Questions tagged [analytic-continuation]

For questions related to analytic continuation

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Deriving the logarithm of negative numbers

is this approach correct? $$\log_a b = \frac{\ln b}{\ln a} \tag{1}$$ but from the famous formula $e^{i\pi} = -1$ we can extend the domain of the natural logarithm and have $i\pi = \ln (-1)$ therefore ...
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3 votes
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Description of complete analytic function for $\sqrt{4z-\sqrt[3]{z}}$.

The problem is to describe all branches and all the curves of analytic continuation of $\sqrt{4z-\sqrt[3]{z}}$. I started with representing function in a way $\sqrt{4w^3-w}\circ\sqrt[3]{z}$ So, there ...
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Question on analytic continuation of the principal branch of logarithms on $B(1, 1) $

(Excercise $6$ ,Conway ,page $217$ ) Let $D_0=B(1, 1) $ and $f_0$ be the restriction to $D_0$ of the principal branch of logarithm. For $n\in \Bbb{Z}$ let $\gamma(t) =e^{2\pi i nt}$ for $0\le t\le 1$ ....
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4 votes
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Fitting a statement in a long list of equivalent results

Theorem. Suppose that $D\subset \mathbb C$ is a connected open set. The following are equivalent. Either $D=\mathbb C$ or $D$ is conformally equivalent to $\mathbb D$. $D$ is homeomorphic to $\mathbb ...
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1 answer
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Recursive analytic continuation of Riemann zeta function

If you read pages 51-55 of the book The Theory Of Functions by Konrad Knopp (Publication date 1947) and you are patient enough to overcome a very bad made digital copy of the book (I do not understand ...
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1 vote
0 answers
23 views

Continuing $\log(z)$ analytically along the curve $\gamma(t)=e^{it}$

Show that $\log(z)=\sum_{n=1}^\infty \frac{(-1)^{n-1}(z-1)^{n}}{n}$ can be analytically continued along the curve $\gamma(t)=e^{it}$. Compute $f_t$ and their radii of convergence. What is $f_{2\pi}$? ...
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For an arc,locally analytic imply analytic?

I'm stuck on a problem from Complex Analysis by Marshall. Problem:$\gamma:[0,1]\rightarrow\mathbb{C}$,is a simple arc.If $\gamma$ is covered by finite many open analytic arc,then it is analytic itself....
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Anayltic Continuation on complement of countable set

I'm reading Reconstructing a neural net from its output, and there's this remark. Let $F_1$ and $F_2$ be analytic on $\Omega_1, \Omega_2$, with $\Omega_1$ connected and $\mathbb{C} \setminus \Omega_2$...
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  • 1,352
2 votes
1 answer
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Open Sets in Schwarz Reflection Principle

The Scwarz Reflection Principle says that if $A$ is an open, connected set in the upper half plane of $\mathbb{C}$ whose boundary intersects the real axis in an interval $[a,b]$, then we can extend a ...
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Convergence issues with this derivation of the functional equation of the Riemann zeta function

I am using Riemann's Zeta Function by H.M. Edwards as a reference for the derivation of the functional equation of the Riemann Zeta Function, as well as this pdf. I am not very familiar with the ...
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Can Branch Points Move?

I am aware that poles of a function can move in different branches of a holomorphic function as it is analytically continued. I was wondering if the same applied to branch points. Formally, suppose $f$...
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Theorem about a continuous function that does not have global minimum at (a,b) [closed]

Theorem: If $f(x)$, which is continuous and non-constant on $(a,b)$, does not have a global minimum in $(a,b)$ then if the limits exist we have : $\lim_{x\to a} f(x) < f(y)$ for every $a<y<b$,...
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Does the analytic continuation method for infinite series always work and is unique?

