Questions tagged [analyticity]

A function is analytic if it has a converging power series expansion. Please also use one of (real-analysis) or (complex-analysis) to specify real or complex analyticity: though the definition is the same, the properties of functions analytic over real numbers is quite different from the properties of functions over the complex numbers.

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65 views

Riemann zeta function is analytic for $\operatorname{Re}(z)>1$

I want to show that, Riemann zeta function is analytic on the domain $\operatorname{Re}(z)>1$. I know that it is absolutely and uniformly convergent on the right half of the line $Re(z)=1$. Now, I ...
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Analyticity of a complex function..

A paper I am reading says that $e^{-i q \left(-\log (-q+i 0^+ )+1+\frac{i \pi }{2}\right)}$ is a function analytic on upper half plane and $e^{-i q \left(-\log (q+i 0^+ )+1+\frac{i \pi }{2}\right)}$ ...
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Sequential approach to differentiability of complex valued function

Let $f$ be analytic function on an open set $G$. Let $z\in G$. Let $\{z_n\}$ and $\{w_n\}$ be two sequences in $G$ which converges to $z$. Then show that $lim_{n \to \infty} \frac{f(z_n)-f(w_n)}{z_n-...
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Unsure about a step in the analyticity of a holomorphic function

Consider the following section in a proof in my notes: We show here that, as a consequence of Cauchy's integral formula any holomorphic function g(z) in an open domain D is analytic. Consider a circle ...
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What is an analytic coefficient? (Kashiwara)

In Kashiwara's thesis he uses $\mathcal{D}$-modules to investigate (systems of) linear PDEs with analytic coefficients. There are also mentions of analytic manifolds analytic linear PDEs analytic $\...
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Conjecture on openness of an analytic mapping

Consider a real analytic function $g: \mathbb{R}^m \rightarrow \mathfrak{M}_{\mathbb{R}}(n, k)$ to the set of $n \times k$ matrices where $k \ge n + 1$ such that $\forall x \in \mathbb{R}^m \quad \...
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Analytic region of the function $z^{2 \alpha}$

This is a paragraph from Ahlfors Complex Analysis It states that for the function $z^{2 \alpha}$ where $0<\alpha<1$, it is possible to choose an analytical branch of the function whose argument ...
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What is the analytic nature of ln z within a region? [closed]

Is ln z analytic within the region |z-1|<1 and if so, how does one prove its analytic nature within the region?
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Bolzano-Weierstrass and zeros of complex analytic function

I am working on a textbook exercise. A similar question: An analytic function in a compact region has finitely many zeros, but it's not quite clear to me and I also have possibly another approach? I ...
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Real-valued analytic function (complex)

I am studying elementary complex variable topic. An exercise in Brown-Churchill book asks us to show that if $f$ is analytic in a domain $D$ and real-valued at all points then $f$ must be constant ...
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Asymptotics in generating functions, am I doing right?

For each generating function $A(x)$, find c and ρ such that the coefficient $a_n$ of $x^n$ in the generating function $A(x)$ satisfy: $a_n \sim c * ρ^n$. $$A(x)= \frac{1-2x}{(1-x)(1-3x)}$$ $$A(x)=\...
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The coefficient and asymptotic in generating function

Let $L_n$ be the set of all the paths from $(0,0)$ to $(n,0)$ such that every step is $u=(1,1)$ , $d=(1,-1)$ and $r=(2,0)$. Notice that the path could go under the $x$ axis. a. Write a generating ...
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If $\sum_{n=1}^{\infty} 1/a_n $ diverges, can $\prod_{n=1}^{\infty} (1-z/a_n) $ be analytic anywhere

This question was asked in my complex analysis assignment. Question :If $\sum_{n=1}^{\infty} 1/a_n $ diverges, can $\prod_{n=1}^{\infty} (1-z/a_n) $ be analytic anywhere? I think it can't be as we ...
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On the class of real-valued functions which extensible to entire functions

Let $\mathcal{D}$ be the class of real-valued functions $f(x)$ defined on $\mathbb{R}$, which are extensible to entire functions $\bar{f}(z)$ on $\mathbb{C}$ such that $\bar{f}(x)=f(x)$, for all $x \...
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Function having all order derivatives that are positive is analytic

If a real function $f$ on an interval $I$ has derivatives of all orders which are positive at every point of $I$, how can I prove that $f$ is real analytic? (That is, it equals the Taylor expansion of ...
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Fourier transform of analytic functions and radius of convergence

Let $\phi$ be a fonction in the Schwartz space $\mathscr S(\mathbb R)$ which is also analytic on the real line and such that the radius of convergence on the real line is bounded below by $\rho>0$. ...
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Why does this weird iteration converge to the square root ??

