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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23 views

Verification of a simple identity regarding $z \in \mathbb{C}$ and $t \in \mathbb{R}^{\geq 0}$

There is a simple identity used in the study of complex numbers which is given by $$ |t^z|= t^{Re(z)} $$ I'm just curious, how does one actually go about showing this? I had used this in a proof ...
2
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1answer
68 views

Evaluating a holomorphic function at $\pi$

$f(z)$ is a holomorphic function over $\Bbb C$. $f(0)=1$. and $|f(z)| \le 1$ for all $z \in \Bbb C$. find $f(\pi)$. I guess intuitionally that $f(\pi)=1$. But I don't know how to prove!
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2answers
540 views

Why is $\log(e^z)\ne z$ in complex analysis?

I always know that the natural logarithm $\log_e$ is the inverse of the exponential function $e$, but to my surprise when reading elementary functions in complex analysis, I discover that that is not ...
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2answers
93 views

How does someone topologically prove that a function is analytic on some curve

Let us say for the sake of an example, that I want to evaluate $\int_\gamma \! \frac{1}{z-5} \,$ where $|\gamma| = 2$. I know that this integral gives me zero which means that the function $f(z) = ...
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1answer
343 views

Complex Analysis /Calculus III question

I want to integrate $\int_\gamma \! \frac{dz}{z^2}\,$ where $\gamma(t) = e^{it}$ from $0 \leq t \leq \pi$ which is the top half of the unit circle. I keep on getting zero as an answer. Does this mean ...
3
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1answer
557 views

Why is every holomorphic bijection of the Riemann sphere a Möbius transformation?

Just based on some reading, I know that every Möbius transformation is a bijection from the Riemann sphere to itself. I'm curious about the converse. For any holomorphic bijection on the sphere, why ...
4
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1answer
226 views

Analytic in $\mathbb{C}$ implies $\left|\frac{f'(x)}{f(x)}\right|$ is bounded in $\mathbb{R}$?

If $f(z)$ is an analytic function in the complex plane, $z=x+iy$, and $f(x)\neq 0$ for all $x\in \mathbb R$, does this imply that $\frac{f'(x)}{f(x)}$ is bounded on $\mathbb R$?i.e., ...
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498 views

Another residue theory integral

this is the last from me I need to evaluate the following real convergent improper integral using residue theory (vital that i use residue theory so other methods are not needed here) I also need to ...
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1answer
518 views

Evaluating $\int\limits_0^\infty \frac{\log x} {(1+x^2)^2} dx$ with residue theory

I need a little help with this question, please! I have to evaluate the real convergent improper integrals using RESIDUE THEORY (vital that I use this), using the following contour: ...
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103 views

Evaluate integral: $ \int_{-1}^{1} \frac{\log|z-x|}{\pi\sqrt{1-x^2}}dx$

Show that $$ \int_{-1}^{1} \frac{\log|z-x|}{\pi\sqrt{1-x^2}}dx = \log{\frac{|z+\sqrt{z^2-1}|}{2}},\quad z \in \mathbb{C} $$ How can I apply the Joukowski conformal map to this problem? Thanks.
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65 views

characterizing the boundary of the convergent region of $f(z)= \sum_{n=1}^{\infty} z^{(1/z)^{n}}$

Let $$f(z)= \sum_{n=1}^{\infty} z^{(1/z)^{n}}$$ A domain colored portrait (with artifacts) for $f(z)$ on the unit disk looks like: The gray and white regions are where the software package had ...
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1answer
145 views

Why is holomorphically convex hull is contained in the convex hull?

Let $K$ be an arbitrary compact subset of domain $\Omega$. Why is the holomorphically convex hull of $K$ is contained in the convex hull of $K$? Holomorphically convex hull of $K$ is defined as ...
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3answers
906 views

Show that $f$ maps the entire unit disc onto itself.

Suppose $f$ is analytic in the unit disc $D(0,1)$ and maps the unit circle into itself. Show then that $f$ maps the entire disc onto itself. So the outline wants us to use the Max Modulus Theorem to ...
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415 views

Laurent Series and finding values of the specified sum.

Comparing the coefficients in the Laurent developments of $cot(\pi z)$ and its expression as a sum of partial fractions, find the values of $\sum_{n=1}^\infty$ $\frac{1}{n^{4}}$ and ...
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1answer
767 views

If $f$ is a non-constant analytic function on a compact domain $D$, then $Re(f)$ and $Im(f)$ assume their max and min on the boundary of $D$.

