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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30 views

Calculate a sum $\sum_{k=0}^{\infty}\cos\frac{k\pi}{6}$

I have to calculate a sum $\sum_{k=0}^{\infty}\cos\frac{k\pi}{6}$. Our lecturer told us we should use de Moivre formula. But i think this sum deosnt even converge...
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1answer
32 views

Show that $\Delta u$, $\Delta v= 0$

Let $f=u+iv$ be a complex differentiable function. Then $u$ and $v$ are harmonic. My solution: We have Cauchy-Riemann equations held: $u_x=v_y$ and $u_y=-v_x$. Now: $u_{xx}=v_{yx}$ and ...
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1answer
69 views

What is wrong in this calculation $\int_{-\infty}^{\infty}\frac{\cos x}{1+x^2}dx$?

$\def\Res{\operatorname{Res}}\def\Re{\operatorname{Re}}$ First solution: Complex function $f(z)=\frac{\cos z}{1+z^2}$ has a pole $z=i$ on the upper complex plane. $\Res (f,i)=\frac{e+1/e}{4i}$, so $$ ...
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2answers
59 views

$\int_{|z|=2}^{}\frac{1}{z^2+1}dz$

I tried finding the integral of $\int_{|z|=2}^{}\frac{1}{z^2+1}dz$ but not sure whether it is correct. $\gamma(t)=2e^{it},t\in[0,2\pi]$ ...
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2answers
41 views

$\int_{\gamma}^{}\frac{1}{z}dz$, $\gamma$ is the ellipse $x^2+4y^2=1$ traversed once with the positive orientation

I am unable to find the integral $\int_{\gamma}^{}\frac{1}{z}dz$, $\gamma$ is the ellipse $x^2+4y^2=1$ traversed once with the positive orientation. This maybe possible to be done using Cauchy-Goursat ...
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1answer
63 views

Find the value of the integral on the contour C

Ok, so I'm trying to figure out this problem. It asks to find the value of the contour integral $\dfrac{e^z}{z^2(z-\pi i)}$ on the contour $C$ shown in the following figure I believe that in order ...
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1answer
144 views

Rudin's Proof on Riesz Representation Theorem

The proof is from Rudin's Real and Complex Analysis. I am having a hard time understanding part of the Riesz Representation Theorem The Theorem states: Every open set $E$ satisfies ...
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0answers
66 views

Show that the integral of Riemann function is analytic

I'm trying to resolve this problem. Let $\Omega$ be an open set no empty of $\mathbb C$, $[a,b]$ a compact interval of $\mathbb{R}$, further $f,\ g\colon[a,b] \to \mathbb C$ two integrable Riemann ...
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1answer
46 views

Integral of a complex function over semicircle enclosing a pole

I have a function with a pole $x_0$ on the real axis. Why would the integral of that function over the countour that a semicircle that is the upper half of the circle enclosing $x_0$. I cannot figure ...
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1answer
119 views

Arzela-Ascoli equivalent theorems

The following theorems are equivalent? Is the Theorem 2 false? Theorem 1 (Arzela-Ascoli): Let $X$ be a compact metric space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb ...
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1answer
44 views

maps that preserve harmonic functions

Is there a theory of the type of maps between domains that preserve harmonic functions? For instance, in the 2-dimensional case, we know that conformal maps (or even just holomorphic ones) are such ...
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235 views

Harmonic functions in the upper half plane

It is a cautionary remark that is often made that solutions to the Dirichlet problem (with continuous boundary conditions) are not unique when the domain in question is the upper half plane. Yes, you ...
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0answers
36 views

Existence of a sequence of polynomials that approximate a holomorphic function uniformly on the closed unit disk [duplicate]

Let $f:D(0,1) \to \mathbb C$ be continuous on the closed unit disc $D(0,1)$ and holomorphic on the open unit disc. Show that there exist a sequence of polynomial that converge uniformly in the closed ...
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3answers
88 views

Stitching two analytic functions?