Let $1 < x < 4$ , $a_1 = x$ and $b_1 = 0$. Now consider the (conditional) iterations if $a_n > b_n$ then $a_{n+1} = 4(a_n - b_n - 1)$ $b_{n+1} = 2(b_n + 2)$ else ( $a_n = b_n$ or $a_n < ...
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How to prove that this condition implies that the function is constant

I am trying assignments of my complex analysis course which will not be discussed and I got struck on this question: Let D be the open unit disc centred at origin and $f : D \to \mathbb{C}$ be an ...
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A question in a solved example related to differentiability of an analytic function

This question was part of an assignment and its solution was given but I am having trouble understanding a conclusion. So, I am asking for help here. Question is: Let f: $\mathbb{C} \to \mathbb{C}$ ...
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Examples of non-polynomial analytic homeomorphisms

For any odd positive integer $k$, the map $f_k(x):=x^k$ is both an analytic function and a homeomorphism on $\mathbb{R}$, since $f_k^{-1}(x) = \sqrt[k]{x}$ is bijective an analytic. In fact, this ...
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When Non-Constant Analytic Functions are Real-Unbounded

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be a real analytic function which is non-constant. Are there simple conditions on $f$ to deduce that $$ \sup_{x \in \mathbb{R}^n} \|f(x)\|=\infty ? $$ ...
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Let $f$ have a pole of order $m$ at $z_0$ and let $g$ have a pole of order $n$ at $z_0$. Classify the isolated singularity of $f/g$ at $z_0$.

Please help me understand and how to start. Let $f$ have a pole of order $m$ at $z_0$ and let $g$ have a pole of order $n$ at $z_0$. Classify the isolated singularity of $\displaystyle{f/g}$ at $z_0$....
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Does there exists a sequence of polynomials and rational functions approximating an analytic function uniformly?

This question was asked in my complex analysis quiz and I was absolutely confused on which result to use. Consider the function $f(z)=1/z$ on the annulus $A=[{z \in \mathbb{C} : 1/2 < |z|<2}]$. ...
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Particular example of a non-vanishing analytic function f in unit disc |z|<1.

This question is from Ponnusamy and Silvermann complex analysis section : Cauchy inequality. Also please note that they are not homework problems. I am trying myself. Question: Give an example of a ...
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Complex Analysis and Bernoulli Numbers from $\frac{z}{2} \cot (\frac{z}{2})$

Define the Bernoulli numbers $B_n$ by $\frac{z}{2} \cot (z/2) = 1 - B_1 \frac{z^2}{2!} - B_2 \frac{z^4}{4!} - B_{3} \frac{z^6}{6!} - ...$ Explain why there are no odd terms in this series. What is the ...
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If $u+\iota v$ is analytic function, then $dv$ is equal to?

If $u+\iota v$ is analytic function, then $dv$ is equal to: My attempt: $dv=v_xdx+v_ydy$ and since $u+\iota v$ is analytics, C-R eqns are satisfies and we have $u_x=v_y$ and $u_y=-v_x$. So, $dv=-u_ydx+...
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Does an analytic function $f(x|y)$, $x,y \in C$ holds its analyticity after conditioning on a random variable $y$ e.g. $g(x,y)=E[f(x|y)]$?

Let I have a function $f(x|y)$ is holomorphic (and Analytic in the complex plane) where $y$ is deterministic and $x,y\in C$. In this part $y$ is no more determinstic. $h$ is a random variable; both ...
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Consider the functions $f(z)=x^2+iy^2$ and $g(z)=x^2+y^2+ixy$. At $z=0$,

Consider the functions $f(z)=x^2+iy^2$ and $g(z)=x^2+y^2+ixy$. At $z=0$, (a) $f$ is analytic but not $g$ (b) $g$ is analytic but not $f$ (c) both are analytic (d) neither $f$ nor $g$ is analytic ...
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How do we use limits to prove differentiability of a complex function and then prove the limit unique within the neighborhood of the point?

With complex functions, there are several paths that can be taken to approach a point $z_0$. We usually take the limit $$ f(z)' = \lim_{z \to 0} \frac{f(z_0+z) - f(z_0)}{z} $$ If I take an example, $w ...
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Differentiability versus analyticity domains for complex functions

I'm having some trouble understanding the differences between the concepts of differentiablity and analyticity domains of a complex function. I know that when a complex function $f(z)$ have a complex ...
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Checking the analyticity of a complex function.

This was a question in an assignment that demands the use of Cauchy's Integral Formula in the questions. The question goes like this: Integrate $\displaystyle{g(z)=\frac{e^z}{ze^z-2iz}}$ over $\...
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Mittag-Leffler star of the hypergeometric function $F(a,b;c;z)$

I am trying to determine the Mittlag-Leffler star of the (Gauss) hypergeometric function $F(a,b;c;z)$ centered at $z=0$ for arbitrary $a$, $b$ and with $c\neq 0,-1,-2,\dots$ Here is what I have ...
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Is this approach correct in finding the largest open set on which this function is analytic

This question was part of my assignment in complex analysis . Find the largest open set on which $\displaystyle \int_{0}^{1} \frac{1}{1+tz} dt $ is analytic . I wrote $F(t)= \displaystyle \int_{0}^{...
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Analytic characteristic function convergence

This feels a bit silly, but I can't seem to find a nice reference. I have a weakly converging sequence of random variables $X_n \Rightarrow X$. The characteristic functions $\phi_n(s) := \mathbb{E}[e^{...
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Harmonic and analytic function on complex numbers

If $f(z) = u(x, y)+ iv(x, y)$ is an analytic function in a domain $D$ and $f(z) \ne 0$ for all $z \in D$, show that $φ(x, y) = \ln |f(z)|$ is harmonic in $D$. The above question was taken from Dennis ...
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Integral of an analytic function, also analytic?

