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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3
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1answer
352 views

contour integral with rational and cosh

Here is a fun looking integral. $$\int_{0}^{\infty}\frac{1}{(4x^{2}+{\pi}^{2})\cosh(x)}dx=\frac{\ln(2)}{2\pi}$$. I rewrote it as $\frac{2e^{z}}{(4z^{2}+{pi}^{2})(e^{2z}+1)}$ It would appear there ...
2
votes
1answer
193 views

Prove that there is an unique $z$ s.t. $f(z) = z$ where $z$ is a complex number

Let $f$ be analytic on the closed unit disk centered at the origin and $|f(z)| < 1$ for $|z| = 1$. Show that $f$ has exactly one fixed point inside the open unit disk. That is, there exists a ...
2
votes
1answer
397 views

Application of Jensen's formula

While studying for an upcoming complex analysis qualifying exam, I found the following problem in Conway's Functions of One Complex Variable (XI.1 exercise #2). Let $f$ be an entire function, ...
2
votes
1answer
115 views

Problem on analytic function.

Let $f(z)$ be analytic function on $D = \{z\in C : |z-1|<1\}$ such that $f(1) = 1$. If $f(z) = f(z^2)$ for all $z\in D$. Then which of the following statement is not correct. 1-$f(z) = [f(z)]^2$ ...
1
vote
1answer
115 views

Residues - Which coefficients to choose?

When finding a residue, how am I to know which coefficient to choose? For instance, if I have, let's say three poles, which coefficients of the Laurent series do I choose to calculate the residue? ...
0
votes
1answer
123 views

Trouble with Cauchy Riemann…not sure which law to use?

I'm unsure of which Cauchy-Riemann law to use when I'm given either a real or imaginary function. For instance. I might be given a real function and asked to work out the imaginary part. For ...
1
vote
1answer
87 views

A simple Riemann mapping question

Let $\Delta$ denote the open unit disc. Let $G$ be a simply connected region and $G\neq\mathbb{C}$. Suppose $f:G\rightarrow\Delta$ is a one-to-one holomorphic map with $f(a)=0$ and $f'(a)>0$ ...
0
votes
1answer
228 views

Problem on Complex Analysis.

If $f(z)= u - iv$ is an analytic function of $z = x + iy$ and $u-v=\Large\frac{e^y-\cos x+\sin x}{\cosh y - \cos y}$, find $f(z)$ subject to condition, $f\left(\frac{\pi}2\right)=\large \frac{3 - ...
1
vote
1answer
89 views

Prove $2|f'(0)| \leq \sup_{z_1,z_2\in D} |f(z_1)-f(z_2)|$

Let $f:D\to \mathbb{C}$ be a holomorphic function where $D$ is the open unit disk. Then prove $$ 2|f'(0)| \leq \sup_{z_1,z_2\in D} |f(z_1)-f(z_2)| $$ I can show that $$2f'(0) = \frac{1}{2\pi i} ...
4
votes
0answers
350 views

Intuition for the Poisson kernel

The derivation of the Poisson kernel for a disc seems to involve a trick, and I don't really understand how one would come up with it. Let $f$ be a holomorphic function on a disc $D_{R_0}$ centered ...
1
vote
1answer
118 views

Analytic bijection preserves order of poles

If $f:U\rightarrow\mathbb{C}$ has a pole at $z$ of order $n$ and $\phi:V\rightarrow U$ is an analytic bijection with $\phi(w)=z$, show that $w$ is a pole of $f\circ\phi:V\rightarrow\mathbb{C}$ of ...
0
votes
3answers
150 views

For which simple closed curves $\gamma$, $\int\limits_{\gamma} z^{2}+z+1\, dz=0$?

For which simple closed curves $\gamma$ is $\displaystyle\int_{\gamma} z^{2}+z+1\, dz=0$ Could someone help me through this problem?
0
votes
1answer
359 views

Describe the set whose points satisfy the following relation.is region?

