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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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 ...
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1answer
78 views

Let $f(z)=e^x + ie^{2y}$ where $z=x+iy$. Where does $f'(z)$ exist?

Let $f(z)=e^x + ie^{2y}$ where z=x+iy is a complex variable defined in the whole complex plane. a)Where does f'(z) exist? b) Where is f(z) analytic? Answer: a) I used the Cauchy Riemann to test ...
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1answer
68 views

Tricky question regarding holomorphy

Question: Find all holomorphic functions $f(z)$ on $C \setminus \{0\}$ such that $$f(1) = 1,\ \ \ \ \ \ \ |f(z)| \le \frac{1}{|z|^3}$$ Attempt at solution: I've discovered that $f(z) = ...
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1answer
84 views

Two reflections will evidently result in a (fractional) linear transformation

I'm having hard time figuring out the following sentence in my textbook. "Two reflections will evidently result in a (fractional) linear transformation" I'm confused because I don't know if the two ...
2
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1answer
173 views

Definite integral involving hyperbolic cosine

I have had no experience so far with hyperbolic functions so any help will be appreciated. This is on the chapter of complex integration but I would especially appreciate it if you could turn this ...
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2answers
148 views

Computation of a certain integral

I would like to compute the following integral. This is for a complex analysis course but I managed to around some other integrals using real analysis methodologies. Hopefully one might be able to do ...
2
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1answer
171 views

The differentiability of $f(z)$ if $ \lim\limits_{z \to z_0} \frac {|f(z)-f(z_0)|}{|z-z_0|}=k$?

Let $f:\Omega \to \mathbb C$, where $\Omega$ is a region in $\mathbb C$, and $z_0 \in \Omega$. Suppose that $\displaystyle \lim_{z \to z_0} \frac {|f(z)-f(z_0)|}{|z-z_0|}=k$ for some constant k ...
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58 views

If $h(k)$ preserves the complex conjugation property, does $ikh(k)$ preserve it too?

My question is this: if $h(k)$ preserves the complex conjugation property (in other words, $h(k) = h(-k)$, $k$ can be just $-n$, $-n+1$, ..., $0$, $1$, ..., $n-1$),then $ikh(k)$ also preserves complex ...
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1answer
39 views

A simple computation of log function

I just want to make sure I got the right calculation. $$\log[(1+i)^{2i}]=\log[e^{i\ln2-\pi/2-4k\pi}]=i\ln2-\pi/2-4k\pi=i\ln2-\pi/2.$$
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2answers
257 views

Locate the singular points and state whether it is a pole, a removable singularity, or an essential singulatity: $z/(e^z-1)$

Locate the singular points and state whether it is a pole, a removable singularity, or an essential singulatity: $$f(z) = \frac{z}{e^z - 1}.$$ I obtained $z=0$. But I don't understand how to ...
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1answer
103 views

what is the difference between the Argument Principle and Rouché's Theorem

What is the difference between the Argument Principal and the Rouche's Theorem. I am not sure when to use which one when I have a question for example: How many roots of $z^4 + z^3 + 1=0$ lie ...
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0answers
66 views

Type of isolated singularity of holomorphic funtion unique?

I am using "Real and complex analysis" by rudin. I have just read the proof concerning the three different types of isolated singularities, and that one of them must occur. The book doesn't say ...
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4answers
313 views

Why $\displaystyle f(z)=\frac{az+b}{cz+d}$, $a,b,c,d \in \mathbb C$, is a linear transformation?

Now I'm confused with what "a linear transformation" means. In linear algebra textbook, I learned that a linear transformation is $T:V \to W$, where V,W are vector spaces, which satisfies additivity ...
3
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2answers
51 views

Is $f(z)=z^n+nwz$ one-to-one?

Let $f(z)=z^n+nwz$ be a complex function with $|w|=1$ and $n>1$ a natural number. Is this function one-to-one inside the unit circle ($|z|<1$)? ATTEMPT I didn't have a lot of luck checking ...
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133 views

Prove Complex Function Is Holomorphic

Prove for $a\gt0$ that following series is holomorphic $$ \sum_{n=1}^\infty \frac {1}{(a+n)^z} \quad \textrm{for} \quad \operatorname{Re}z \gt 1 $$ I'm trying to prove this given that $Re \quad z ...
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125 views

Need some help with my Complex Analysis textbook

I'm confused with a proof in "Complex Analysis" by Ahlfors.(P.73 2.3 comformal mapping) I need some help for the last part of the proof. "Let $f:\Omega\to \mathbb C$ and fix $z_0\in \Omega$. ...
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2answers
389 views

Non-existence of a bijective analytic function between annulus and punctured disk

Suppose $A=\{z\in \mathbb{C}: 0<|z|<1\}$ and $B=\{z\in \mathbb{C}: 2<|z|<3\}$. Show that there is no one -to-one analytic function from A to B. Any hints? Thanks!
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0answers
181 views

Problem calculating with the residue theorem

I've come across this integral and I'm having some problems with it. I get to a solution, but looks a bit weird and I may be doing something wrong. $\int_C\cos(e^{(1/z)})dz $ Being $C$ the unit ...
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2answers
1k views

How to find all Laurent series for $\frac{1}{z^2 - z}$ centered at $1$?

