1
vote
1answer
18 views

Taylor series of an analytic function that maps the unit disk surjectively onto the upper half plane

Given only that $f(z)$ is analytic and maps the unit disk $|z| < 1$ surjectively to the upper half plane $\Im(z) > 0$, how much can we deduce about $f(z)$? In particular, can we find the radius ...
2
votes
1answer
21 views

Manipulating Taylor series with $\operatorname{Log} z$

I'm trying to prove the following: $$ \operatorname{Log}z = \sum_{n=1}^{\infty} \frac{\left(1-\frac1z\right)^n}{n} $$ for $\left|1-\frac1z\right|<1$. Does anyone have advice on where to ...
1
vote
1answer
24 views

Laurent series of $f(z)=(z^2-1)\mathrm{cos}\frac{1}{z+i}$ in $z_{0}=-i$ and $Res[f(z), -i]$

This is how I've done so far: $$ f(z)=(z^2-1)\mathrm{cos}\frac{1}{z+i} \\w=z+i \;\;\;\; \Rightarrow \;\;\;\; z=w-i \\f(w)=((w-1)^2-1)\mathrm{cos}\frac{1}{w}=(w^2-2wi-2)\mathrm{cos}\frac{1}{w} ...
0
votes
1answer
58 views

Calculating Laurent Series of Complex Function

How does one alternate the Bernoulli number series expansion $$\frac x{e^x - 1}=\sum_{n=0}^{\infty}\frac{B_nx^n}{n!}$$ To calculate the Laurent Series centered at 0 in the annulus of convergence of ...
0
votes
1answer
17 views

Showing uniform continuity of function giving radius of convergence

Let $f$ be an analytic function on an open disk $D$ and let $R(z)$ denote the radius of convergence of the power series of $f$ about a point $z$. Is there an easy way to show that $|R(z_1) - R(z_2)| ...
1
vote
0answers
61 views

To show a power series is a Taylor series

Is it possible to prove if $f(x) = \sum_{n = 0}^\infty a_n(x - a)^n$ then the series is the Taylor series of $f$ without using complex analysis, as done here?
1
vote
1answer
22 views

Finding the convergence radius of a complex laurent series

Find the maximal ring where the following series converges: $$\sum_{n=1}^\infty\frac{3^n+2^n}{(z-5)^n}+\sum_{n=0}^{\infty}\frac{n^2}{20^n}(z-5)^{2n}$$ I think that taking the minimum between the ...
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 $|z-a|>r$. ...
0
votes
0answers
24 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange B├╝rmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
1
vote
1answer
77 views

Quick question on convergence

Shouldn't the radius of convergence be defined as: $$\frac{1}{R} = \lim_{n\rightarrow \infty} \left(a_n \right)^{\frac{1}{n}}$$ Not sure what they are doing above..
1
vote
1answer
34 views

Prove the taylor series of $ \cos(2z)$

First i turned $$\cos(2z) = \frac{e^{2iz} + e^{-2iz}}{2}$$, then using the taylor series of $$e^{z}$$I calculated the taylor series of both arguments. $$\frac{e^{2iz}}{2} = \sum_{n=0}^{\infty ...
0
votes
2answers
28 views

Complex functions and Taylor series

Find the Taylor series arround $z_0=0$ write radius of convergence a) $f(z)=\cosh(z)$ b) $f(z)=\log(z+1)$ I don't know how it works with the complex functions. Could you show me the workflow? I ...
2
votes
3answers
52 views

Convergence of Taylor Series

My professor made this claim about Taylor Series convergence in my Complex Variables class and I am still not entirely convinced (he said it's explained in the textbook and textbook states, "we will ...
3
votes
0answers
65 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
1
vote
1answer
78 views

Complex Taylor and Laurent expansions

Let $f(z):=\dfrac{1}{2-z-z^2}, z\in\mathbb{C}\setminus\left\{ {1, -2}\right\}$. i) Express $f$ in the form $\dfrac{A}{1-z}+\dfrac{B}{2+z}$. [Answer to this is $\dfrac{1/3}{1-z}+\dfrac{1/3}{2+z}$]. ...
1
vote
1answer
72 views

Laurents Series Expansion Complex Analysis

So here is the problem, I am having a lot of trouble with laurents expansions and if you guys even know any sources where I can learn these really well and very simply then that would be a great help. ...
1
vote
1answer
40 views

Taylor Series & complex analysis

I am taking complex analysis. There's a question in the book when trying to prove the theorem, and the theorem goes like this: If $f$ is analytic in the disk $|z-z_0|<R$,then the taylor ...
0
votes
0answers
176 views

Laurent Series and Taylor Expansion of $ 1 / (e^z - 1) $

Could someone please assist me with the second part of the second paragraph, from "By expanding $f_1$..."? I am not convinced that my expansion for $f_1$ is right - I used the standard binomial, ...
1
vote
1answer
55 views

Convergence question and degree of polynomial

I'm currently teaching myself power series and Taylor's theorem for complex analysis and I'm having trouble answering questions of the following form: $1)$ Suppose the power series ...
0
votes
1answer
41 views

What is the series expansion of $f(z)\cdot\exp\left({s\,\log(z)}\right)$?

