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Questions tagged [analytic-continuation]

For questions related to analytic continuation

Analytic continuation is a technique used in complex analysis to expand the domain of an analytic function.

If $f$ is analytic on some open $U\subseteq \Bbb{C}$, then it can usually be extended to an analytic function $F$ on a larger connected domain. That extension, when it exists, is unique.

574 questions
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Analytic continuation of $(x-a)^{-1/4}$?

In Landau's book on quantum mechanics an expression of the form $$ (x-a)^{-1/4} $$ is analytically continued around the upper half-plane along the semicircle from $0$ to $\pi$. The result stated is $$(a-x)^{-1/4}e^{-i\pi/4}$$ How does one derive…
hhh3
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How to find the analytic continuation of this function

Lets say that I have the following function $$f(x,\gamma)=\frac{x^{\gamma-2}}{(\gamma-2)!}+\frac{x^{\gamma-4}}{(\gamma-4)!}+...$$ $$f(x,\gamma)=x^\gamma\sum_{k=1}^\infty \frac{x^{-2k}}{(\gamma-2k)!}$$ When gamma is an…
Joshua Pasa
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Does an analytic continuation for a particular Leibniz series exist?

Define a Leibniz series as follows, \begin{eqnarray*} L(x) & = & \sum_{k=1}^{\infty}(-1)^{k}e^{-kx}\ln k,\ \ x>0 \end{eqnarray*} I have two questions: (I) Is there an analytical form for $L(x)$? (II) Does the analytic continuation for $L(x)$ from…
Ren-Hong Fang
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Can $f(z)=\int_E \frac{dt}{t-z}$ be extended to an entire function?

Suppose that $E \subset \mathbb{R}$ is compact and $m(E)>0$. Let $\Omega=\mathbb{C} \setminus E$ and \begin{equation} f(z)=\int_E \frac{dt}{t-z} \end{equation} for all $z \in \Omega$. I think that $f$ cannot be extended to an entire function…
KK Kwok
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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 like $e^{i\pi}$? Why is this the chosen way to give…
Pineapple Fish
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holomorphic continuition of harmonic function

Let f(x) be a harmonic funtion, can it be extended to a analytic function? I know it is true if f(x) is analytic, but for harmonic function, is it still true? Thanks!
fengpeng wang
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