This is a homework problem I got, my attempted proof is: Since $f$ is non constant and analytic, $f=u(x)+iv(y)$ where neither $u$ nor $v$ is constant(by Cauchy Riemann equations) and $u v$ are both ...
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201 views

Where have I gone wrong? Contour integration $\int_{-a}^a {u\over 1+u+u^2} du$ as $a\to \infty$

I would like to integrate $\int_{-a}^a {u\over 1+u+u^2} du$ as $a\to \infty$. So I thought I might use the residue theorem. In the complex plane, the singularities occur at $z=e^{\pm i2\pi\over 3}$. ...
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2answers
179 views

Integration of two dilogarithms: $\int_{0}^{\frac{1}{B_{1}}}\frac{dx}{x}\ln(x-1)+\int_{0}^{\frac{1}{B_{2}}}\frac{dx}{x}\ln(x-1)$

I have an integral: $\int_{0}^{\frac{1}{B_{1}}}\frac{dx}{x}\ln(x-1)+\int_{0}^{\frac{1}{B_{2}}}\frac{dx}{x}\ln(x-1)$ where $1<\frac{1}{B_{1}}\leq2 , 2\leq\frac{1}{B_{2}}$ I need to find some ...
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502 views

Convergence of the infinite product $\displaystyle{\prod_{n = 1}^{\infty} \frac{z - \alpha_n}{z - \beta_n}}$

I've been trying to solve this homework problem for a while but I can't seem to get any significant ideas about how to approach it, so I would really appreciate any hints that could help me solve it. ...
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2answers
84 views

Power series, range

How can we find the first few terms of the Laurent series of $f(w)={1\over \cos w-1}$ where $w\in \mathbb C, |w|<2\pi$? I am wondering if there is another way maybe exploiting one of the more ...
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1answer
270 views

Absolute Convergence of an Infinite Product based on Weierstrass's Factor Theorem

I am trying to show that $\left\{ \left( 1-\dfrac {z} {\pi }\right) e^{\left( \dfrac {z} {\pi }\right) }\right\} \left\{ \left( 1+\dfrac {z} {\pi }\right) e^{\left( -\dfrac {z} {\pi }\right) }\right\} ...
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97 views

Calculate $\#(Per_n(f))$ when $f(z)=z^2$

DEFINITIONS: A complex number $z_0$ is called a fixed point of $f$ if $f(z_0)=z_0$. It is called a periodic point of period $n>1$ of $f$ if $f^i(z_0)\neq z_0$ for $1\leq i \leq n-1$ but ...
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156 views

Represent $\mathbb{R}^3$ as an union of disjoint circles using stereographic projection

I have begun to learn complex analysis and have solved a few problems on stereographic projection and Riemann sphere but can't solve the problem in the subject. Could you help please?
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1answer
200 views

Showing a function is holomorphic

$f$ is continuous on $\gamma$ which is smooth & bounded. Let F be a function such that $$F=\int_\gamma \frac{f(\beta)}{\beta-z} d\beta$$ for all $z$ not in $\gamma$ Show F is holomorphic at $z$ ...
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1answer
137 views

What to do when integration boundary is on pole.

What to do when integration boundary is on pole. I want to integrate $(dx/x)\log(x-1)$ from $0$ to, lets say, "$a$", where "$a$" is arbitrary $a>1$, $a\le 2$. $x$ is real.
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2answers
341 views

Yet another complex analysis problem

I need help computing an integral. My motivation is to understand a standard way to build the holomorphic functional calculus for unbounded operators, though the actual question could probably be a ...
4
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1answer
156 views

help with a bizarre integral

i've been trying to do this integral , but with no luck . $$\int_1^\infty \frac{\left \{x \right \}}{x}\left(\frac{1}{x^{s}-1}\right) \; dx$$ $\Re(s)>1 $ , $\left \{x \right \} $ is the ...
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1answer
374 views

Is the function $f$ differentiable at $z=0$?

Is the function $f$ given by $$f(z) =\left\{ \begin{array}{ll} \frac{(\bar{z})^2}{z}, & \hbox{if }z\neq 0; \\ 0, & \hbox{if }z=0. \end{array} \right.$$ differrentiable at ...
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2answers
465 views

how to use schwarz lemma

Let $f(z)$ be an analytic function on $D=\{z : |z|\leq 1\}$. $f(z) < 1$ if $|z|=1$. How to show that there exists $z_0 \in D$ such that $f(z_0)=z_0$. I try to define $f(z)/z$ and use Schwarz Lemma ...
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1answer
2k views

Analytic complex function which is constant.

I come across these question when I am studying George Cain Complex analysis. Suppose $f$ is analytic on a connected open set $D$, and suppose $f^{'}(z)=0$ for all $z\in D$. Prove that $f$ is ...
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1answer
301 views

The derivative of a complex function.

Question: Find all points at which the complex valued function $f$ define by $$f(z)=(2+i)z^3-iz^2+4z-(1+7i)$$ has a derivative. I know that $z^3$,$z^2$, and $z$ are differentiable everywhere in ...
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2answers
233 views

Showing $\int_0^{2\pi} \log|1-ae^{i\theta}|d\theta=0$

This is a homework problem for a second course in complex analysis. I've done a good bit of head-bashing and I'm still not sure how to solve it-- so I might just be missing something here. The task is ...
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2answers
73 views

polynomials on the complex numbers

Let p be a complex polynomial $ p\left( {a + bi} \right) = p\left( z \right):{\Bbb C} \to {\Bbb C} $ How can I prove the following? $$ \lim_{\lVert z\rVert\to\infty} \Vert p(z)\rVert = \infty.$$ I ...
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2answers
399 views

Is there an elementary method for evaluating $\int_0^\infty \frac{dx}{x^s (x+1)}$?

