# Tagged Questions

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 ...

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### Analytic Functions

Prove or give a counter-example: If $f_j(j=1,2,...,n)$ is analytic on the domain $D$ such that $\sum_{j=1}^n |f_j(z)|^2$ is constant on $D$. Then each $f_j$ is a constant function. Inputs: We know ...
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### Linear combination of a real-valued function and its inverse is analytic Implies the real-valued function is analytic.

If $u$ is a real-valued function on a disc $\Delta_R$ such that $u^{-1}+iu$ is analytic on $\Delta_R$, then does this imply that $u$ is analytic on $\Delta_R$? I am actually trying to prove some ...
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### Symmetry in Analytic Continuation of $\sum_{n=0}^{\infty} e^{-x E_n}$

Suppose we have the following function: $$F(x)=\sum_{n=0}^{\infty} e^{-x E_n}$$ Where $E_n$ is a positive monotonically increasing sequence, bounded from below. Is there a general condition on $E_n$ ...
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### Analytic function zero in the given disk

I need to show that f(z)=0 for all z \in D(0,2). From the analyticity of f in D(o,2), I know by Cauchy's theorem it's integral in |z|<2 is zeros. And clearly the integrand has a pole at 1/(n+1) ...
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### $(B_t)_{t\ge 0}$ be Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t}$ is an analytic function. [closed]

Let $(B_t)_{t\ge 0}$ be a one-dimensional Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t} \; \text{for all} \; t\ge 0, \xi \in \mathbb{R}$ is an analytic function. A more general question ...
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### Proving directly (without integration) that differentiable implies analytic for one complex variable

From Titchmarsh, Theory of Functions, 2nd ed 1939: [His definition of an analytic function appears to be that $f(z)$ is analytic at $z_0$ if $\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}$ exists (...
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### Describe all real-valued functions which are analytic on $\mathbb{C}$

This is a homework question, so if I am wrong please do not explicitly give me the answer. Question: Describe all real-valued functions which are analytic on $\mathbb{C}$. My Answer: Given that we ...
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### Analyticity of $\dfrac{1}{z}$ vs. $\dfrac{1}{z^2}$

I am learning complex analysis on my own. I am familiar with the theorems, and I am able to compute by hand and get correct results. But there is something that escapes me. What is the criteria for ...
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### Help to understand this step of the proof of holomorphic functions are analytic

I'm trying to understand the proof of the analyticity of holomorphic functions. The step I don't understand is when one interchanges the series and the integral. In every source I have read, it says ...
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### Find an analytic continuation

Let $f(z)=\sum_{j=0}^{\infty}z^j$ for $|z|<1$. For what values of $\alpha$ ($|\alpha|<1$) does the Taylor expansion of f(z) about $z=\alpha$ yield a direct analytic continuaton of f(z) to a disk ...
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### Analyticity and differentiability of complex functions

I understand what analytic functions are and what differentiability of a complex function means but I have been reading "advanced engineering mathematics by kreyszig" and it says that the concept of ...
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### Finding residues at a point $a$ where $a$ is a pole.

I am faced with the following problem: (a) Find $\displaystyle res_{a} \frac{\varphi(z)}{(z-a)^{n}}$ where $\varphi$ is a given function analytic at $a$, $\varphi(a) \neq 0$, and $n$ is a positive ...
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### Why is $\{(1/(n+1),n)\mid n\in\mathbb{N}\}$ a subanalytic set?

I have a question about subanalytic sets. Why is $\{(1/(n+1),n)\mid n\in\mathbb{N}\}$ a subanalytic set but its projection onto $\mathbf{R}\times \{0\}$ is not? Thanks in advance,
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### Finding an analytic function

enter image description here I cannot find any such function. Also, why would a function that is analytic at 0 following these criteria not be analytic on (-2,0). Thanks in advance for your help.
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### Analytic continuation of a real function

I know that for $U \subset _{open} \mathbb{C}$, if a function $f$ is analytic on $U$ and if $f$ can be extended to the whole complex plane, this extension is unique. Now i am wondering if this is ...
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### Identity theorem for forms (on analytic manifolds)

It is well-known that for holomorphic functions the Identity theorem holds: if two holomorphic functions agree on an open subset, they agree everywhere (assuming the manifold connected). I would ...
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### Can a function “grow too fast” to be real analytic?

Does there exist a continuous function $\: f : \mathbf{R} \to \mathbf{R} \:$ such that for all real analytic functions $\: g : \mathbf{R} \to \mathbf{R} \:$, for all real numbers $x$, there exists ...
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### Application of Schawarz lemma??

Let $f$ be an analytic function defined on the unit disc $D=\{z:|z|<1$. If $|f(z)|\leq 1-|z|$ for all $z\in D$ then show that $f$ is a zero function on $D$. We have $|f(z)|\leq 1-|z|$ for all $z$ ...
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### an analytic function being zero

Let $f$ be an analytic function defined on the unit disc $D=\{z:|z|<1\}$. If $|f(z)|\leq 1-|z|$ for all $z\in D$ then show that $f$ is a zero function on $D$. Please give only hints.
### $f$ has pole of order $m$ and $g$ has a pole of order $n$ at $z_{0}$, show $f+g$ has isolated singular point there
I am faced with the following problem: Suppose $f(z)$ and $g(z)$ have poles of order $m$ and $n$ respectively, at a point $z_{0} \in \mathbb{C}$ with $m \neq n.$ Show that $z_{0}$ is an isolated ...
### Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$.
Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$. I'm thinking about a contradiction proof. Assuming ...