# Questions tagged [analytic-continuation]

For questions related to analytic continuation

101 questions
318 views

### Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$\theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) ,$$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
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### Show that $f(z)=\sum_{n=0}^{\infty}z^{2^n}$ can't be analytically continued past the unit disk.

I'm reading the problems of Stein and Shakarchi's Complex Analysis, Chapter 2 Problem 1 asks to show that $$f(z)=\sum_{n=0}^{\infty}z^{2^n}$$ cannot be analytically continued past the unit disk. (...
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### Is $\int_0^\infty \frac{dt}{e^t-xt}$ analytic continuation of $\sum_{k=1}^\infty \frac{(k-1)!}{k^k} x^{k-1}$?

The following power series apparently converges only for $-e \leq x <e$: $$f(x)=\sum_{k=1}^\infty \frac{(k-1)!}{k^k} x^{k-1}$$ We can use it to define a real function $f(x)$, analytic in that ...
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### Translation of integration contour

I have an argument where, with a meromorphic function $\phi$ satisfying $\phi(s) = \phi(1-s)$, the following equality appears: $$\int_{(2)} \phi(s) y^{-s} \, ds = \int_{(1/2)} \phi(s) y^{-s} \, ds$$ ...
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### Is there a natural topology in which the analytic continuation of a series converges everywhere it is defined?

The expression $$\sum_{n=1}^\infty n = -\frac{1}{12}$$ has been met with considerable controversy. Previous questions (such as this and this) searching for a way of rationalizing the formal ...
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### Black magic behind Padé approximation

In condensed matter physics one usually calculates the so-called Matsubara Green's function in the set of discrete points $G (i \omega_n)$, where $$\omega_n = (2n+1) \pi T$$ Physically significant ...
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### Analyitc Continuation of Partial $\zeta$ function

Let $A\subset \mathbb{N}$. Consider the partial $\zeta$ function $$\sum_{a\in A} \frac{1}{a^s}$$ We know this converges for Re$(s)>1$. Under what conditions of $A$, can this be analyically ...
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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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### Dirac delta from poles of a function

Suppose we are given the simple expression $$F(k) = \frac{1}{E^2-E(k)^2}$$ which has a pole when $E^2 = E(k)^2$ and where $E, E(k)$ are real numbers. When working with this expression (e.g. inside ...
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### Riemann zeta meromorphic cont. using Abel summation formula

In Stein&Shakarchi, Complex Analysis, chapter 6, problem 2-3 (p. 180), they hint at a method to meromorphically continue the zeta function to the entire complex plane. I can see from Abel's ...
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### What is the relationship between the asymptotic series outside the radius of convergence and the analytic continuation?

In his post about computing the zeta function by smooth cutoff regularization of the series $\sum\frac{1}{n^2}$, Terry Tao shows how $-{1\over 12}$ is the finite part of the asymptotic series for the ...
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### Comparison of entire functions with $f=cg$

Let $f,g:\mathbb C\to\mathbb C$ be holomorphic with $|f(z)|\leq|g(z)|$ for all $z\in\mathbb C$. Show that there is a $c\in\mathbb C$ with $|c|\leq 1$ such that $f=cg$. If $g\equiv 0$ then we ...
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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 ...
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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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### Abel-Plana Analytic Continuation of the Partial Sums of the Zeta Function

So I am trying to analytically continue the formula for the partial sums of the Riemann zeta function by using the fact that, if $\zeta_{k}$ represents the $k^{\text{th}}$ partial sum of the zeta ...
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### Analytic continuation for $\sum_{n=0}^{\infty}(\sqrt n+1/3)^{-s}$

Define a function $F(s)$ by: $$F(s)=\sum_{n=0}^{\infty}(\sqrt n+1/3)^{-s}$$ Is there a closed form expression for the analytic continuation of $F(s)$ to $F(-s)$?
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### Analytic Continuation of Sum $\sum_{n=0}^{\infty} e^{-b \sqrt n}$

Suppose we have the following function: $$f(b)=\sum_{n=0}^{\infty} e^{-b \sqrt n}$$ Is there a closed form expression for the analytic continuation of $f(b)$ to $f(-b)$?
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### Natural Boundary of Sum $\sum_{n=0}^{\infty} e^{-b n^p}$

Is it possible to prove for which $p$ does the sum $$\sum_{n=0}^{\infty} e^{-b n^p}$$ have a natural boundary on the imaginary b axis?
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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 ...