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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103 views

Uniform convergence of complex exponent derivative

I'm trying to prove the following: Let $\Re z > 0$. Then $$\lim_{\varepsilon \to 0} \frac{t^{z + \varepsilon} - t^z}{\varepsilon} = t^z \log t$$ uniformly in $t \in [0,1]$. I've tried to ...
3
votes
2answers
100 views

Problem concerning pole singularity.

I am stuck on a problem which is given as an exercise in my book. I really would like to solve it primarily by myself, but unfortunately I don't "see" the solution intuitively and therefore I have no ...
4
votes
2answers
609 views

Help with an irregular integral

I am looking for help with doing the following integral : $$\frac{1}{2\pi i}\int_{1}^{\infty}\ln\left(\frac{1-e^{-2\pi i x}}{1-e^{2\pi i x}} \right )\frac{dx}{x\left(\ln x+z\right)}\;\;\;\;z\in ...
3
votes
1answer
227 views

general form of Schwarz's Lemma

I'm reading Ahlfors, Complex Analysis, pag. 135....he's generalizing Schwarz' Lemma, which states that if $f$ is analytic in the unit disc with $f(0)=0$ then $|f(z)|\leq |z|$. He says...."still more ...
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4answers
442 views

Counterexamples in complex analysis

In contrast to other topics in analysis such as functional analysis with its vast amount of counterexamples to intuitively correct looking statements (see here for an example), everything in complex ...
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1answer
196 views

linear fractional transformation with two fixed point on the unit circle

Let $f$ be a linear fractional transformation of the unit disc in itself, fixing points 1 and -1. Can i conclude that $f$ fixes the real axis?
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1answer
150 views

how can i approach to prove the function is one-to-one given specific condition?

In the book(complex variable ;herb silverman), there is a proof about univalent function. My question is how to prove the proposition in special case that $f(z) = f(z_0)$. I take several approachs ...
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1answer
212 views

composition of non analytic functions can be analytic?

I was wondering if the following scenarios are possible: 1) composition of an analytic function with a non-analytic continuous function being analytic (except for trivial cases) 2) composition of two ...
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0answers
275 views

Generalized Cauchy Integral Formula to find coefficients of Taylor Series

I have a complex analysis question that I am stuck on. Let $\sum_{n=0}^{\infty}a_{n}z^{n}$ be the Taylor series expansion at 0 for $(1-z)^{-a}$ where $a>0$. Using the generalized Cauchy integral ...
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2answers
447 views

Conformal mapping of two circles into a line and a circle

I have the following conformal mapping: I need to find $\lambda = f(\zeta)$ and its reverse. Zeros on the figure are given, the axis are oriented as usual. The resulting distance between the line ...
2
votes
1answer
243 views

contour integration $\int_0^\infty \frac{dx}{x^p(x^2+2x\cos{\phi}+1)}$

I'm supposed to verify that this: $$\int_0^\infty \frac{dx}{x^p(x^2+2x\cos{\phi}+1)}=\pi\frac{\sin{p\phi}}{\sin{p\pi}\sin{\phi}}$$ where $0<p<1$ and $0<\phi<\pi$ How do I do this with a ...
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9k views

The Integral that Stumped Feynman?

In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods ...
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1answer
77 views

linear fractional transformations fixing a line

I want to find all linear fractional transformations that fix the points 1 and -1. In particular i'd like to give this set a group structure and see if it is some familiar group or not. I wrote ...
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2answers
181 views

Let $S$ be the open unit disk and $f: S\to \Bbb C$ be a real-valued analytic function with $f(0)=1$

I was thinking about the problem that says: Let $S$ be the open unit disk and $f:S\to \Bbb C$ be a real-valued analytic function with $f(0)=1$.Then which of the following option is correct? The set ...
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3answers
181 views

Cauchy's integral theorem, versions.

I am currently aware of the following two versions of the global Cauchy Theorem. Which one is stronger? 1.)If the region $U$ is simply connected, then for every closed curve contained therein, the ...
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1answer
96 views

Calculate $\dot{\overline{z}}$.

Knowing the expression of $\dot{z}$: \begin{equation} \dot{z}=\lambda z+ \frac{g_{20}}{2} z^2+g_{11} z \overline{z}+\frac{g_{02}}{2}\overline{z}^2, \ \ \ z,\lambda\in C \end{equation} How do I ...
2
votes
2answers
174 views

$f$ be an entire function whose value lie in a straight line in the complex plane

I was thinking about the problem that says: Let $f$ be an entire function whose values lie in a straight line in the complex plane. Then which of the following option(s) is/are correct? (a) $f$ ...
0
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1answer
404 views

Cauchy Riemann conditions

I am a bit confused about differentiability in complex analysis. We showed in class that if a function $f$ is differentiable at $z_0$, then the Cauchy-Riemann equations hold at $z_0$. We also showed ...
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2answers
340 views

A Fourier transform using contour integral

I try to evaluate $$\int_{-\infty}^\infty \frac{\sin^2 x}{x^2}e^{itx}\,dx$$ ($t$ real) using contour integrals, but encounter some difficulty. Perhaps someone can provide a hint. (I do not want to use ...
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1answer
306 views

Let $f$ be a non-constant entire function such that $\left \lvert f(z) \right\lvert=1$ for every $z$ with $\left \lvert z \right\lvert=1$.