I have two questions. The definition I learned defined a laurents for a function analytic in some annalus? I guess this question is asking annali centered at $1$? How do I determine all the ...
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1answer
96 views

Can we give a definition of the cotangent based on a functional equation?

I've recently learned that the cotangent satisfies the following functional equation: $$\dfrac1{f(z)}=f(z)-2f(2z)$$ (true for $f(z)\neq 0$). Can we solve this equation for real or complex ...
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502 views

Show that the partial derivatives of a harmonic function fit into a holomorphic function

Let G be a map in $\mathbb C$ ( $G \subseteq \mathbb C $) and $u: G\mapsto \Bbb C$ is harmonic function. Then show that $f: G\mapsto \Bbb C$ $$ f(x+iy)=\frac{du}{dx} - i\frac{du}{dy}$$ is ...
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195 views

How to find the number of roots using Rouche theorem?

Find the number of roots $f(z)=z^{10}+10z+9$ in $D(0,1)$. I want to find $g(z)$ s.t. $|f(z)-g(z)|<|g(z)|$, but I cannot. Any hint is appreciated.
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1answer
281 views

How to calculate this integral using residue theorem?

$f,g$ are holomorphic in $D(0,1)$. $P_1,P_2,...,P_k$ are roots of $f$ in $D(0,1)$. their orders are $n_1,...,n_k$. Compute $$\frac{1}{2\pi i}\oint_\gamma\frac{f'(z)}{f(z)}\cdot g(z)dz.$$ Using ...
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1answer
200 views

Is there a conformal self-map on the upper half-plane that swaps two points?

Again, is there a conformal self-map that interchanges two points in the upper half-plane? I'm beginning to think this isn't so. Such a map would be a FLT $\frac{az+b}{cz+d}, ad-bc=1$, with real ...
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1answer
152 views

problems related to poles and essential singularity

I was thinking about the problem which says: Let $f(z)$ be a function defined by: $$f(z)=\frac{\sin(1/z)}{z^{2}+11z+13}.$$ Then which of the following is correct? (a) No singularity. (b) Only ...
3
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2answers
227 views

Proving that $\sum_{|j| < n} (n-|j|) \exp(ij\lambda)= \frac{\sin^2(\frac 1 2 n\lambda)}{\sin^2(\frac 1 2 \lambda)}$

I want to show that $\sum_{|j| < n} (n-|j|) \exp(ij\lambda)= \dfrac{\sin^2(\frac 1 2 n\lambda)}{\sin^2(\frac 1 2 \lambda)}$. I know from Proving ...
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315 views

Taylor Series of the Complex Log and Contour Integration

Write the Taylor series of $\text{Log}(1+w)$ with center at $w=0$ on $|w|<1$; check that if $|z-2|<1$, then $|z|>1$. (If you have difficulties in checking this formally, try to draw a ...
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1answer
96 views

The problem and definition of principal part

In James Brown's complex variables and applications, there is an exercise: Let $f \left( z \right) = \frac{8 a^3 z^2}{\left( z^2 + a^2 \right)^3}$ with $a > 0$. Show that the principal part of $f ...
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1answer
699 views

Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer.

Using residues, try the contour below with $R \rightarrow \infty$ and $$\lim_{R \rightarrow \infty } \int_0^R \frac{1}{1+r^n} dr \rightarrow \int_0^\infty \frac{1}{1+x^n} dx$$ I've ...
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1answer
280 views

Given two fixed points on unit disk,find analytic functions from unit disk to unit disk that maximize the distance between values at the two points

Given $z, w \in D$ (unit disk, open), what are the functions (analytic, from unit disk to unit disk) $f$ that maximize the norm of $f(z)-f(w)$? My attempt: We have that $$|f(z)-f(w)| \leq ...
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1answer
120 views

Quasiconformal map between the complex plane and a disk

According to the Poincaré-Koebe theorem, it is known that the unit disk $\mathbb D$ and the complex plane $\mathbb C$ aren't conformally equivalent. My question is maybe naive, but I was wondering if ...
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5answers
3k views

What are the solutions to $z^4+1=0$?

I can't seem to find the solutions to $z^4+1=0 $. $z$ is in the complex plane. The solutions show four roots; however, how do I find them once $z^4 = -1$?
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1answer
52 views

Calculate $\sum_{n=0}^{\infty} z^{n-2}/5^{n-1}$ for $0<|z|<5$ [duplicate]

Possible Duplicate: Complex series: $\sum_{n=0}^\infty\left( z^{n-2}/5^{n+1}\right)$ for $0 &amp;lt; |z| &amp;lt; 5$ I don't even know where to start. I can't think of any formulas ...
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3answers
244 views

Is the complex derivative “speed”?

