The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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Definition and analyticity of $T^z$ where $T$ is a positive operator

Let $H$ be a Hilbert space. Suppose that $T\colon D(T) \to H$ is a positive selfadjoint operator where $D(T)$ is the domain of $T$. The spectrum $\sigma(T)$ of the operator $T$ is a subset of ...
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18 views

Showing that two given functions are harmonic

I'm preparing for my complex analysis midterm on Thursday and our professor gave us the following as a practice problem: I'm a bit confused on how to approach part (a). Here's my train of thought: ...
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17 views

an inequality derived from conformal automorphisms of unit disk

Let $f$ be a holomorphic function on $D(0,1)$ such that $|f(z)|<1$ for all $z\in D(0,1)$. I have obtained $$ \frac{|f(0)|-|z|}{1+|f(0)||z|}\leq |f(z)|\leq \frac{|f(0)|+|z|}{1-|f(0)||z|}. $$ Is it ...
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1answer
31 views

Pair of functions $F(x)$ (transcendental),$A(x)$ (algebraic) with expanded series of positive integer coefficient linked by derivative

$$F(x)=\sum_0^{\infty}b_k x^k,b_k\in \mathcal{N} \bigcup 0,\exists M \space b_k \leq M^k$$. $$A(x)=\sum_0^{\infty}a_k x^k,a_k\in \mathcal{N} \bigcup 0,\exists M \space a_k \leq L^k$$ where $F(x)$ is ...
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1answer
23 views

Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Also note that though I were able to resolve the first problem the second one is still ...
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1answer
31 views

Showing that $u(x, \, y) = \ln(x^2 + y^2)$ is harmonic without computing partial derivatives

I'm trying to show that $u(x, \, y) = \ln(x^2 + y^2)$ is a harmonic function, without explicitly computing the partial derivatives and showing that $u_{xx} + u_{yy} = 0$. I believe that it would ...
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8 views

Identifying holomorphic sections of a line bundle with homogeneous functionals

I was told that given a holomorphic line bundle $L \to X$ we can identify the space of sections $\Gamma(X,L^k)$ with the space of $k$-homogeneous holomorphic functions on $L^*$ -- let's call the ...
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2answers
110 views

roots of a polynomial inside a circle

I am asked to show that for $n$ larger or equal to $2,$ the roots of $1 + z + z^{n}$ lie inside the circle $\|z\| = 1 + \frac{1}{n-1}$ Attempt1: Induction for the case $n = 2,$ the roots of $1 + z + ...
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1answer
30 views

The functions $\{f_n(x) = n\}$ are analytic and each miss the points $-2, -3$. But, they are not a normal family. So what am I missing. Thanks.

Here is a theorem of Montel: Let $\mathcal{F}$ be a family of analytic functions defined on a domain $\Omega$ . If there are two fixed complex numbers $a$ and $b$ that are omitted from the range of ...
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1answer
22 views

Riemann removable singularity theorem for annuli

Let $\mathbb{D}^*=\{z \in \mathbb{C} \ | \ 0 < |z| < 1 \}$ denote the unit punctured disk in the complex plane. Riemann's theorem about removable singularities in particular implies the ...
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2answers
44 views

complex variable integral using residue theorem [duplicate]

I am asked to calculate a complex integral. how to compute $\displaystyle \int \limits_{-\infty}^{\infty}\frac{x^4}{1+x^8}dx$ with residue theorem?
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1answer
48 views

Let $f(z) = \sum_{n = 0}^\infty a_nz^n$ be the Taylor series around $0$. Prove that lim $a_n/a_{n+1} = z_0.$ [duplicate]

Let $f(z) = \sum_{n = 0}^\infty a_nz^n$ be the Taylor series around $0$ of a function which is analytic in $\mathbb{C}$ \ ${z_0}$, $z_0\neq 0$ and has only a simple pole at $z_0.$ Prove that $lim_{n ...
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46 views

Any other operators that may convert algebraic function into transcendental ones

As we know, the integral may convert or map a rational function or algebraic function into a transcendental one. Are there any other operators that may convert a rational function or algebraic ...
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0answers
17 views

Complex structures on punctured disks.

Let $X$ be a smooth surface diffeomorphic to the punctured unit disk $\{(x,y)\in \mathbb{R}^2 \ | \ 0<x^2+y^2<1\}$ in the plane. It admits a lot of non equivalent complex structures, for example ...
2
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1answer
35 views

Fail to see the mistake when applying Cauchy's integral formulae

We need to find $\;f(z)$ with the property that $$f''(a)=\oint_{\partial C_1(0)}{\sin^2z\over (z-a)^3}dz, \quad \forall\;|z|<1$$ Could someone explain me why I cannot do it this way: ...
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1answer
41 views

Prove that $f$ has a removable singularity at $z_0$, and compute $\lim_{z\to z_0} f(z)$

I am again stuck on a qual question while I am preparing for my upcoming exam: Let $W$ be analytic in a domain $D$. Let $z_0\in D$ be such that $W'(z_0)\neq 0$. Define $$f(z) = ...
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0answers
16 views

Complex contour integral properties

Do I understand correctly, that for complex line integrals the properties of common integrals (e.g. Riemann-integrals) cannot be applied? Neither linearity: $$\int_\gamma \beta \;f(z)dz \not = ...
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3answers
56 views

Does there exist an analytic function that satisfies these properties?