Let $f$ be an analytic function on the open unit disc and let $g$ be an analytic function on the complement of its closure. Further assume that the two functions have a the same continuous limit on ...
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1answer
28 views

Finding the value of a contour integral comprised of line segments

I am attempting to work the following problem but I think am just forgetting a few things in order to answer it. The question asks to find the value of the integral of $(2z+1)dz$ on the contour $C$, ...
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1answer
78 views

Find complex power series expansion for $\int e^{-w^2} dw$

If a function $E(z)$ is defined on $\mathbb{C}$ by $$ E(z) = \int_0^z e^{-w^2} dw,$$ find a power series expansion for $E(z)$ about $0$. What does this power series converge? I know how this ...
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1answer
135 views

Equicontinuity and pointwise bounded implies locally uniformly bounded

Is the following proposition true? Let $\mathcal{F}\subset H(\Omega,\mathbb{C})$ be a family of holomorphic functions such that $\bullet$ $\mathcal{F}$ is equicontinuous $\bullet$ ...
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1answer
36 views

The Laurent series of $f(z)=\frac{1}{e^z-1}$

I am trying to expand $f(z)=\frac{1}{e^z-1}$ in Laurent series. One approach I tried involved writing $f(z)=e^{-z} \frac{1}{1-e^{-z}}$, expanding the fraction as a geometric series: ...
2
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1answer
115 views

Proof that $p:\mathbb{C}\setminus\{0\}\rightarrow\mathbb{C}\setminus\{0\}$ is a covering map, with $p(z)=z^2$

Prove that $p:\mathbb{C}\setminus\{0\}\rightarrow\mathbb{C}\setminus\{0\}$ is a covering map, with $p(z)=z^2$ Let $X=\mathbb{C}\setminus\{0\}$ Let $b\in X$ write $b=re^{j\theta}$ (with ...
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2answers
111 views

Looking for intuïtive explanation why contour integral of $\frac{dz}{z} $equals $2\pi i$ in complex analysis

$$\oint \frac{dz}z = 2\pi i$$ I've seen the derivation of it using the parametrisation. Since this result is used all the time in my complex analysis course, i'd like to understand this ...
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1answer
73 views

How to integrate $\int_{-1}^1\frac{1}{a + bx }dx$, where $a,b\in \mathbb{C}$ without using branch cuts.

Is there a way to integrate $$\int_{-1}^1\frac{1}{a + bx }dx,\,\,\,\,(*) $$ where $a,b\in \mathbb{C}$ without using branch-cuts? I was approached with such an integral relatively early in my text, and ...
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258 views

Calculating the abscissa of convergence for general Dirichlet Series

I'm currently interested in proving this theorem which I have been thinking for quite a while: Define a Dirichlet Series $$\sum_{k=1}^{\infty}a_k e^{-\lambda_k z}$$ where $\lambda_k$ is a strictly ...
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0answers
58 views

Conditions on an analytic function to make it constant

$f:\mathbb C\rightarrow C$ is an analytic function.The question gives 3 conditions and asks to conclude whether it is possible to conclude that $ f$ is constant from here. a.Im$(f^{'}(z))>0$ $ ...
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2answers
47 views

solution of $\bar{z}=\xi z$

Does $\bar{z}=\xi z$ has solution in $z$ (complex number) for all values of $\xi \in \mathbb{C}$? I was trying to use normal method but its coming quite complicated.
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1answer
18 views

Why is $f^{(n)}(z)$ real when z is real for all $n$, if this is true for $n=0$?