I was thinking about the problem that says: Let $f$ be a non-constant entire function such that $\left | f(z) \right |=1$ for every $z$ with $\left \lvert z \right \lvert =1$. Then which of the ...
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1answer
70 views

Suppose that $f(z)$ is meromorphic on a disk, show that negative powers in the Laurent series of $f(z)$ is the sum of the principal parts of its poles

More specifically: Suppose that $f(z)$ is meromorphic on a disk $\{\lvert z\lvert <s \}$, show that the Laurent decomposition of $f(z)$ on the annulus $\{s-\epsilon<\lvert z\lvert <s\}$ has ...
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1answer
265 views

sum of the residue at all poles of the rational function p/q is 0.

We are given two polynomials p, q with deg(q)>1+deg(p). Show that the sum of the residue at all poles of the rational function p/q is 0. How do I go about solving this?
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0answers
47 views

How I can prove that singularities are outside or within the region?

I can not find a way to justify rl as the singularities $z = -1 + i$ and $z = 1 + i$ are inside the semicircle. Can you help me please? Consider $~\int\limits_C\frac{z^3e^{iaz}}{z^4+4}\,\mathrm ...
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1answer
44 views

How I can justify that the residue is -1/2?

I've been trying to develop this excercise but can not find a way to justify the fact that the residue is -1/2 Could anyone help me please?
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1answer
217 views

How to calculate the residue of this function?

The residue of $f(z)=z\sin\frac{1}{1-z}$ at $z=1$. Any hint is appreciated.
2
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1answer
395 views

Branch Points of Function

Consider $$f(z)=\int_0^{\infty} \frac{e^{-t}}{t+z}dt.$$ I'm trying to determine where this function has branch points, define suitable branch cuts, and determine the discontinuity across the cut. ...
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1answer
42 views

Continous function defined by supremum of Radius of disc

Let $S\subset \mathbb{C}$ be a simply connected domain (i.e. every point in complement of $S$ can be connected to $\infty$). Let $C=\{z:|z-\alpha|=r\}$. Would you help me to show that for all $z\in ...
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votes
2answers
156 views

Residue Theorem Troubles

I'm trying to compute $$\int_{0}^{\infty} \frac{\cos(ax)}{1+x^4}dx$$ using the residue theorem. I've split this integral into two contour integrals in the complex plane--one along the real axis and ...
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2answers
77 views

A question about a complex variable function

My question is about the function $f(z)=e^{-z^2}$. Is it everywhere continous? Holomorphic? Thanks, Dan
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1answer
227 views

Phragmen-Lindelöf theorem, question from Conway, chapter VI

Page 141, Question 3: Let $G=\{z:|\operatorname{Im} z| < \pi/2\}$ and suppose $f:G\rightarrow C$ and $\limsup|f(z)| \leq M$ on $w$ in the boundary of $G$. Also, suppose $A < \infty$ and $a ...
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1answer
68 views

A function in the intersectino of Hardy spaces for $p < 1$

I'm working on an exercise from Peter Duren's Theory of $H^p$ Spaces. The question is: Show that $(1-z)^{-1}$ is in $H^p$ for every $p < 1$, but not in $H^1$. I have been able to show the ...
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0answers
53 views

$U \subset \mathbb{C}$ open, how do I construct $f$ holomorphic, such that there is no continuation of $f$ on any neighbourhood of $z \in \partial U$?

Let $U \subseteq \mathbb{C}$ be open. I want to construct a holomorphic function $f: U \to \mathbb{C}$, such that for all $z \in \partial U$ and for all $\varepsilon > 0$, there is no holomorphic ...
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1answer
50 views

Find condition for positivity of complex-valued functions

The question is as follows. the rational function defined on complex plane $ \displaystyle R(z) = c \cdot \prod_{i=1}^{n} \frac{(z- \alpha_i)(1-\bar \alpha_i z)}{(z-\beta_i)(1-\bar \beta_i z)} $ ...
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1answer
87 views

question about solution of equation complex variable

A friend just told me that the equation $e^{z^2}=0$ has solution. Is it true? Thanks, Dan
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2answers
94 views

Is it possible to calculate this integral using complex analysis?

Evaluate $$ \int_0^\infty\frac{dx}{x^2-2x+4}. $$ I cannot figure it out. Any hint is appreciated.
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1answer
211 views

Extension of Liouville's Theorem?

