# Questions tagged [analytic-continuation]

For questions related to analytic continuation

101 questions
0answers
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 ...
0answers
612 views

### 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. (...
0answers
125 views

### 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 ...
0answers
34 views

### Translation of integration contour

I have an argument where, with a meromorphic function $\phi$ satisfying $\phi(s) = \phi(1-s)$, the following equality appears: $$\int_{(2)} \phi(s) y^{-s} \, ds = \int_{(1/2)} \phi(s) y^{-s} \, ds$$ ...
0answers
87 views

### 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 ...
0answers
142 views

### 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 ...
0answers
44 views

### 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 ...
0answers
25 views

### Meromorphic continuation of the multifactorial

The double factorial $z!!$, like the normal factorial function, can be extended to the complex plane using $$z!! = 2^{z/2} \left(\frac{\pi}{2}\right)^{(\cos\pi z-1)/4} \left(\frac{z}{2}\right)!$$ ...
0answers
41 views

### 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 ...
0answers
72 views

### 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 ...
0answers
120 views

### 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 ...
0answers
143 views

### 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 ...
0answers
75 views

### 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
44 views

0answers
13 views

### 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 ...
0answers
38 views

0answers
115 views

### 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 ...
0answers
40 views

### 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 $F(-s)$?
0answers
42 views

### 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 $f(-b)$?
0answers
40 views

### 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?
0answers
57 views

0answers
28 views

### 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
11 views

### 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 ...
0answers
14 views

### Understanding analytic continuation and swapping integration and analytic continuation.

This question is partly Physics motivated, I am trying to study real-time dynamics of a system, but the theory is well-defined in Euclidean space-time. I have a complicated integrand, function of ...