Sorry if this is worded poorly, but I don’t know enough in the subject to word it better: $$\sum_{n=0}^{\infty}n=-1/12$$ according to $$\zeta(-1)=\sum_{n=1}^{\infty}\frac{1}{n^s}=-1/12$$ Is there ...
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Abusing analytic continuation

I'm playing with math I only partially understand, which means I know enough to be dangerous. Why doesn't this transformation work? I wasn't necessarily expecting it to but I also don't understand ...
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1 vote
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Extension of a real analytic function to a complex analytic one in a bounded domain

Let $f$ be a real analytic function defined on an open interval $I=(a,b)$ with $a,b$ finite. Is it always possible to extend $f$ to a complex analytic function defined on an open subset $U\subseteq \...
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How to do analytic continuation?

I have some trouble with analytic continuation, even for a simple question. For example, I want to analytically continue the following function from negative real axis to the whole plane, $$\frac{1}{\...
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24 views

Can we use power mean to generalize min and max for complex numbers?

Power mean $M_p(a,b)$ of order $p \in \mathbb{R}$ for a pair $(a,b) \in \mathbb{R}^+$ is defined as $M_p(a,b)= \Big(\frac{a^p+b^p}{2}\Big)^{\frac{1}{p}}$. For example $p = 1$ gives arithmetic mean, ...
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A curious Mellin pair

While thinking about the relation: $$\zeta(s)\Gamma(s)=\int_0^\infty \frac{x^{s-1}}{e^x-1}~dx $$ I started with a Mellin transform on bounded support: $$ f(s)= \int_0^1 x^{s-1}\bigg(-\frac{1}{e^{\...
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How to guarantee uniqueness of analytic continuation of the zeta function

The Riemann zeta function has one analytic continuation to $\Re(s)>0$ given by: $$\zeta(s)=\frac{s}{s-1}-s\int_{1}^{\infty}\frac{\{x\}}{x^{s+1}}dx \space \space (1)$$ It also has anther analytic ...
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Evaluate a complex integral. Does it agree with a previous result?

Let $$ \Phi(s)=\sum_{n=1}^\infty e^{-n^{s}} $$ converging iff $s>0$ is real. So far the best method for an analytic continuation of $\Phi(s)$ is to use the Cahen-Mellin integral outlined below (see ...
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Analytic continuation from real axis?

I was given the following question. Use the theorem on analytic continuation discussed in class, perform an analytic continuation for each of the following functions from the real axis to the ...
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  • 1,057
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How to find the analytic continuation of this function

Lets say that I have the following function $$f(x,\gamma)=\frac{x^{\gamma-2}}{(\gamma-2)!}+\frac{x^{\gamma-4}}{(\gamma-4)!}+...$$ $$f(x,\gamma)=x^\gamma\sum_{k=1}^\infty \frac{x^{-2k}}{(\gamma-2k)!}$$ ...
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3 votes
0 answers
140 views

Does this function come up anywhere in mathematics? $f(s)=\sum_{n=0}^\infty\frac{(-1)^n}{n!}\zeta(-ns)$

I'm wondering if anyone knows if this function comes up anywhere in mathematics: $$f(s)=\sum_{n=0}^\infty\frac{(-1)^n}{n!}\zeta(-ns)$$ where $\zeta(s)$ is the Riemann Zeta function. I'm asking because ...
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1 vote
1 answer
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derivative of $\left| x^x \right|$ in the entire domain?

$x^x$ is my favorite "misbehaving" function. Nearly all exercises, examples and calculations of it involve its relatively tame positive side only. There's also the derivative, $x^x (\ln x +...
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2 answers
98 views

Natural Boundary of Euler's Partition Generating Function

Let $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$. Let's consider the analytic function $f:\mathbb{D}\to\mathbb{C}$ given by, for all $z\in\mathbb{D}$, $$f(z)=\prod_{n=1}^\infty (1-z^n)^{-1}.$$ For each ...
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  • 3,049
1 vote
1 answer
60 views

