1
vote
2answers
19 views

Series Coefficient Convergence implies Uniform Convergence

Trying to find a reference for the following. Define the entire functions, $$f_n(x)=\sum_{k=0}^\infty a_{n,k}x^k\ \ \ \ \ \ \ \ \ \ \ f(x)=\sum_{k=0}^\infty a_kx^k.$$ If for each $k$, ...
0
votes
1answer
47 views

Find the radius of convergence of the Power series $1+z+(z^2)/(2^2)+(z^3)/(3!)+(z^4)/(2^4)+…$

Find the radius of convergence of the Power series $$1+z+\frac{z^2}{2^2}+\frac{z^3}{3!}+\frac{z^4}{2^4}+\frac{z^5}{5!}\cdots $$ Put the series in the form ...
1
vote
1answer
43 views

Determining a power series of $f(z)=\exp(z^{2})$

Let $f(z)=\exp(z^{2})$. Determine a power series of $f$, i.e the coefficients $a_{k}$ such that $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ for all $z\in \mathbb{C}$, where $a_{k}=f^{(k)}(0)/k!$. First ...
0
votes
0answers
26 views

Power series with complex variables inequality

I am struggling to prove the following inequality: For $z \in \mathbb{C}, r \in \mathbb{R}, n \in \mathbb{N}$, if $|z| \leq r$ and $1 \leq r < n$ then ...
1
vote
1answer
27 views

Radius of convergence of powerseries containing $(\log n)^n$

$$ \begin{align} \sum_{n=2}^\infty (\log n)^n(z+1)^{n^2} \end{align} $$ What is the radius of convergence of this power-series? I tried applying the root test and the ratio test , but I couldn't ...
0
votes
2answers
47 views

Disk of convergence of the series $ \sum\limits_{n=1}^\infty n!\,(z-i)^{n!} $

$$ \sum_{n=1}^\infty n!(z-i)^{n!} $$ Find the disk of convergence of this powerseries. Can I set $n!=k$ and then deal with $\sum_{n=1}^\infty k z^k$ . On another note $\frac{z^{(n+1)!}}{z^{n!}}$ ...
0
votes
2answers
39 views

Radius of convergence of powerseries $\sum_{n=1}^\infty \frac{(-1^n)}{n!}z^n$

$$ \begin{align} \sum_{n=1}^\infty \frac{(-1)^n}{n!}z^n \end{align} $$ Find the radius of convergence of this powerseries. To determine the radius of convergence should I split it into two separate ...
2
votes
0answers
49 views

$f\in C^\omega ((a-R,a+R),\mathbb{R})$ [closed]

We discuss the following question in the field of real numbers . A a power series $f(x)= \sum_{n=0}^\infty a_n (x-a )^n$ converges in $(a-R,a+R).$ Prove: $$\forall x_0\in\left(a-R,a+R \right), ...
0
votes
1answer
36 views

Upper bound for modulus of a function

Let $f(t,x)$ be a bounded and continuous function on $\mathbb{R}_t \times \mathcal{U}$ where $\mathcal{U}$ is an open neighborhood of $0 \in \mathbb{C}_x$. Moreover, assume that for each fixed $t$, ...
3
votes
1answer
42 views

Showing the following function is entire…

The full problem asks about the following function using it's Maclaurin series: $$f(x)=\left\{ \begin{array}{lr} \frac{\sin(z)}{z} & : z \neq 0\\ \;\;\;\;1 & : z=0 \end{array} \right.$$ I've ...
14
votes
3answers
342 views

What is the Riemann surface of $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$?