Describe the set whose points satisfy the following relation.is region? |z − 2| > |z − 3|. My atempt The open half-plane: Re z > 5/2; a region, My guess is that if this region takes all the ...
1
vote
1answer
192 views

Representation of Holomorphic Functions By Exponential

Let $f$ be holomorphic and nonzero on $D_{1}(0)$ the open unit disc. Can we write (for the given domain) $f(z) = e^{h(z)}$ where $h$ is holomorphic? This seems clear using a naive log argument but I'm ...
1
vote
0answers
65 views

How to evaluate this in the complex plane?

How to evaluate this in the complex plane? $\int_{\gamma}^{} z^{e^{z^{2}}}\, dz$ when $\gamma$ is the unit circle.
2
votes
2answers
196 views

how to evaluate this integral by considering $\oint_{C_{(R)}} \frac{1}{z^{2}+1}$

Consider the integral $I=\int_{-\infty}^{\infty} \frac{1}{x^{2}+1}\, dx$. Show how to evaluate this integral by considering $\oint_{C_{(R)}} \frac{1}{z^{2}+1}, dz$ where $C_{R}$ is the closed ...
4
votes
3answers
2k views

Entire function bounded by a polynomial is a polynomial

Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k>0$ is constant. Show that $f$ is ...
2
votes
2answers
376 views

Laurent series for an even function

Show that if the Laurent series $\sum_{n=-\infty}^{\infty}a_n(z-z_0)^n$ represents an even function, then $a_{2n+1}=0$ for $n=0,\pm 1,\pm 2,\ldots$, and if it represents an odd function, then ...
1
vote
1answer
53 views

Minimize a function in polar coordinates along a curve

Consider the line $\theta - \ln r = c$ where $-\pi < \theta \leq \pi$ and $c$ is a fixed real constant. How would I find the point $(r, \theta)$ which minimizes $1/r$? Alternatively, what is the ...
1
vote
1answer
315 views

Uniform convergence of sequence to the exponent function

Let $f_n(z) = (1-z^2/n)^n$, and let $f(z)=\operatorname{exp}(-z^2)$. I need to show that $f_n$ converges uniformly to $f$ in any closed disc. I saw this: Uniform Convergence of an Exponential ...
2
votes
1answer
87 views

Show that $\lim_{\varepsilon\to 0}I_\varepsilon = 0$

Consider $$I_\varepsilon :=\oint_{C_\varepsilon} z^αf(z)\,dz,$$ where $\alpha>−1$ is real, where $C_\varepsilon$ is a circle of radius $\varepsilon$ centered at the origin and $f(z)$ is analytic ...
1
vote
1answer
73 views

Show that $\lim_{R \to{+}\infty}{I_{R}}= 0$.

Consider $$\displaystyle I_{R}=\int_{C_{R}}^{} \frac{e^{iz}}{z^{2}}\, dz,$$ where $C_{R}$ is the semicircle with radius R in the upper half plane with endpoints $(-R,0)$ and $(R,0)$ $(C_{R}$ is ...
5
votes
2answers
220 views

Julia Set of polynomials

If $f$ is a polynomial and $z\in\mathbb{C}$, show that either $f^n(z)\rightarrow\infty$ or $\{f^n(z) : n\geq 1\}$ is a bounded set. Here, $f^2(z)=f(f(z))$ and $f^n(z)=f(f^{n-1}(z))$ for $n\geq 2$ ...
0
votes
1answer
244 views

Confusion about $\log(\frac{1+z}{1-z})$ being Analytic on $\mathbb{C}-[-1,1]$

Observe that: $\log(\frac{1+z}{1-z}) = -2\int\frac{dz}{1-z^2}$. (Not precisely true but read on) Supposedly this function is analytic on the domain $\mathbb{C}-[-1,1]$, despite the fact that it's ...
1
vote
1answer
256 views

Power Series Definition

What does it mean for a series to be centered around a number? I'm taking complex analysis and am suddenly very confused. I didn't have this explanation, or proof of taylor and power series in ...
1
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3answers
284 views

How can I show that arc length $L(\gamma)$ of a curve is unchanged after reparametrization?