Meromorphic continuation of L-functions

I am following these notes and on page 2 the claim is that if we have an $L$-function $$L(s) = \sum_{n=1}^{\infty}\frac{a_n}{n^s}$$ with $a_n=O(n^r)$ and if $L$ has a meromorphic continuation and ...
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Analytic Continuation for a finite integral

In an earlier post (Assigning values to a divergent integral?), it was claimed in one of the answers that 'analytic continuation' can be used to prove that $$ I = \int_0 ^\infty dx (\frac{\sqrt{x}}{1+...
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Analytic Continuation of Parametric Integral

Suppose I have two parametric integrals $F_i$ with $i=1,2$ defined as $$ F_i(z) := \int_\mathbb{R} dx f_i(x,z)$$ and suppose that $f_i$ are nice enough to ensure that the $F_i$ are holomorphic on some ...
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-3 votes
1 answer
116 views

1-1+1-1+... = k+1/2?

Let $$F(s) = \sum_{n=0}^{\infty} f_{n}(s)$$ be a complex analytical function defined by a series (not necessarily a power series) that absolutely converges on an open set $s \in U$. Assume that the ...
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1 vote
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29 views

Analytic continuation / equivalence based on both value and complex derivate at 2 isolated points.

This is is a question which will help me ensure I have understood complex analysis and analytic continuation. We start with an analytic function $f(z)$, holomorphic in the entire complex plane for ...
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1 vote
0 answers
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Question regarding the fundamental system of an DE

We consider an differential equation $$ d_A: \frac{dW}{dz} = A(z) W$$ with $A(z) \in \mathit{Mat}(n, \mathcal{O}_{\mathbb{C^*}})$ and $W(z)$ a fundamental system of this equation. If I now consider ...
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0 answers
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Needham's explanation of rigidity leading to analytic continuity.

I am self-teaching, and would like to make sure I have understood Needham's explanation of the rigidity of analytic functions, and then going further to explain analytic continuation. Needham is the ...
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Find smooth version of $g(x) = \sum_{i=0}^x a^x$

I want to find a smooth function $f$ such that for all positive integers $x$, $f(x)=g(x)$ where $g(x)$ is given below. In less mathy terms I want a smooth version of $g(x)$ constrained by its integer ...
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0 votes
1 answer
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Identity of a modification of the primorial function

The primorial function $n\#$ is defined as the product of all primes less than or equal to $n$. If $p_i$ is the $i$th prime, I define a function $P(n)$ such that $P(n) = p_n\#$. E.g., $P(1) =2, P(2)=6,...
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2 votes
1 answer
80 views

Prove that every holomorphic function has a maximal continuation.

I have to show that there exists a maximal holomorphic continuation for every $f : \Omega \rightarrow \mathbb{C}$ holomorphic, with $\Omega \subseteq \mathbb{C}$ open, connected. We defined a ...
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1 vote
0 answers
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Existence and Uniqueness of ODE solution

Consider the initial value problem: $\frac{dy}{dt} = (t^2 + y^4)^{\frac{1}{2}}, \; y(t_0) = y_0$ Discuss existence, uniqueness, and continuation of solutions through each point $(t_0, y_0) ∈ \mathbb{R}...
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-1 votes
1 answer
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Analytic Continuation of a function defined in integral

Let $C$ be the boundary of a unit circle. The function $f:\mathbb{C} \backslash C \to \mathbb{C}$ is defined by $f(z)=1-\frac{1}{2\pi i} \int_C \frac{1}{\zeta-z} d\zeta$. I found that $f(z)=0$ when $z$...
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1 vote
1 answer
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Can this type of function, defined on the closed unit disk, be analytically continued into the complex plane?