The definition of an analytic function is: A function $f$ is (real) analytic on an open set $D$ in the real line if for any $x_0\in D$ one can write $$f(x) = \sum_{n=0}^\infty a_n(x-x_0)^n,$$ in which ...
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Proof that poles on real axis count as “half” a residue when evaluating real integrals [duplicate]

There is a small formula for finding definite integral by complex analysis methods : If $f(x)$ contains cosine and sine functions along with polynomial functions then $f(x)$ can be treated as a real ...
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Justify the equation $\sum_{n=0}^{\infty} \frac{(-1)^n}{m+nk} = \int_{0}^1 \frac{t^{m-1}}{1+t^k} dt$, $m > 0$.

I am trying to solve this problem from my exam prep under real analytic function section. Justify the equation $\sum_{n=0}^{\infty} \frac{(-1)^n}{m+nk} = \int_{0}^1 \frac{t^{m-1}}{1+t^k} dt$, $m > ...
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Proof for sufficient conditions for analyticity

In proof for sufficient condition for analyticity one of the step is The assumption that the first-order partial derivatives of u and v are continuous at the point $(x_0,y_o)$ enables us to write $\...
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Find all functions $f$ analytic in the open unit disk and such that $f(\frac{1}{n})= e^{-n} $ [duplicate]

Find all functions $f$ analytic in the open unit disk and such that $$f(\frac{1}{n})= e^{-n} $$ I don't know how to start
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$f$ holomorphic in $\mathbb{D}$ extends to Analytic function [duplicate]

I have the following question: Prove of give a counterexample: Suppose that $f$ is holomorphic on $\mathbb{D}$ and continuous on its closure. Then, $f$ extends to a analytic map on $B_{(0,R)}$ for ...
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Discussion of the nature of an analytic Function on a changed domain

Suppose I have an analytic Function $f(z)$ on an open unit disk $\mathbb{D}$ centred at origin. But it has zero of order $m$ at $z_0$ in $\mathbb{D}$. So it's quite difficult to me to discuss the ...
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An analytic continuation of the square root along the unit circle

I want to find an analytic continuation of the square root along the unit circle but I am not sure whether I am doing it correctly. Let $C_0$ be the open disk of radius $1$ around $1$, and let $f_0:...
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1answer
39 views

uniform convergence of derivative of the series of analytic functions

Suppose $D\subseteq \mathbb{C}$ is a domain(open and connected set).$f_n:D\to \mathbb{C}$ is a sequence of analytic functions and $f:D\to \mathbb{C}$ is continuous. (i) If $\sum f_n$ converges ...
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Real Analytic continuation [closed]

For which values of $p,q\in[1,\infty)$ the following functions have a real analytic continuation to the whole real line. 1.$f:(0,\infty)\to \mathbb R$ where $f(t)=(1+t^p)^{\frac{q}{p}}$. 2.$g:(-\...
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1answer
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Bounded number of zeros derivatives can have implies analyticity

I'm trying to prove that if $f\in C^{\infty}(]-1,1[,\mathbb{R})$ and there exists a $p \in \mathbb{N}$ so that for all $n\in\mathbb{N}$, $f^{(n)}$ has at most $p$ zeros in $]-1,1[$ then $f$ is ...
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1answer
78 views

Holomorphic function$f: D \to D$ with $f(\frac{1}{2}) =-\frac{1}{2}$ and $f '(\frac{1}{4}) =1$

Does there exist a holomorphic function$f: D \to D$ with $f(\frac{1}{2}) =-\frac{1}{2}$ and $f '(\frac{1}{4}) =1$ where $ D= \{ z \in \mathbb{C} : |z|<1\}$. I cannot use any of Schwarz lemma or ...
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34 views

Use of the Identity Theorem?

I have the following question: Let $s(y)$ and $t(y)$ be real differentiable functions on $y$ with $-\infty < y < \infty$ satisfying $s(0) = 1$ and $t(0) = 0$, with the property that the ...
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25 views

Why is this function defined by iteration analytic?

The function I'm dealing with is reached by solving iteratively the following integral equation ($x$ is real, and $ u $ is a real smooth function that goes to zero on infinity): $$ \chi(x) = 1-\int_{...
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1answer
33 views

Am I applying MAX Modulus principle correctly here?

$(1)$ Find all possible entire functions s. t. $$|f(z)|\leqslant2|z|+1.$$ $(2)$ Prove you have found all such functions. According to my understanding of $MAX$ modulus principle, $|f(z)|$ doesn' t ...

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