Show that the arc length $L(\gamma)$ of a curve $\gamma$ is unchanged if $\gamma$ is reparametrized Can you help me please?
4
votes
2answers
150 views

Property of Analytic Function

If $f:\mathbb{C}\rightarrow\mathbb{C}$ is analytic and $Im(f(z))\neq 0$ whenever $|z|\neq 1$, show that $f$ is a constant. It sounds familiar but not so trivial at all...
10
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1answer
231 views

Cauchy's Integral Formula and Green's Theorem

I have been re-reading through my complex analysis text and wanted to try something different. Cauchy's Integral Theorem is typically proved using an application of Green's Theorem and then by virtue ...
3
votes
1answer
143 views

Singularity of Generating Function

Given $E' = (E^2 + E - x)/2xE$ $xF = E^3 E' + 2xE^3 E'' + E^2 - x^2$ where $E = \sum_{n > 0}{e_n x^n}$ with $e_n = (n-1) \sum^{n-1}_{i = 1}{e_i e_{n-i}}$ for $n > 1$ and $e_1 = 1$ I am ...
3
votes
1answer
119 views

How do I prove that $\sin(π/2+iy)=1/2(e^{y}+e^{−y})=\cosh y$?

How do I prove that $\sin(π/2+iy)=1/2(e^{y}+e^{−y})=\cosh y$? Can you help please?
8
votes
1answer
180 views

Maximum distance between images of two points under an analytic function

Let $z$ and $w$ be two points in the complex unit disk, and let $f$ be a holomorphic function from the unit disk to itself (i.e. $|f| < 1$). Intuitively, it seems that the maximum value of $|f(z) - ...
3
votes
1answer
311 views

If $f$ is analytic in a region and at every point either $f\,' = 0$ or$ f = 0$, then $f$ is constant

Assume that $f$ is analytic in a region and that at every point, either $f\,'= 0$ or $f = 0$. Show that $f$ is constant. My attempt: $[f^{2}(z)]\,'=2f(z)f\,'(z)≡0$, so it would only be ...
7
votes
2answers
375 views

Using conjugate differential to determine existence of a harmonic conjugate?

Consider $u(z)=\ln(|z|^2)=\ln(x^2+y^2)$. I know that $u$ does not have a harmonic conjugate from $\mathbb{C}\setminus\{0\}\to\mathbb{R}$ but playing around with partial derivatives and integrating ...
5
votes
1answer
210 views

Why is this function odd?

Suppose a complex valued function $f$ is entire, maps $\mathbb{R}$ to $\mathbb{R}$, and maps the imaginary axis into the imaginary axis. I see that $f(x)=\overline{f(\bar{x})}$ on the whole real ...
4
votes
0answers
71 views

Convex pentagons are similar if conformally equivalent.

The problem: Suppose two convex pentagons $A$ and $B$ have equal interior angles (that is, $A=A_1A_2A_3A_4A_5$ and $B=B_1B_2B_3B_4B_5$) with $\angle A_j =\angle B_j$ for each $j\in\{1,\ldots,5\}$). ...
1
vote
1answer
85 views

how evaluate $\int_{-1}^{1} z^{\frac{1}{2}}\, dz$?

How can evaluate $\int_{-1}^{1} z^{\frac{1}{2}}\, dz$ with the main branch of $z^{\frac{1}{2}}$? Thanks for your help
6
votes
2answers
2k views

Prove that the composition of differentiable functions is differentiable.

Prove that the composition of differentiable functions is differentiable. That is, if $f$ is differentiable at $z$, and if $g$ is differentiable at $f (z)$, then $g\circ f$ is differentiable at $z$. ...
12
votes
5answers
683 views

Is it true that $\lvert \sin z \rvert \leq 1$ for all $z\in \mathbb{C}$?