In order to prove the Schwarz Reflection Principle using the Taylor expansion, we need the information that $f^{(n)}(z)$ is real if $z$ is real for all $n$. We have $f(z)$ real if $z$ is real, and ...
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2answers
52 views

Find the complex function f(z) given the following properties

Find the complex function f(z) given the following properties $(z^2+1)f(z)$ is entire $f(z)$ is an even function $Res(f;i) = i$ limit as abs(z) approaches infinity of f(z) = 1 So based on 3. it ...
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1answer
32 views

Holomorphic Function in $\infty$

I have a question and would like someone to help me, I need to know: How to define a holomorphic function at $z= \infty$ ?
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3answers
245 views

Steepest descent method

I really don't understand how we generally choose the contour for the steepest descent method in complex analysis? I approximate the Fresnel integral $$ \int_{0}^{\infty}\cos{x^2}dx$$ and I found it ...
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1answer
58 views

Function holomorphic in the neighb. of zero, bounded by exponent is equal 0

I want to prove that if $f$ is a holomorphic function in a neighbourhood of $0$ and $|f(\frac{1}{n})| \le \frac{1}{e^n}$ for $n$ sufficiently big, then $f =0$. I know that if $f$ is holomorphic in a ...
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0answers
32 views

Order of pole(s) in $f(z) = \frac{1}{1+ z^3}$?

I just want to check that I understand the meaning of the order of poles of a complex variable. $f(z) = \frac{1}{1+ z^3}$ I believe $f$ has $1$ pole of order $1$ and two poles of order $3$. Ie. the ...
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1answer
29 views

Uniform Convergence of an infinite series

Show that for each $r> 0$, $\sum_{k=0}^{\infty} \frac{1}{k^2 -z}$ converges uniformly on the set $E_r = \{z: |z| \leq r, z\neq k^2$ for $k = 0,1,2,3, ...\}$. I tried to use the Weierstrass M-test ...
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1answer
80 views

Negative index coefficients of Laurent series for 1/sin(z)

Given $f(z) = \dfrac{1}{\sin(z)}$ a) Give singularities b) Determine coefficients $a_{-1}$ and $a_{-3}$ of the Laurent series So I thought: a) $n \pi$, where $n$ is an integer b) ...
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1answer
47 views

Contour integral of convergent power series

Given that $\frac{e^z}{z^k} = z^{-k} + z^{1-k} + \frac{z^{2-k}}{2!} + \frac{z^{3-k}}{3!} + ...$ converges uniformly on any set $\{z \in C: r \leq |z| \leq Z\}$ (where $0 < r < R$), show that for ...
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1answer
54 views

Two disk automorphisms are agree at a point of the open unit disk .

I want to prove the conjecture, If two disk automorphisms are agree at a point of the open unit disk, then they must be identical. I think this can be prove only using Schwartz lemma. Here ...
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3answers
53 views

Why is this true - $\int_C \frac{1}{z^n} dz = 0$ if $n \ne 1$

So I was looking up the reasoning behind the residue theorem and was wondering what was so significant about the $a_{-1}$ coefficient of the Laurent series and I came across this result - $$\int_C ...
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1answer
23 views

how to show that the set where Real part of an analytic function vanish contains arcs

Consider a function $f$ given by $f(z)=z^3g(z)$, where $g(0)\neq 0$. Then clearly $z=0$ is a zero of function $f$ of multiplicity $3$. Let $A$ be the set given by $\{z: \Re f=0\}$. How do i show that ...
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0answers
52 views

Sum of Lomax random variables

Suppose $X_1,X_2,\cdots X_n$ are $n$ i.i.d Lomax random variables with pdf $f(x)=\frac{m}{(1+x)^{m+1}},x\geq 0,m\in \mathbb N$. I need to determine the pdf (or cdf) of the sum $S_n=\sum_{i=1}^{n}X_i$. ...
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1answer
38 views

Is converse of Lewy theorem true?