Does there exist an analytic function $f:\{z\in\mathbb{C}:0<|z|<1\}\to\mathbb{C}$ such that $\displaystyle\lim_{z\to0}[z^{-3}f^2(z)]=1$? I'm assuming that there is not such a function, so I've ...
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give the details answer [on hold]

for any integer n, let $f_n:[0,1]\to\mathbb{R}$ be defined by $f_n(x)=\frac{x}{nx+1}$ for $x \in[0,1]$ then (a) the sequence $f_n$ converges uniformly on $[0,1]$ (b) the sequence $f_n'$ of ...
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1answer
42 views

The complex equation

In solving $|z|i +2z =1$, I seem to be constantly getting two solutions while both answer key and Wolfram claim to be only one. What am I doing wrong? Let's share the fun: $(\sqrt{x^2 +y^2}) i +2x ...
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44 views

Infinite radius of convergence

If the root test limit tends to infinity, is that sufficient to say we have an infinite radius of convergence? Thanks
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1answer
73 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
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2answers
25 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 ...
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1answer
41 views

Complex analysis without Cauchy's theorem

Is there an approach to complex analysis that is fundamentally different from the usual route via Cauchy's theorem? For example, can one prove that a complex-differentiable function is given locally ...
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1answer
17 views

differntiability of the following function [on hold]

Let f(x)=sinx/x,x≠0 =1 ,x=0, then f is a)discontinuous b)continuous but not differentiable c)differentiable only once d)differentiable more than once
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1answer
37 views

Computing a contour integral over curve not centered at origin

Consider the integral $$ \int_C \frac{1}{z} \, dz $$ where $C$ is the circle of radius $R$ centered at the point $z_0 \in \mathbb{C}$. We parametrize the curve by $z(\theta) = z_0 + Re^{i\theta}$ ...
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27 views

real anaylsis for twice differentiable function [on hold]

Let f:[0,1] tends to [0,1] be any twice differentiable function satisfying f(ax+(1-a)y) less then or equals to a(f(x)+(1-a)f(y)) for all x,y belongs to [0,1] and any a belongs to [0,1].then for all x ...
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18 views

The proof of the Area Theorem for Conformal Maps

The Area Theorem: Suppose $f(z)$ is one-to-one and analytic on the punctured unit disk, and is given by $f(z) = 1/z + \sum_0^\infty a_nz^n$ Then $\sum_0^\infty n|a_n|^2 \le 1$ I'm reading the ...
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15 views

Prove that $|Im\{f(z)\}|≤ $$\frac{2}{\pi} \log \frac{1+|z|}{1 - |z|},$ $z \in \mathbb{D}$. [duplicate]

Let $f(z)$ be an analytic function defined on the unit disk $\mathbb{D} = \{z : |z| < 1\}$ so that $f(0)=0$ and $−1<Re\{f(z)\}<1$ for all $z \in \mathbb{D}$. Prove that $|Im\{f(z)\}|≤ ...
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79 views

Which contour is best for $\int_0^\infty\frac{1}{x^2 + x + 1}dx$

The following is a complex analysis problem. Does anyone have any idea what contour would be good to use? $$\int_0^\infty\frac{1}{x^2 + x + 1}dx$$ Its roots on the bottom are are $\frac{-1 \pm ...
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1answer
25 views

Entire functions satisfying $|\mathrm{Re}\, f(z)| \geq c|\mathrm{Im} \,f(z)|$ [on hold]

Suppose $f$ is entire and there exists a constant $c > 0$ such that $|\mathrm{Re}\, f(z)| \geq c|\mathrm{Im}\, f(z)|$ for all $z \in \mathbb{C}$. Must $f$ be a constant?
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58 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?
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97 views

From the series $\sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)$ to $\zeta(\frac{1}{2}+it)$

Here is a pretty series $$ \displaystyle \sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)=\frac{1}{2} \left(1-\ln (2\pi)+\gamma\right) \quad (*) $$ where $H_{n}:=\sum_{1}^{n} ...
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16 views

Problem on Conformal transformation

An angular domain in the complex plane is defined by $0< \phi<\pi/4$. The mapping which maps this region onto the left half plane is....(fill in the blanks)... My thoughts: The transformation ...
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58 views

