# Questions tagged [analytic-functions]

For questions about analytic functions, which are real or complex functions locally given by a convergent power series.

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### Functional equation $F(x+1) = x + F(x)$ and analytical expansion of Riemann zeta-function

I have found a solution for functional equation $F(x+1) = x + F(x) \tag1$ $F(x) = \frac{x(x-1)}{2}+C$, where $C$ - some constant However, I have also tried to recursively expand equation $(1)$ at $0$...
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### Analytic function on the complex plane except the negative real axis

Let $G$ and $H$ be defined by $G=\mathbb{C}\setminus\{z=x+iy: x\le 0, y=0\}$ $H=\mathbb{C}\setminus\{z=x+iy: x\in \mathbb{Z},x\le 0, y=0\}$ Suppose $f:G\to \mathbb{C}$ and $g:H \to \mathbb{C}$ ...
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### Complex function with values on the unit circle copied everywhere

If $f:\mathbb{C}\setminus \{0\}\to \mathbb{C}$ is a function such that $f(z)=f(\frac z{|z|})$ and its restriction to unit circle is continous,then $(1)\lim _{z\to 0} f(z)$ exist. $(2)f$ is analytic ...
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### An application of Rouché's theorem

i need help to proof the next: Let $f$ be analytic at $D$ minus a finite numbers of interior points where $f$ has poles. Show that if $0<|f(z)|<1$ over $\partial D$, then the number of poles of ...
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### Bounding an analytic function on the closed unit disk [duplicate]

I have just encountered the following exercise which has me quite stumped: Let $f$ be analytic on the closed unit disk, and we assume that $| f(z) | \leq 1$ for all $z$'s in this set. We also ...
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### Cauchy product between a power series and a vector of power series

From here, it is known that given any formal power series in the following form ($a_0$ may or may not be zero), $$g(z) = \sum_{i=0}^\infty a_i z^{i} \quad \quad (1)$$ there exists another non-...
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### Prove that all singularity of $\frac{1}{e^z+3z}$ is of order 1

This is a problem from my past QUal: "Prove that all singularity of $$\frac{1}{e^z+3z}$$ is of order 1. You don't need to find the singularities." Usually this kind of problem is easy to me. My ...
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### A subsequence of $f^n$ converge pointwise where $f$ is analytic on the unit ball

This is a problem from my past Qual: "Let $\Omega$ be the unit disk and $f:\Omega\to \Omega$ be an analytic function. COnsider the sequence $\{f^n(z)\}$ where $f^n=f\circ\ldots\circ f$ ($n$--times). ...
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### pole and zero cancellation for power series

I typically work on rational functions. There, the pole and zero cancellation can be done straightforwardly. For example, consider $$G(z)= \frac{z-1}{z+2},$$ I can cancel the pole $-2$ of $G(z)$ by ...
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### Non-constant analytic function on an open connected subset $U$ of $\mathbb{C}$

Let $U\subset \mathbb{C}$ be an open connected set and $f:U\rightarrow \mathbb{C}$ be a non-constant analytic function. Consider the following sets:- $X=\{z\in U:f(z)=0\}$ $Y=\{z\in U:f$ ...
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### Analytic function on an open subset $U$ of $\mathbb{C}$

Let $U$ be an open subset of $\mathbb{C}$ and $f:U \rightarrow \mathbb{C}$ be an analytic function.Then which of the following are true? $(a)$ If $f$ is one-one, then $f(U)$ is open in $\mathbb{C}$ ...
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### derivative of hypergeometric function

I am doing an integral in Mathematica and I find the solution contains derivatives of hypergeometric functions. I would like (ideally) a simple analytic form for these. I have tried HypExp mathematica ...
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### Regular singular, Irregular singular & Ordinary points of an equation

I'm currently trying to re pick up physics after a few years off.. but I'm not sure on the current question and I have no idea what it actually means. My mathematics is not great but it reminds me of ...
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### zeros of a function in a simple closed contour

i would really appreciate if you can help me with the following demonstration please: Let $C$ be a closed simple contour such that $| f (z) | = cte$ ($f$ is analytic inside $C$ and is not an ...
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### Derivative of function f at origin provided 2Ref + 3Imf = 1.

Let $f = u + iv$ be analytic in a connected open set. If $u$ and $v$ satisfies $2u + 3v = 1$, then what will be the value of $f'(0)$ ? Since $f$ is analytic we know it satisfies Cauchy Riemann ...
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### Use Schwarz' Lemma and/or Maximum Modulus Principle to Prove This Proposition

Suppose the following conditions hold true: (1) The function $f$ is analytic and contractive in the open unit disk. (2) $f(0)=0$. (3) $\exists z_1 \ne z_2 \in B(0,1)$, $|z_1|=|z_2|$, $f(z_1)=f(z_2)$...
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### Dilogarithm of a negative real number outside unit circle

The dilogarithm is defined in $\mathbb{C}$ as $$Li_2(z) = -\int_0^1 \frac{\ln(1 - zt)}{t} dt$$ Because $1-zt \in \mathbb{C}$, then you can write $\ln(1 - zt) = \ln|1 - zt| + i·\arg(1 - zt)$ As ...
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### Analytic function's proof verification - I didnt use all the given condition. Where's the mistake?

let $f\left(x\right)=\sum_{n=0}^{\infty}a_{n}x^{n}$ be with positive convergence radius $R>0$ and assume that exists sequence $y_n \in(0,R)$ decreasing monotonic such that $y_{n}\to0$ and ...
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### 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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### Given a smooth, non analytic curve, obtain a non analytic smooth function.

I managed to prove (by methods of complex analysis) that the curve $$f:[0,1]\to \mathbb{C};\\ x\mapsto e^{2\pi ix}+\sum_{n=5}^\infty \frac{\left(e^{2\pi ix}\right)^{(2^n)}}{n!}$$ is a smooth, non ...
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### Analytic function $\sum\limits_{n=1}^{\infty}\frac {z^{2n-1}}{2n-1}.$

Let $f:\{z:\|z\|<1\}\rightarrow \{z:-\frac{\pi}4<\operatorname{Im}(z)<\frac {\pi}4\}$ such that $f (z)=\sum\limits_{n=1}^\infty\dfrac{z^{2n-1}}{2n-1}$. How to prove that $f(z)$ is analytic ...
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### How to prove that $\frac{1}{x^{k}}$ is analytic in $\mathbb{R} \setminus $

I want to prove that $\frac{1}{x^{k}}$ is analytic in $\mathbb{R} \setminus $ for |x|<1 its obvious. i want to prove for the rest of $\mathbb{R}$ so i tried that: I proved by ...
let $f$ be analytic function in segment A. and assume $\in A$ and $f(0)=0$ prove that exists $n \in N$ and analytic function $g(x)$ in segment A such that: $f(x)=x^n\cdot g(x)$ for ...