In complex analysis, there is a result named Lewy's theorem, which states that: If $u=(u_1,u_2):\subseteq \mathbb{R^2}\to \mathbb{R}^2$ is one-one and harmonic in a neighborhood $U$ of origin ...
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0answers
23 views

Doubt in a step of the proof of Rado-Kneser-Choquet theorem

I am trying to prove Rado-Kneser-Choquet theorem, which states that if $f$ is sense preserving self homeomorphism of the unit circle $\partial D$. Then harmonic extension $F$ of $f$ is self ...
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1answer
46 views

Deflating (factoring) a 6th degree polynomial

What is the procedure to factor a 6th degree polynomial of a complex variable? $$P(z)=1+x^2+x^3+x^4+x^5+x^6$$ I do have the correct answer but no idea how to get to it. The answer is: ...
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1answer
56 views

Are these the correct residues?

$$\int_C \frac{z+1}{z^2-2z} dz$$ for the circle of $\lvert z \rvert = 3 $. Poles are obviously at $ z = {0,2}$. Can I calculate the residues by viewing the fraction in the integral as either $$\int_C ...
2
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1answer
47 views

Supremum of the function of a sum for Weierstrass M-test

I have to prove the uniform convergence of $\sum_{k=1}^\infty \frac{k+z}{k^3 + 1}$ on the closed disc $D_1(0)$. Using the M-test, $|\frac{k+z}{k^3 +1}| \leq |\frac{k+1}{k^3 +1}| = \frac{1}{k^2 - k + ...
2
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1answer
48 views

Compute an integral..

$\int_\gamma z^n dz$ where $\gamma$ is the unit circle $|z|=1$ oriented counter clockise and $n$ is an integer. Hint: the answer will depend on $n$. What I don't get about it is, if I just apply ...
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1answer
33 views

derivative of a map $f(A)=AA^*$

Can someone help me with the derivative of this function. I am getting confused $f:GL(n,\mathbb{C}) \rightarrow GL(n,\mathbb{C})$ $A \rightarrow AA^{*}$ where $A^*$ is the conjugate transpose of A. ...
4
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1answer
104 views

Please calculate $\sum _{ k=0 }^\infty\left[ \tan^{ -1 }\left( \frac { 1 }{ k^{ 2 }+k+1 } \right) -\ldots \right] $

Not many math problems stump me, but this summation has me stumped. Can someone provide a solution to this summation: $$\sum _{ k=0 }^{ \infty }{ \left[ \tan ^{ -1 }{ \left( \frac { 1 }{ k^{ 2 }+k+1 ...
2
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0answers
74 views

Asymptotics of inverse Laplace transform of a function with an essential singularity?

Let $h$ be the function $$ h(x) = \sum_{k\geq0} \frac{(ix)^k}{k!}\zeta(2k), $$ with the Laplace transform $$ \tilde h(s) = -\frac{\pi}{2s}\sqrt{i/s}\cot\left(\pi\sqrt{i/s}\right), $$ which has an ...
0
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1answer
92 views

Conformality of a map

A conformal mapping is a map $f:U\to V$ with $U,V\subseteq\mathbb{C}$ such that the angles are locally preserved. This can be reformulated saying the jacobian matrix is everywhere a scalar multiple of ...
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1answer
38 views

A sufficient condition for $f$ to have polynomial growth

Let $f(z)=\alpha z\bar z+\beta z+\bar \beta \bar z+\gamma\geq 0, \forall\ z\in\mathbb C$, where $\alpha,\gamma \geq 0$, $\beta\in\mathbb C$. Show that $$f(z)\leq (1+z\bar z)(\alpha+\gamma).$$ I ...
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1answer
94 views

Harnack's inequality

Let $u$ be harmonic on $\{|z|<1+\epsilon\}$ for some $\epsilon>0$ and $u \geq 0$ on $\{|z|=1\}.$ Could anyone advise me how to show $\dfrac{1-|z|}{1+|z|}u(0) \leq u(z) \leq ...
1
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1answer
25 views

Contracting closed contours that enclose poles

If $f(z)$ is analytic in a simply connected domain $D$, its integral over any closed contour is $0$. I don't quite understand how the idea of contracting the contour that encloses a pole follows from ...