Questions tagged [analytic-continuation]

Question 1

I would like to understand how to make sense of the following divergent series, or at least to identify the appropriate analytic continuation of its general term: \begin{align} \sum_{n=0}^{\infty}\...

Question 2

Calculate: $${\int\limits_0^\infty{\left[{\frac{x^{s-1}}{e^x-1}-\frac{(e^x-1)^{s}}{(e^x-1)^2}}\right]dx=}{\,\,\color{red}{?}}}\tag{1}$$ For $0\lt\Re(s)\lt1$. An integral definition of $\zeta(s)$ Zeta ...

Question 3

So consider the following sum $$ H(n) = \sum_{k=1}^{n} \frac{1}{k} $$ If you consider this sum as a function of n, the domain of this function is natural numbers. However, the domain of this function ...

Question 4

I am studying the analytic continuation of complex functions and series beyond their analytic domain but currently it is not totally clear for me when exactly the analytic continuation of an infinite ...

Question 5

The standard definition of the analytic-continuation of the Beta function by way of a Pochhammer contour integral is $$ (1-e^{2\pi i\alpha})(1-e^{2\pi i\beta})\text{Beta}(\alpha,\beta)=\int_{P} z^{\...

Question 6

The following is one version of Morera's theorem from complex analysis, as presented by Theodore W. Gamelin. Theorem (Morera’s Theorem). Let $f(z)$ be a continuous function on a domain $D$ (defined as ...

Question 7

I am considering the following problem from an introductory course on elliptic PDE theory. 'Suppose that $\Omega$ is a domain in $\mathbb R^n$ (that is to say, a non-empty connected open subset of $\...

Question 8

I am trying to use polynomial roots to approximate this ratio $R(p)$: $$R(p)=\Re\left(\frac{\underset{k\to \infty}{\text{lim}}\left(\left(H_k^{(s)}\right)^{1/p}+\left(\frac{k^{1-s}}{s-1}\right)^{1/p}\...

Question 9

I recently discovered that the equation $$ \lim_{N\to\infty}\sum_{n=1}^{\infty}\frac{\pi^2}{n^{s-2}(2N+1)^2}\cot^2\left(\frac{n\pi}{2N+1}\right) $$ Converges to $\zeta(s)$ for $s>1$, I wonder if ...

Question 10

I don’t have enough reputation to comment, so I’m posting a follow-up to Can every real-analytic function be extended to one holomorphic outside a discrete subset of $\mathbb{C}$? What about a ...

Question 11

Background. In physics, experimental observables are often related to real-time or real-frequency Green’s functions, $G(t)$ or $G(\omega)$, which are (typically) real-analytic. Physicists often extend ...

Question 12

For this one loop integral, I have approached it using Feynman Parametrization, $$\int_{}^{}\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}+m_{1}^{2})[(k-q)^{2}+m_{2}^{2}]}$$ The parametric integral that I ...

Question 13

I am trying to understand the computation of the monodromy group of the differential equation given by $$x^2f''(x)-f(x)=0.$$ The differential equation has one singular point in $\mathbb C$, namely, $0$...

Question 14

I'm remember seeing a proof in my first complex analysis course that bounded harmonic function on a punctured disk extend harmonically over the puncture, using the fact that such a function is locally ...

Question 15

Let’s say we have a power series $$ f(z) = \sum_{k=0}^{\infty} a_k z^k $$ with radius of convergence $\rho$. Then we define a new function $$ \varphi(zt) = \sum_{k=0}^{\infty} \frac{a_k}{k!} (zt)^k $...

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Questions tagged [analytic-continuation]

analytic continuation of the series $\sum_{n=0}^{\infty}\frac{n^2}{\sqrt{a^2+n^2}}$

On $\int_0^\infty\left[\frac{x^{s-1}}{e^x-1}-\frac{(e^x-1)^s}{(e^x-1)^2}\right]dx$

How to extend this function into positive real numbers from natural numbers

Analytic continuation of complex series with circle method (Taylor series expansion) beyond the region of convergence

Ambiguity of the analytic continuation expression for the Beta function

Can Analyticity Extend to the Boundary in Morera’s Theorem?

Unique continuation for elliptic partial differential equation with C^1 coefficients: an elementary approach?

Using the analytic continuation of the Riemann zeta function to approximate some polynomial roots.

How do I analytically continue this function??

Follow-up to "Can every real-analytic function be extended to one holomorphic outside a discrete subset of ℂ ? What about a meromorphic one?"

How to extend a function in real axie to a meromorphic function in complex plane? [duplicate]

Parametric Integral Analytic Solution

Monodromy group of the differential equation $x^2f''(x)-f(x)=0$

Why must the monodromy constant satisfy $e^C=1$ in the exponential proof of the removable singularity theorem for harmonic functions?

Convergence of the Borel summation integral

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