Complex derivative of log(z)/z

If we use the principal branch of the log function, at which points of $\mathbb{C}$ does $\frac{\log z}{z}$ have a complex derivative? What is its derivative at these points. This is what I have so ...
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1answer
36 views

let $f : U \rightarrow U$ be an analytic function, whose Taylor series at $0$ is $f(z) = z + a_2z^2 + a_3z^3 + …$

Let $U \subset \mathbb{C}$ be a bounded open set containing $0,$ and let $f : U \rightarrow U$ be an analytic function, whose Taylor series at $0$ is $f(z) = z + a_2z^2 + a_3z^3 + ...$ Prove that ...
3
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1answer
31 views

Let $P_N(z) = \sum_{n = 0}^N\frac{z^n}{n!}$. Calculate $\lim_{N \rightarrow \infty}\int_{|z|=2}\frac{1}{P^3(z) - 1}dz$.

Let $P_N(z) = \displaystyle\sum_{n = 0}^N\frac{z^n}{n!}$. Calculate $\displaystyle\lim_{N \rightarrow \infty}\int_{|z|=2}\frac{1}{P^3(z) - 1}dz$. I am not sure how to do this. If I could figure ...
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1answer
43 views

Derivative calculation $\frac{d}{dz}\cos(yz)$

Perhaps the question that I am about to write may seem trivial, but I just started to study the course of Complex Analysis. The question is the following. I have to calculate the derivative ...
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33 views

Real-valued Fourier series representation

I have got stuck on the following task: Find the value of the series $${4\over \pi^2}\sum_{k=1}^\infty {1\over k^2}-{1\over \pi^2}\sum_{k=1}^\infty{(-1)^k\over k^2}$$ using real-valued Fourier ...
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51 views

Clarification of Contour Integration [duplicate]

I apologise if this seems like an elementary and silly question, but I am confused about the integral $$I=\int^{\infty}_{-\infty}\frac{\cos{x}}{1+x^2}dx=\frac{\pi}{e}$$ If I consider a semicircular ...
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2answers
45 views

Types of singularities of a function

How can I determine the type of a singularity of a function $$f(z)={e^{1/z} \over z-1}+{\pi z \over 2\sin(\pi z)}$$ at $z_0=0$? I don't see an easy way to represent it using Laurent series, neither ...
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1answer
21 views

Euler's Formula, from Needham's Visual Complex Analysis

I'm having trouble understanding this passage. http://imgur.com/a/ao68Q#2 I don't understand the last paragraph. Why is it that $|Z(t)|$ remains equal to 1 throughout the motion?
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1answer
28 views

Ratio of convergence

Let $ \phi(z)= \log (1 + \sin z) $ for a small disk, (the origin is the center of the disk) Find the ratio of convergence of Taylor series? If we consider that $\log(1 +z) = \sum_0^\infty ...
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1answer
65 views
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When functional equations determine a unique global analytic function?

I'm learning about analytic continuations and global analytic functions which were seen to be connected components of the sheaf of analytic germs. Sometimes we get problem sets in which we are ...
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27 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 ...
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3answers
48 views

Showing that $\frac{y}{x^2+y^2} \, dx - \frac{x}{x^2+y^2} \, dy = d\left(\tan^{-1}\left(\frac{x}{y}\right)\right)$

I'm trying to show that $$ \frac{y}{x^2+y^2} \, dx - \frac{x}{x^2+y^2} \, dy = d\left(\tan^{-1}\left(\frac{x}{y}\right)\right) $$ but am having trouble figuring out exactly how to approach the ...
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1answer
60 views

Why is the algebraic number a whole number.

Assume the function $f$, analytic on some domain has a non-essential singularity $a$. Define the algebraic order $h$ of $f$ at $a$ to be the real number such that $\lim_{z\to a}|z-a|^k|f(z)|=0$ for ...
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0answers
11 views

If an analytic function has an algebraic order $h$ at infinity then $\lim_{z\to\infty}z^{-h}f(z)$ is not zero nor is it infinity

Assume infinity is not an essential singularity of the analytic function $f$. Then how is $h$, the algebraic order of $f$ such that $\lim_{z\to\infty}z^{-h}f(z)$ is not zero nor is it infinity? p.s. ...
3
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3answers
59 views

Bound in Complex Analysis

Can someone direct me towards the right way to approach this problem? Show $$\displaystyle \left|\int_{|z|=R} \frac{Log{z}}{z^2} dz\right| \leq 2\sqrt{2}{\pi}\frac{\log{R}}{R},\; \text{ for } ...
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1answer
21 views

Images of Regions Under Cayley's Transformation

I'm working on the following problem for my complex analysis course: I can't seem to find Cayley's transformation anywhere in our textbook - could someone clarify to me what it is? I've done a ...