The first thing I was told about the real derivative is that it's "how fast the function is growing" at a given point. This interpretation wasn't addressed in my complex analysis classes. Can the ...
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1answer
111 views

exercise on complex numbers

Let $$f(z)=\frac{z-a}{z-b}$$ with $a,b\in D(0,r)$ and $r>0$. Let $$E=\{z\in\mathbb C: f(z)\notin N\}$$ $$N=\{Re(z)\leq 0;Im(z)=0\}$$ How can i find $E$ in terms of $r$?
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1answer
139 views

rational function with special properties on unit disk.

I'm now solving the following complex analysis problem. "determine the form of rational function in a plane hat has a positive value on unit circle." hint suggested me that such a rational function ...
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0answers
212 views

Three complex analysis problems

Find an open connected set $G$ and two continuous functions $f$ and $g$ defined on $G$ such that $f(z)^2 g(z)^ 2 =1- z^2$ for all $z\in G$. Can you make $G$ maximal? Are $f$ and $g$ analytic? Give ...
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2answers
86 views

If $f$ is entire and $z=x+iy$, prove that for all $z$ that belongs to $C$, $\left(\frac{d^2}{dx^2}+\frac{d^2}{dy^2}\right)|f(z)|^2= 4|f'(z)|^2$

I'm kind of stuck on this problem and been working on it for days and cannot come to the conclusion of the proof.
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1answer
62 views

Let $S=(a_n)_{n=1}^{\infty}$ be a sequence in $\mathbb C$ and $S'$ the set of limits of $S$. Prove that every limit point of $S'$ is a member of $S'$

It seems obvious that a limit point of $S'$ should be a member of $S'$ but I have no idea how to even begin with a proof of this.
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1answer
128 views

Find all functions analytic in $D \subset \mathbb{C}$.

Suppose a complex-valued function $f$ is analytic in the domain $D$ where $D$ is the disk $|z| < R$. If $f(0) = i$, and $|f(z)| \le 1$, what is $f$? I'm thinking that $f$ is just the constant ...
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2answers
150 views

Why does this integral (of the Schwarz kernel) define a holomorphic function in the unit disc?

(This may look very silly to you but I don't understand the reason that was given to me, nor do I have the knowledge to find out by myself). The domain for $z$ is the open unit disc ...
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1answer
551 views

Can the real and imaginary parts of $\dfrac{\sin z}z$ be simplified?

I have calculated the real and imaginary parts of $\dfrac{\sin z}z.$ I've obtained $$\begin{eqnarray} \frac{\sin z}z&=&\frac{\sin(x+iy)}{(x+iy)}\\ &=& ...
3
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1answer
510 views

Doesn't a function with constant modulus on the boundary of a bounded domain have constant modulus over the entire domain?

I'm having difficulties with a question from Complex Analysis (Gamelin). The question has been asked before, but I still have some difficulties with it. It asks to show that a function continuous on ...
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1answer
143 views

Fouriercoefficient of the sawtooth wave help to find that the bessel equation gives $\sum \frac{1}{k^2}=\frac{\pi^2}{6}$

What are the complex fourier coefficients of the function $f:\mathbb{R}\rightarrow \mathbb{R}$ defined by the $2\pi-periodic$ continuation of $f(x)=\pi-x$ , for x $0\le x < 2\pi$ ? And how can ...
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2answers
66 views

Results of dot product for complex functions

Suppose we are given a $C^1$ function $f(t):\mathbb{R} \rightarrow \mathbb{C}$ with $f(0) = 1$, $\|f(t)\| = 1$ and $\|f'(t)\| = 1$. I have already proven that $\langle f(t), f'(t)\rangle = 0$ for all ...
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473 views

contour integral with singularity on the contour

I want to compute the following integral $$\oint_{|z|=1}\frac{\exp \left (\frac{1}{z} \right)}{z^2-1}\,dz$$ The integrand has essential singularity at the origin, and $2$-poles at $\pm 1$,which lie ...
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2answers
292 views

Removable singularity of $f(z)=\dfrac{\sin^2 z}{z}$

I am having trouble understanding when a function might have a removable singularity over a pole. For example: $$f(z)=\frac{\sin^2 z}{z}$$ I believe the pole is at $z=0$. However, if we take the ...
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1answer
77 views

Question about derivatives of complex-valued functions

For $ z \in \mathbb{C}, t \in \mathbb{R}, \\f : \mathbb{C} \times \mathbb{R} \to \mathbb{C}, \\a : \mathbb{C} \times \mathbb{R} \to \mathbb{R}, \\b : \mathbb{C} \times \mathbb{R} \to \mathbb{R}$ And ...
3
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0answers
61 views

Perturbations of algebraic varieties

Let $P(z,w):\mathbb C^2\to\mathbb C$ be a certain polynomial, and consider $p(s,t)=P(e^{is},e^{it})$ its restriction to the torus. In the specific problem I'm considering, the set $Z=\{(s,t): ...
2
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1answer
287 views

Conformal mapping from triangle to upper half plane in terms of Weierstrass $\wp$

I'm trying to explicitly compute a conformal map $f:\Delta \rightarrow \mathbb{H}$ where $\Delta$ is a triangle and $\mathbb{H}$ is the upper half plane, in terms of the Weierstrass $\wp$ function. I ...