The following appears as the second-to-last problem of Stewart's Complex Analysis: Describe the Riemann surface of the function $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$. This problem ...
1
vote
1answer
30 views

Analytic Extension: Imaginary Stripe

I was always wondering the following: Given a real analytic function there exists a positive radius of convergence for every point. This won't be affected by allowing complex numbers so it extends ...
1
vote
1answer
36 views

Question on radius of convergence

Can anyone help me with the following problem: I have a solid geometric picture of what is going on in my head, but I can't seem to turn that into a proof.
1
vote
1answer
37 views

An example of a complex power series. [closed]

I am looking for a complex power series which is convergent for some $z\in\Bbb{C}$ but not absolutely convergent. In other words, $a_0+a_1z+a_2z+\dots$ is convergent but ...
0
votes
0answers
50 views

Find the Laurent Expansion of $f(z)=\frac{1}{z+i}$

Find the Laurent Expansion of $f(z)=\frac{1}{z+i}; f(z)=\frac{1}{(z-i)^2}$ and $f(z)=e^{(z-1)^-1}$ Good evening, I have been trying to solve the above exercises. However, I'm not sure if my procedure ...
2
votes
1answer
57 views

given analytic $f(z)$ in $f(z)/(1-z)$ , derivative $f '(z)$ seems to have singularity at $z=1$

Quick version: I want $f'(1)$, where $$F(z)=\frac{f(z)}{1-z}$$ with $f$ analytic at $z=1$. But when I follow a seemingly valid line of reasoning, I reach the conclusion that $f'(z)$ is not analytic ...
0
votes
2answers
65 views

Complex Power Series

So, I'm trying to find the power series of ${1\over 1-z+z^2} around the point z=0.$ After some rather easy algebra I've determined the expression to be $${1\over z-(1+i\sqrt{3})/2} {1\over ...
2
votes
3answers
54 views

meromorphic function with a pole on the unit circle diverges

Let $f$ be a meromorphic function in a neighborhood of the closed unit disk $\bar{\mathbb{D}}$. Suppose that $f$ is holomorphic in $\mathbb{D}$ and $$ f(z) = \sum_{n=0}^\infty a_n z^n $$ for $z \in ...
2
votes
1answer
77 views

Power series difficulty

How would I find the region of convergence of the series of $\frac{1}{n^3}(\frac{z+1}{z-1})^n$. I thought about rewriting $\frac{z+1}{z-1}$ as $\frac{2}{z-1}+1$ but I don't think that helps. Thanks
0
votes
2answers
28 views

Radius of convergence query

Find the radius of convergence of the series of $\frac{2^n(4z-8)^n}{n}$ My answer: $(4z-8)^n=4^n(z-2)^n=2^{2n}(z-2)^n$. Let $c_{n}=\frac{2^{3n}}{n}$. Then $\frac{c_{n}}{c_{n+1}}=\frac{n+1}{2n}$ so ...
0
votes
0answers
36 views

Confused by a Laplace transform of $f(t)=t^ne^{at}$

Having looked at my lecture notes I was confused by the following part of a derivation of a Laplace transform for the function $\;f(t)=t^ne^{at} ,\quad n\ge0,\; a \in \mathbb{C}, \; f(t)=0 \;\forall ...
6
votes
1answer
43 views

Divergence set at radius of convergence

I came up with this question on my own while I was musing around reviewing notes. After unsuccessful Google search (thwarted by a deluge amount of webpages on basic calculus), I decided to ask here. ...
2
votes
2answers
43 views

how to find convergence and divergence of the series [closed]

consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then find whether the above series ...
2
votes
1answer
22 views

Application of Rouché: Equality of a power series and a finite series

Let $f(z) = \sum_0^\infty{a_n z_0^n}$ be a complex power series with radius of convergence $R>0$ and let $z_0 \epsilon \, \mathcal{U}_R(0)$ an arbitrary point. I need to show with $Rouché$ : For ...
1
vote
0answers
12 views

Taylor's expansion of the singular part of an analytic function

Assume $f$ is analytic on the annulus $R_1<|z-a|<R_2$. Assume $R_1<r<|z-a|$. Define $f_2$ by $$f_2(z)=\frac1{2\pi i}\int_{|x-a|=r}\frac{f(x)dx}{x-z}$$ $f_2$ is analytic on $|z-a|>r$. ...
1
vote
1answer
32 views