Liouville's Theorem states that if a function is bounded and holomorphic on the complex plane (i.e. bounded and entire), then it is a constant function. What if we consider the following, slightly ...
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votes
2answers
154 views

Evaluate $ \ \int_{- \infty}^{\infty} \frac{x e^{2ix}}{x^2 - 1}\,dx \ $ using given contour

The question is: Evaluate $\displaystyle \ \int_{- \infty}^{\infty} \frac{x e^{2ix}}{x^2 - 1}\,dx \ $ using the contour below. (Explain what happens on each part of the contour.) First of all, ...
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votes
1answer
257 views

integration of complex trigonometric function

Compute $$\oint_{|z-\frac{\pi}{2}|=\pi+1}z\cdot \tan(z)dz$$ My solution: the integrand is a meromorphic function with simple poles at points: $\frac{\pi}{2}+n\pi$, with $n$ integer. Among these ...
4
votes
3answers
64 views

Laurent expansion of $\frac{e^{iz}}{z\,(z^2+1)^2}$ at $z=i$.

I would like to find the residue of $$f(z)=\frac{e^{iz}}{z\,(z^2+1)^2}$$ at $z=i$. One way to do it is simply to take the derivative of $\frac{e^{iz}}{z\,(z^2+1)^2}$. Another is to find the Laurent ...
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0answers
55 views

Lower-bounding the distance between zeros of a continuous function

Consider a continuous function of the form: $L(v) = \sum_{i = 0}^{m}[vA_{i} - B_{i}]p^{i}$ where $p$ is the root of the polynomial equation: $vf(p) - g(p) = 0$ with $f(p)$ and $g(p)$ being two ...
5
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1answer
228 views

Showing that infinite product $\prod{(1+\frac{i}{k})}$ diverges

In Bak and Newman's Complex Analysis they ask to show that the infinite product $\prod_{k \ge 1}{(1+\frac{i}{k})}$ diverges (with $i$ being the imaginary unit). My intuition is that it does not ...
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3answers
147 views

Evaluate $ \ \frac{1}{2 \pi i} \oint_{|z|=3} \frac{f' (z)}{f(z)}\,dz ,~~\mbox{ where }~ f(z) = \frac{z^2 (z-i )^3 e^z}{3 (z+2 )^4 (3z - 18 )^5} . \ $

How do I evaluate $ \ \frac{1}{2 \pi i} \oint_{|z|=3} \frac{f' (z)}{f(z)}\,dz ,~~\mbox{ where }~ f(z) = \frac{z^2 (z-i )^3 e^z}{3 (z+2 )^4 (3z - 18 )^5} \ $ ? The singularities are z = -2 and z = ...
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1answer
216 views

Questions regarding finite Blaschke product

This problem is totally out of my ability. Not even sure what it is talking about. Somebody please help me to solve this... A finite Blaschke product of degree $n-1$ is a function of the form $ \ ...
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3answers
2k views

Some way to integrate $\sin(x^2)$?

Because the straight forward approach involves Fresnel integrals I thought about a different approach of taking the imaginary part of $\int_{-\infty}^{\infty}\exp(ix^2) $ but have no idea how to ...
0
votes
2answers
62 views

complex analysis

evaluate the principal value of $i^{i+1}$ and derive a general expression for $\arccos(A)$, where $A$ is a real number $> 1$ and hence find $\arccos (3)$, writing your answer in the same form, ...
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1answer
272 views

Books on complex analysis

Is there any book on $1$-dimensional complex analysis, where all is written in the language of sheaf theory? It's clear, that a lot of constructions can be formulated in simplier way using it. There ...
3
votes
1answer
192 views

Multivariate analytic function property

Suppose function $F: \mathbb{C}^n \to \mathbb{C}$ is analytic everywhere and in every coodinate; i. e. for any $q \in \mathbb{C}^n$ and for any $j \in \{1,2,\ldots,n\}$ function $f_{q,j}: \mathbb{C} ...
3
votes
1answer
185 views

How to prove the identity $\pi^{s/2}=e^{(\log(2\pi)-1-\gamma/2)s}\prod_{\rho}e^{s/\rho} $?

In the wikipedia the Hadamard product for the Riemann's zeta function has two forms. The first one is ...
5
votes
6answers
378 views

The notion of complex numbers

How does one know the notion of real numbers is compatible with the axioms defined for complex numbers, ie how does one know that by defining an operator '$i$' with the property that $i^2=-1$, we will ...
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0answers
150 views

Is this a valid proof of the orthogonality of harmonic conjugates?

My textbook (Churchill) is asking me to prove that the contours $u(x,y) = c_1$ and $v(x, y) = c_2$, where $u$ and $v$ are the real and imaginary components of an analytic function $f(z)$, are ...