Let ${\Bbb D}:=\{z\in\Bbb C:|z|<1\}$ and $f:\overline{\Bbb D}\to\Bbb C$ be given by $f(z):=\sum_{n=0}^\infty a_nz^n$, where the constants $a_n\in\Bbb C$ are such that $\sum_{n=0}^\infty |a_n|$ is ...
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1 vote
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Analytic continuation of integral type function

Consider we have a function that is defined in a unit disk in complex plane. Let it be $$F_\alpha(z)=\displaystyle\int\limits_{0}^{1} (1-zt)t^\alpha dt,\alpha>0$$ I have to evaluate $F_\alpha (x_1)-...
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1 vote
0 answers
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Region of convergence for $\Gamma(s)\zeta(s)$ in analytic continuation of Riemann zeta function

One way to determine values of the (analytically continued) Riemann zeta function is to $\zeta(s)$ is to use the product $\Gamma(s)\zeta(s)$ and use our knowledge of the poles of $\Gamma(s)$. We can ...
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Are there "half step", or other subdivisions, of the Mandelbrot iteration function?

The Mandelbrot Set is generated by iterating $f(z)=z^2+c$. Is there a "half-step" function $g(z)$ such that $g(g(z))=f(z)$? What is the method for finding such half-iteration functions? ...
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2 votes
0 answers
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trying to understand analytic continuation (2 scenarios)

The following two scenarios I hope will help me fill in gaps in my understanding of analytic continuation. Scenario 1 We have a function $f(z)$ of a complex variable $z$. It is known to be zero at $z=...
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  • 1,756
2 votes
2 answers
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Does analytic continuation preserve $\zeta(\overline{s})=\overline{\zeta(s)}$?

It is easy to show that the for the series representing the Riemann zeta function for $\Re(s)>1$ $$\zeta(s)=\sum\frac{1}{n^s}$$ the symmetry $\zeta(\overline{s})=\overline{\zeta(s)}$ holds. ...
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Do the poles of the $\Gamma(z)$ multiply out to 1 in some way?

Is there a meaningful way to assign values $\Gamma^\star(z)$ related to the values of the gamma function $\Gamma(z) = \int_0^\infty {e^{-t}t^{-z}dt}$ such that $\prod\limits_{ \Bbb Z^-} \Gamma^\star (...
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2 votes
1 answer
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A new(?) analytic continuation for the Riemann zeta function.

While tweaking the definition for the Euler gamma constant I found that the following appears to be true: $$\zeta(s)=\lim_{n\to \infty } \, \frac{a^{s-1} \sum\limits_{k=1}^{a n} \frac{1}{k^s}-b^{s-1} \...
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2 votes
2 answers
537 views

An Exercise from Chapter 2 of Stein's Complex Analysis

I have been working through Stein and Shakarchi's Complex Analysis and I'm stuck on Exercise 15 from Chapter 2. The question states: Suppose $f$ is a non-vanishing continuous function on $\overline{D}$...
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Analytic continuation in plane isomorphic to $\Bbb C$

Define $\varphi(s)=\prod_{n=1}^\infty \exp\big(n^{-s}\big)$ where $s=\exp a + i\exp b$ for $a,b \in \Bbb R.$ Does there exist an analytic continuation of $\varphi(s)?$ For real $s>1$ the product ...
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5 votes
1 answer
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Under what conditions, or how can be done that, one says that something divergent can have a value?

Excuse my question, I just don't get it. In this proof, is mentioned: Now if you write formally the derivative of the Dirichlet-series for zeta then you have $$ \zeta'(s) = {\ln(1) \over 1^s}+{\ln(1/...
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1 vote
0 answers
71 views

Is this a valid example of analytic continuation?

I am trying to understand (and then explain) analytic continuation as simply as possible. Question 1: Is the following an example of analytic continuation? The following function is defined for all $...
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  • 1,756
2 votes
1 answer
180 views

About the unicity of the analytic continuations of $\zeta$ and the continuation used to have $\zeta'(0) = -(\ln(1) +\ln(2) +\ln(3) + \ldots )$

The analytic continuation of an analytic function is unique. I've come across different representations of those, for example: $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-...
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