A holomorphic function $f$ has an essential singularity in $0$ iff $\exists(z_k)_k$ s.t. $z_k\to 0$ and $|z_k^mf(z_k)|\to\infty$ for all $m$

Let $f:\mathbb{C}\setminus\left\{0\right\}\to\mathbb{C}$ be a holomorphic function $\Rightarrow$ $f$ has an essential singularity in $0$ if and only if $\forall m\in\mathbb{N}:\exists ...
2
votes
1answer
45 views

Convergence of a Complex Power Series at the radius of convergence

I am currently reviewing some complex analysis, and have come across this question which I absolutely have no idea on how to attempt: Suppose the radius of convergence of the power series $f(z) = ...
0
votes
1answer
92 views

If $f(z):=\sum_{n=0}^\infty a_nz^{-n}$ is compact convergent, then $f$ is holomorphic

Let $\left(a_n\right)_{n\in\mathbb{N}}\subset\mathbb{C}$ such that $$f(z):=\sum_{n=0}^\infty a_nz^{-n}$$ is compact convergent on $B_r(0)\setminus\left\{0\right\}$. I want to show: $f$ is ...
0
votes
0answers
27 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
0
votes
1answer
34 views

Radius of convergence of entire function

Let $f$ be an entire function on the complex plane. Is the radius of convergence of $f$ around any point $z_0$ infinite? If so, why? Thank you.
2
votes
1answer
46 views

Showing integral on contour tends to zero

I'm trying to prove: $$\int \frac{e^{t(z+\frac{1}{z})}}{z^2} = \sum_0 ^{\infty} b_m t^{2m+1}$$ Where the integral is over a contour centre the origin, radius R, and the $b_m$ are some coefficients. ...
1
vote
3answers
68 views

Laurent series and residue of $f(x)=\frac{1}{1+e^z}$

I am having trouble trying to expand this function using Laurent series, and finding the residue$$f(x)=\frac{1}{1+e^z}$$ If I replace $e^z$ with its series I get ...
0
votes
2answers
28 views

finding the residue of the following

I can find $$Res_{z=0} \frac{\sin z}{z^4} $$ but stuck with finding $$Res_{z=0} \frac{\cot z}{z^4} $$ so please help me
3
votes
0answers
74 views

Series that diverges at infinitely many points on the unit circle

Initial problem Given $A=\{\alpha_1,\dots,\alpha_k\}$ with $|\alpha_i|=1$, does there exist a power series $\sum a_nz^n$ that converges everywhere on the unit circle except when $z\in A?$ ...
2
votes
0answers
105 views

Prove that periodic analytic function can be written as $\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$

This question involves the following homework problem: PROBLEM Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form ...
3
votes
1answer
32 views

Radius of Convergence for a Complex Function

I'm really rusty on my series convergence, and I guess I was more asking for a bit of clarification on a question I'm working on. I've been asked to find the power series expansion for $\frac{1}{3 - ...
1
vote
2answers
34 views

a series derived from a holomorphic function converges implies that the coefficients converge to $0$

Let $D=\{z\in\mathbb{C}\mid |z|<2\}$. Let $f:D\setminus\{\frac{i}{2}\}\longrightarrow \mathbb{C}$ be holomorphic with $f(z)=\sum_{n=0}^\infty a_nz^n$ for any $|z|<\frac{1}{2}$. Suppose $a_n\neq ...
4
votes
1answer
49 views

Complex power series divergent and convergent on dense subsets of the circumference of convergence?

Is it possible to have a complex power series $ \sum a_nz^n $ with radius of convergence R such that the series diverges on a dense subset of the circumference of convergence and converges on another ...
1
vote
0answers
26 views

How do we show $\ln z=\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}(z-1)^n$ for all $z\in\mathbb{C}$ with $|z-1|<1$?

