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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Does a bounded continuous function map Cauchy sequences to Cauchy sequences?

I only ever see the example of $f:(0,1]\rightarrow \mathbb{R}$ where $f(x)=\frac{1}{x}$as that of a continuous function that does not map Cauchy sequences to Cauchy sequences. Are there examples of ...
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3answers
61 views

poles of a polynomial

What are the poles of a polynomial? Are they the same as the roots?
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1answer
43 views

Computing integral using complex analysis methods

I'm trying to compute the integral $$ \int_0^{\infty} \frac{\ln(x)}{x^2 + 1} \, dx $$ using complex analysis methods. We haven't learned residue calculus yet though, only contour integrals up ...
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1answer
55 views

Computing a very messy contour integral

I'm hoping that someone might be able to help me with the following problem. I'll walk through my current work and indicate where I'm stuck. Compute the contour integral: $$ \oint_{|z-1-i| = ...
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0answers
25 views

Domain of convergence Taylor series [on hold]

What would be the domain of convergence of the taylor series of : 1) $\frac{sinz}{z^2+3}$ at $z=0$ 2) $\frac{z+5}{(z-1)(z-4)}$ at $z=2$ 3) $zcoth6z$ at $z=0$ How do I do this? Thanks
2
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1answer
53 views

Contour integral method

Let $f(z)=z^5-3iz^2+2z-1+i$. Evaluate the integral of $\frac{f'(z)}{f(z)}$ around a contour $C$ where $C$ encloses all the zeroes of $f$. I'm not sure what to do here. It seems unlikely I should be ...
2
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3answers
35 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 ...
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0answers
30 views

Find the set of $z$ which satisfies the given equation

Let $w \to w^{a}$ be the principal branch of the power function defined for $|\mathrm{Arg}(w)| <\pi$. Find the set of all values of $z\in \mathbb{C}$ such that the following identity holds for ALL ...
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26 views

Question about Maximum Modulus Principle [duplicate]

Let $f_1, f_2, \ldots ,f_n$ be holomorphic functions on a region $\Omega$. Show that if $\phi (z)=|f_1(z)|+|f_2(z)|+\ldots +|f_n(z)|$ attains a maximum value on $\Omega$, then $f_i$ is constant for ...
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0answers
37 views

Complex Fourier Series and using the square norm

Find the complex Fourier series of $f(x)=e^{(-πx/2)}$ on $-π < x < π$ Discuss the significance of $|C_n|$ in the solution. I've tried so far Using the Complex Fourier Series: $$ %% ...
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0answers
26 views

Showing that $\log|f|$ is harmonic given that $f$ is analytic [on hold]

Suppose that $f$ is an analytic function. How would I show that $\log|f|$ is harmonic?
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68 views

Partial fractions for $\pi \cot(\pi z)$

I want to derive $$\pi \cot(\pi z) = \sum_{-\infty}^{\infty}\frac{1}{z-n} + \frac{1}{n}$$ without taking derivatives. I know through Mittag Leffler that $$\pi \cot(\pi z) = g(z) ...
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2answers
88 views
4
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1answer
36 views

Boundary behaviour of holomorphic function on unit disk

Let $\mathbb{D}=\{z \in \mathbb{C} \ | \ |z|<1 \} $ be the open unit disk in the complex plane. I would like to see explicit examples of the following phenomena: a holomorphic function $f$ on ...
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1answer
59 views

solving integral with complex analysis

I have problems with understanding of the evaluation of this integral below. It has been a long a time ago since I had complex analysis. where $a = (1-\sqrt y )^2$ and $b = (1+\sqrt y )^2$. Now my ...
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1answer
22 views

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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1answer
30 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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24 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
33 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
24 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
33 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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0answers
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
113 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
31 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
33 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
51 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
50 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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0answers
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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20 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 ...
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1answer
36 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: ...
2
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1answer
43 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
57 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 ...
2
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1answer
44 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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0answers
46 views

Infinite radius of convergence [on hold]

If the root test limit tends to infinity, is that sufficient to say we have an infinite radius of convergence? Thanks
2
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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
26 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 ...
2
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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
19 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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31 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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2answers
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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16 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)\}|≤ ...
3
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2answers
81 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
26 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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59 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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2answers
111 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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0answers
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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0answers
59 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 ...
4
votes
1answer
37 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 ...