{n=0}^{\infty} ...