I found a way to evaluate $\int_0^\infty \frac{dx}{x^s (x+1)}$ using the assumption that $s\in\mathbb{R}$ and $0<s<1$. Apparently it should be easily extended to all $s\in\mathbb{C}$ with ...
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3answers
3k views

Mapping half-plane to unit disk?

Say you have the half-plane $\{z\in\mathbb{C}:\Re(z)>0\}$. Is there a rigorous explanation why the transformation $w=\dfrac{z-1}{z+1}$ maps the half plane onto $|w|<1$?
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2answers
313 views

On the modulus of $\Gamma(z)$

In about two weeks, I'm going to be giving a presentation on the complex-valued Gamma function $\Gamma(z)$. By definition, I know that $$\Gamma(z)= \int_0^\infty e^{-t}t^{z-1}dt.$$ Now if I let ...
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1answer
377 views

Analytic function $f$ agrees with $\tan x$ on $0 \leq x \leq 1$—is $f$ entire?

Suppose an analytic function $f$ agrees with $\tan x$, $0 \leq x \leq 1$. Could $f$ be entire? Since $f$ and $\sin z/\cos z$ agree at a set of points and both are analytic in an open neighborhood of ...
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1answer
2k views

How is Cauchy's estimate derived?

Cauchy's integral formula says $$ f^{(n)}(z)=\frac{n!}{2\pi i}\int_C\frac{f(\zeta)d\zeta}{(\zeta-z)^{n+1}}. $$ If we let $C$ be the circle of radius $r$, such that $|f(\zeta)|\leq M$ on $C$, then ...
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102 views

Length of $\frac{\partial }{\partial z}$ in Kaehler geometry.

I am taking a Kaehler geometry course this semester. The book we use is Tian's Canonical Metrics in Kaehler Geometry. I got a little confused about the calculation there in. For example, ...
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189 views

Complex integral with zeta

this is a homework problem I am stuck on: Compute the following integral for $\sigma > 1$ $$\displaystyle \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T}\left|\zeta{(\sigma + it)}\right|^2dt .$$ I ...
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0answers
239 views

Series expansion about a logarithmic singularity

Let $f : \Omega \to \mathbb{C}$ be holomorphic, where $\Omega$ is an annulus centered at $z=0$. We say that $f$ has a logarithmic singularity at $z=0$ if no analytic continuation of the germ of $f$ ...
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3answers
465 views

Using residue theory to evaluate $ \int_0^\infty \frac{ \sin \pi x}{x(1-x^2)} \;\text{ dx}$

I'm on the last question of my homework and it's involving using the residue theory, which I dont really understand, so could somebody lend me a hand? I have to evaluate the real convergent improper ...
2
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2answers
1k views

Characteristic function of Cauchy distribution.

When computing the characteristic function of Cauchy distribution, we applied the Cauchy Integration theorem: $$ \int_{C_{R}}\frac{e^{i\alpha z}}{z^{2}+1}dz=\int_{-R}^{R}\frac{e^{i\alpha ...
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1answer
177 views

Evaluating the convergence of a complex series

This is a part of the proof on a textbook of the fact that a power series that converges on an open disc defines an analytic function. First note this inequality about a real series: for any ...
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1answer
186 views

How does the exponential function $\operatorname{exp}$ stretch strips to half planes?

Consider the strip in the complex plane defined by $\{z\in\mathbb{C}:\Im(z)\in (0,\pi)\}$. Applying $\operatorname{exp}$ to this strip maps it onto the half plane $\{z\in\mathbb{C}:\Im(z)>0\}$? ...
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3answers
887 views

time-frequency domain

im confused on how these folks seems to like convert a frequency into a time function, and a time function into a frequency function. i know that time function uses amplitude that varies over time, ...
3
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3answers
990 views

Why is there no continuous square root function on $\mathbb{C}$?

I know that what taking square roots for reals, we can choose the standard square root in such a way that the square root function is continuous, with respect to the metric. Why is that not the case ...
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388 views

Guidance on an exercise in Gamelin's $\textit{Complex Analysis}$

I'm trying to prove the following exercise from Gamelin's Complex Analysis: Exercise XII.$1$.$5$ (p. $319$): Fix $M>1$, and let $\{f_n(z)\}$ be a sequence of meromorphic functions on a domain ...
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1answer
139 views

Maximum principle for one point on boundary

Suppose $f$ is bounded and analytic on the open unit disk $\mathbb{D}$. Say that $f$ extends continuously to one point $z_0$ on $\partial \mathbb{D}$, the boundary of $\mathbb{D}$. Now does the ...
4
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1answer
312 views

Local normalization of algebraic curves

I am currently reading about the normalization theorem: Suppose $C$ is an irreducible plane algebraic curve, and let S be the set of singular points. Then there exists a compact Riemann surface $\hat ...
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2answers
207 views

Complex Polynomial transformation

I'm studying for an exam and professor gave us to create a little program that automatically does a transformation for a polynomial with complex coefficients, I don't have many problems doing the ...