Let $$g:B_1(1):=\left\{z\in\mathbb{C} :|z-1|<1\right\}\to\mathbb{C}\;,\;\;\;z\mapsto\ln z-\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}(z-1)^n$$ (1) In a first step, I'm asked to show, that $g$ is ...
1
vote
0answers
35 views

Power series expansion of $e^{\frac{c}{2}(z-1/z)}$

Show that \begin{equation} e^{\frac{c}{2}(z-1/z)}=\sum_{n=-\infty}^{\infty}a_nz^n \end{equation} where \begin{equation} a_n:=\frac{1}{2\pi}\int_0^{2\pi}\cos(n\theta-c\sin(\theta))d\theta ...
0
votes
1answer
20 views

Laurent series of quotient

If I have two functions $f,g$ that are holomorphic around a point $z_0 \in \mathbb{C}$. Assume the Laurent series are known and both $f$ and $g$ have a finite principal part. $$f(z) = ...
0
votes
1answer
70 views

Analysis of singularities and taylor representation of $f(z)=\frac{z^2-1}{\sin \pi z}$

Let $$f(z)=\frac{z^2-1}{\sin \pi z}$$ A) Find all singulartities of $f$ in $\mathbb{C}$ and classify each as a pole (specifying the order), essential, removable, or other. B) Explain why $f(z)$ has ...
3
votes
0answers
58 views

Prove that $e^{\ln{z}}=z$ from the power series

For my course in complex analysis we have to prove that the trivial relation $e^{\ln{z}}=z$. We are given the series for $\ln z$: $$f(w)=\sum_{n=0}^\infty (-1)^{n+1}\frac{w^n}{n}$$ $$\ln z = ...
0
votes
0answers
43 views

Properties of power series and their analytic continuation

Suppose a power series $$\sum_{k=0}^\infty a_k z^k$$ is valid for $|z|<R$, and can be analytically continued to some function $f(z)$, for all $z\in\mathbb{C}$ , except for a finite number of points ...
4
votes
1answer
43 views

Power series with differentiable coefficients

Suppose for each $s$ in an open interval, $P_s(x)=\sum_{k=0}^\infty a_k(s) x^k$ is a power series with radius of convergence greater than R, where each $a_k(s)$ is differentiable. My question is: Is ...
1
vote
1answer
57 views

$\frac{1}{(1+s^{2}) (1+t^{2})}$ real analytic in $\mathbb R^{2}$ but not real-entire; why?

A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point $(s_{0}, t_{0})$ in there corresponds an expansion with ...
0
votes
2answers
78 views

What's the behavior of $\displaystyle\sum_{n=1}^\infty (z+\sqrt{5}+2i)^{n!}$ outside its radius of convergence?

I want to check the behavior of $$\displaystyle\sum_{n=1}^\infty (z+\sqrt{5}+2i)^{n!}$$ outside its radius of convergence. I've tried to use the ratio test as follows: ...
1
vote
1answer
37 views

Moving Center of Power Series

Given a power series: $$\lim_{N\to\infty}\sum_{k=0}^N A_k (z-a)^k$$ I expand the powers: $$\lim_{N\to\infty}\sum_{l=0}^N(\sum_{k=l}^N A_k \binom{k}{l}(-1)^{k-l}a^{k-l})z^l$$ But here I face the ...
1
vote
1answer
24 views

$\left(\frac{a_n}{n^k}\right)_n$ is bounded implies $\sum_{n=0}^\infty a_nz^n$ has a radius of convergence $\ge 1$

Let $$\left(\frac{a_n}{n^k}\right)_n\subset\mathbb{C}\;\;\;\;\;(k\in\mathbb{N})$$ be a boundet sequence. I want to show that the power series $$\sum_{n=0}^\infty a_nz^n\;\;\;\;\;(a_n,z\in\mathbb{C})$$ ...
1
vote
0answers
45 views

A theoretical question regarding Frobenius method

The following is a theoretical question regarding Frobenius method. Let $b(x),c(x)$ be real functions analytic at 0. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. ...