Is it true that $\left\lvert \sin z \right \rvert \leq 1$ for all $z \in \mathbb{C}$ ? I think that is not true, can anyone help me?
2
votes
2answers
735 views

Uniform convergence vs. local uniform convergence for sequences of complex functions

A sequence of complex functions $\{f_n\}$ such that $f_i:D\rightarrow\mathbb C$ for all $i\in\mathbb N$, is locally uniformly convergent to $f$ if for all $P\in D$, exists a neighborhood $U$ of $P$ ...
3
votes
1answer
168 views

Uniform convergent series

Let $\Omega$ be domain in $\mathbb C^2$. For each compact set $K_j$ define the holomorphic function $f_j$ on $\Omega$, such that $$\sup_{k_j}|f_j|<2^{-j}.$$ Define $$f= ...
3
votes
1answer
925 views

Automorphisms of the unit disc

Define $$\phi_a(z) = \frac{z-a}{1-\overline{a}z}, \qquad \rho_\alpha(z) = e^{i\alpha}z,$$ with $|a|<1$ and $\alpha \in \mathbb{R}$, so that $\phi_a \circ \rho_\alpha$ is a holomorphic automorphism ...
0
votes
1answer
329 views

Elegant proof that a rational function has no essential singularity.

Theorem III of Townshends Complex Analysis has a proof that rational functions have no essential singularities. Just wondering, are there any particularly elegant proofs that a rational function has ...
5
votes
2answers
613 views

If $F$ is entire with removable singularity at $\infty$, then $F$ is constant?

On page 24 of Krantz's Complex Analysis, there is the following proof: Proposition 2: If $F$ is entire and $F$ has a removable singularity at $\infty$, then $F$ is constant. Proof: By ...
2
votes
1answer
220 views

Riemann surface diagram of $\frac1{\sqrt z}$

I wish to find what the Riemann surface diagram of $\dfrac1{\sqrt z}$ looks like. The problem I'm having is that this function is not defined for $z=0$, and that this function doesn't map any ...
5
votes
1answer
842 views

Use rectangular contour to integrate $\sin(ax)/(\exp(2\pi x)-1)$

I have been self-studying CA and find it very interesting. So, working through problems in a book I have, I ran across $$\int_{0}^{\infty}\frac{\sin(ax)}{e^{2\pi ...
7
votes
1answer
669 views

The power series $\sum\limits_{n=1}^{\infty} \frac {z^{n} }{ n^{2}} \ $

This is an exercise from Remmert. The power series $\sum\limits_{n=1}^{\infty} \frac {z^{n} }{ n^{2}} \ $ has radius of convergence $1 \ $. Show that the function it represents is injective in $\{ z ...
0
votes
1answer
291 views

Holomorphic functions

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function such that for an open interval $V \subset \mathbb{R}$ the following holds: $f(V)=0$. Does there exist an open set $U \subset ...
1
vote
1answer
365 views

To show given function is smooth

Let $\phi$ be upper semi-continuous function defined on $\Omega\subset \mathbb C^2$. Let $\Omega_n= \{z \in \Omega: d(z,\partial \Omega) > \frac1n \}$. Let $\chi\in C_c^\infty$ of $|z_1|$, ...
1
vote
1answer
184 views

Existence of the absolute value of the limit implies that either $f \ $ or $\bar{f} \ $ is complex-differentiable

This is an exercise from Remmert. Let D be domain in $\mathbb{C} \ $, and $f : D \rightarrow \mathbb{C} \ $ a real-differentiable function. Suppose that for some $ c \in D $ the limit $\lim_{h \to ...
2
votes
0answers
138 views

compute this integration

$$\int_{x_1}^{x_2}\frac{\sqrt{\frac{1}{3}x^3+a}}{(1-x)\sqrt{x}\sqrt{-\frac{4}{3}x^3+x^2-a}}dx$$ where $a\in(0,\frac{1}{12})$ is a constant. In this case, $-\frac{4}{3}x^3+x^2-a=0$ has exactly two ...