For analytic $f$, how can I represent the expression $f(z)\cdot\exp\left({s\,\log(z)}\right)$, i.e. $f(z)\cdot z^s$ in the form $$\sum_{n}^\infty\left(\sum_{k}^\infty a_k s^k\right)z^n,$$ at least ...
4
votes
0answers
91 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
4
votes
2answers
199 views

Express $(1-z)^{-1}$ as a power series around $z_0=-1+i$.

I need to express $(1-z)^{-1}$ as a power series in powers of $(z+1-i)$. I would like some guidance on the complex analogue of power series and in writing out this particular case. Many thanks for ...
3
votes
1answer
67 views

Showing $|a_k | \le 1$

Let $A$ be the closed unit disk $A= \{z \in \mathbb{C}: |z| \le 1 \}$. Suppose $f$ is an entire function whose Taylor series centered at the origin is $$\sum_{k=0}^{\infty} a_kz^k$$ and that $f$ maps ...
1
vote
1answer
62 views

Maclaurin Series Complex Numbers

I'm having trouble getting to the right solution on the function ${z^2\over (1+z)^2}$ ${z^2\over (1+z)^2}$ = ${z^2}$${1\over (1+z)^2}$ = ${z^2}$${1\over (1+z)(1+z)}$ = ${z^2}$${A \over (1+z)}$ + ...
1
vote
1answer
51 views

Taylor series expansion - application

I am working on the following: Let $f : \mathbb C \to \mathbb C$ be analytic. Suppose for all $z \in \mathbb C$ hold $f(2z) = 4f(z)$ and $f(1) = 1$. Then $f(z) = z^2$ for all $z \in \mathbb C$. I ...
2
votes
1answer
63 views

Laurent series for $\frac{z}{z+1}$ when $1<|z|<\infty$

Calculate the Laurent series for $\displaystyle\frac{z}{z+1}$ when $1<|z|<\infty$. There is really no singularity here, right? Can I just use a Taylor series, or what should I do?
3
votes
1answer
132 views

Convergence radius of $\sqrt{\cos(z)}$

Compute the first 3 non zero terms of the Taylor expansion of $\sqrt{\cos(z)}$ at $z=0$ and determine its convergence radius, considering only the principal branch of the square root. I've computed ...
1
vote
2answers
34 views

Derivative of a Taylor series

I have a question about when we compute the derivative of a series. If the original series converges inside a region $R$, must its derivative also converge on the same region $R$?
2
votes
2answers
92 views

Product of two Taylor series

I have the following product of two Taylor series: $$f(x)g(x)=\frac{1}{z-1}\frac{1}{z-2}=\sum_{n=0}^{\infty} z^n \sum_{n=0}^{\infty} \frac{1}{2^{n+1}} z^n$$ I wanted to know 2 things: 1st. How can ...
1
vote
3answers
114 views

Taylor series for $e^z\sin(z)$

How can I write the Taylor series for $e^z\sin(z)$ at $z=0$ without making the procedure too complicated? Isn't there an easier way than to compute it's derivatives and find a pattern?
2
votes
1answer
94 views

Laurent series, radii of convergence.

I'm working on the following exercise: Prove that a Laurent series \begin{align*} \sum_{n = -\infty}^\infty a_n(z-z_0)^n = \sum_{n = 0}^\infty a_n(z-z_0)^n + \sum_{n = 1}^\infty ...
2
votes
1answer
871 views

Finding the taylor series of $f(z) = 1/(1+z^2)$.

I am working on the following exercise: Find the Taylor expansion of the function $f(z) = \frac{1}{1+z^2}$ about $z = 3i$. We had the Taylor Series Theorem in the lecture: Let $D \subset ...
3
votes
1answer
50 views

Counterexample: For real functions existence of all higher order derivatives doesn't imply analycity.

In the lecture we had an example for a function $f: \mathbb R \to \mathbb R$, which is not analytic. We defined, that a function is said to be analytic at some point $x_0$ if a Taylor series expansion ...
1
vote
1answer
110 views

How to prove that a complex power series is differentiable

I am always using the following result but I do not know why it is true. So: How to prove the following statement: Suppose the complex power series $\sum_{n = 0}^\infty a_n(z-z_0)^n$ has radius of ...
1
vote
1answer
61 views

Find “singular expansion” of a function

I have the function $(1-z)^{-z}$, analytic except on $\mathbb{R}_{\geq 1}$ Now in the text, it says the "singular expansion" at $z=1$ is $\displaystyle \frac{1}{1-z} + \log(1-z)+O((1-z)^{1/2})$ I'm ...
1
vote
0answers
53 views

