# 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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### 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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### 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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### 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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### 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 ...