Computation of the remainder term on a Taylor expansion using contour integrals

I am not really used to the methods of complex analysis, I would like to know for basic monotonic functions like exp(x), log(x), sqrt(x), powers (x^n) and trigonometric functions defined on an real ...
0
votes
1answer
80 views

exp(x) for imaginary numbers

Well, I know how to get the $e^x$ function polynomial expansion, but how do I know that this is also valid for imaginary numbers, like $i\pi$? I know that the ...
3
votes
1answer
124 views

Taylor series with Fibonacci coefficients

Let $\{a_n\}$ be the Fibonacci numbers given by $a_0=0,a_1=1,a_{n+2}=a_{n+1}+a_n$ for $n\geq 0$. Prove that $f(z)=a_0+a_1z+a_2z^2+\ldots$ is a rational function, and determine which rational ...
2
votes
1answer
101 views

Evaluate $I=\int_{\gamma (0,1)}\frac{zdz}{(z^2-4z+1)^2}$ using Taylor's Theorem

I've shown that $f(z)=\frac{z}{(z^2-4z+1)^2}$ is holomorphic apart from at points $\alpha=2-\sqrt3$ and $\beta=2+\sqrt3$ and that the talyor coefficient of $g(z)=\frac{z}{(z-\beta)^2}$ centred at ...
1
vote
1answer
77 views

McLaurin series of complex function

I've got a function $g(z) = \frac {(1-z)(e^z + e^{-z})}{e^z - e^{-z}}$. I have to find coefficients $c_0, c_1, c_2, c_3$ of McLaurin's series of function $g$ (which is $\sum_{n=0}^{\infty} c_n z^n ...
1
vote
0answers
39 views

Taylor series convergence

$$f(z)=\int^z_0 \frac{\zeta-\sin(\zeta)}{\zeta^2+4} \, d\zeta$$ I am supposed to find the convergence radius of its Taylor series at point $a=2$. I can find the radius in simple cases by finding ...
7
votes
2answers
129 views

What are the properties of the roots of the incomplete/finite exponential series?

Playing around with the incomplete/finite exponential series $$f_N(x) := \sum_{k=0}^N \frac{z^k}{k!} \stackrel{N\to\infty}\longrightarrow e^z$$ for some values on alpha (e.g. ...
4
votes
1answer
135 views

Maclaurin series for $e^z /\cos z$.

I want to find the Maclaurin series for the function $$f(z)=\frac{e^z}{\cos z}.$$ Right away I can tell that the radius of convergence will be $\pi/2$, since it's the distance to the nearest ...
0
votes
1answer
219 views

What is the difference between Taylor series and Laurent series?

Can someone intuitively describe what is the difference between Taylor series and Laurent series? Also, what is the most general formula for both?
2
votes
1answer
160 views

Complex Analysis - Taylor Series

The question reads as follows: Let the function $f$ be given by $f(z)=\frac{z}{2}+\frac{z}{e^z-1}$ if $z\neq0$ and $f(z)=1$ if $z=0$. Show that $f$ is analytic at $z=0$ and that $f(z)=f(-z)$. ...
2
votes
1answer
211 views

Intuition regarding Taylor series for $\frac{e^z}{1-3z}$.

The question asks me to find the Taylor series for $$f(z)=\frac{e^z}{1-3z}.$$ The radius of convergence is $|z|<1/3$ and I know the expansions for $e^z$ and $1/(1-3z)$ are \begin{align} e^z ...
3
votes
2answers
180 views

Is there a closed form expression for the Taylor series of exp((f(z))?

Given a holomorphic function $f(z) = \sum_{k=0}^\infty f_k z^k/k!$, is there a readable formula for the Taylor series of $\exp(f(z))$? Using the chain and product rules, one can obtain $$\partial_z ...
4
votes
2answers
212 views

Find the principal part of the Laurent expansion of $\frac{(z^{2}-2z-3)^{2}}{\cos(\pi z)+1}$ around $z_{0}=1$

Problem: The function $f(z) = \frac{(z^{2}-2z-3)^{2}}{\cos(\pi z)+1}$ has an isolated singularity at $z_0=1$. a) Find the principal (singular) part of the Laurent expansion of $f$ in a punctured ...
1
vote
0answers
83 views

What is a typical example of taylor series for $f(z)$ that converges iff $\operatorname{Re}(z)>0$

What is a typical example of taylor series for $f(z)$ that converges iff $\operatorname{Re}(z)>0$ where $\operatorname{Re}(z)>0$ is not a natural boundary ?
1
vote
0answers
55 views

Looking for examples where $f(z)=\operatorname{inv} \int_{0}^{z} g(z)\, dz$ with $f(z)$ entire and $g(z)$ not meromorphic.

I'm looking for examples where $f(z)=\operatorname{inv}\int_{0}^{z} g(z) \, dz$ with $f(z)$ entire and $g(z)$ not meromorphic. For clarity, by $\operatorname{inv}$, I mean the functional inverse. ...