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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304 views

A complex polynomial with partial derivatives equal to zero is constant.

There is an exercise in Function Theory of One Complex Variable by Greene & Krantz that is very similar to a Proposition in the book, but I am having trouble getting to the conclusion. Let $f ...
2
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1answer
367 views

Image of a map in the complex plane

Is there an elegant way (either intuitive/ by a series of diagrams or by manipulating numbers/algebra) to find out what the image of $\sin(w)$ where $w\in \mathbb C$ from a domain say $\{w\in \mathbb ...
4
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2answers
424 views

Continuous function on the unit circle must be $c\bar{z}$

This is a homework problem, so hints or rough outlines are strongly preferred to a full solution. Problem. Let $C$ be the unit circle. Suppose the continuous function $f : C \rightarrow \mathbb{C}$ ...
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0answers
74 views

(RESOLVED) Interpreting a holomorphic function

The equation for and electric field is given by $E=−∇ψ$ where $\psi$ is the potential, and in this case $ψ=−Q\ln r$ where $Q$ is just some constant. I have found its harmonic conjugate to be $−Qθ+c$ ...
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2answers
613 views

For what values does $\sum_0^\infty\frac{z^n}{1+z^{2n}}$ converge?

Was playing around the series $\sum_0^\infty\frac{z^n}{1+z^{2n}}$, where $z$ is complex, trying to figure out where it converges. Assuming $|z|>1$ $$ ...
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1answer
126 views

How to do this directly rather than using Pick's Lemma

In the field of complex analysis, suppose the complex-valued function $w= f(z)$ is a conformal self-map of the open unit disk $\mathbb{D}$. Then in this particular case, we have equality in Pick's ...
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2answers
131 views

If $\lim_{n\to\infty}|a_n|/|a_{n+1}|=R$, why does $\sum a_nz^n$ also have radius of convergence $R$?

I'm trying to teach myself complex analysis, and I've been working on this idea. Suppose $\lim_{n\to\infty}|a_n|/|a_{n+1}|=R$, I would like to know why $\sum a_nz^n$ also has $R$ as its radius of ...
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1answer
194 views

Harmonic conjugate

I have been asked the following question and would appreciate an explanation. Suppose we have to find an analytic function $F(z)$ where $z=x+iy\in \mathbb C$ and its real part is $g(x,y)$. Question: ...
2
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1answer
276 views

Why are $\limsup \sqrt[n]{1/n!}=0$ and $\limsup\sqrt[n]{n!}=\infty$?

I was looking at the power series $\sum\frac{z^n}{n!}$ and $\sum n!z^n$, and wanted to compute their radii of convergence. For the first, $\limsup \sqrt[n]{1/n!})=0$, and for the second $\limsup ...
2
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2answers
1k views

Is $|z|^2$ complex differentiable?

I think I am a bit confused about the definition of (complex) differentiability. Yes, I know that's stupid, but I am hoping that someone could clear it up for me. I know that the definition of ...
0
votes
1answer
198 views

Complex differentiability

Would the complex function sech(z) be holomorphic because cosh(z) defined by its power series is holomorphic? I am not sure why sech (z) is even (complex) differentiable everywhere since surely it is ...
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1answer
232 views

Constructing the Riemann Sphere

Problem: In the construction of the Riemann sphere, we begin with the sphere $\mathbb{S}^2$ with two charts: the stereographic projection $\sigma_N : \mathbb{S}^2 \setminus \{N\} \to \mathbb{R}^2 ...
5
votes
2answers
105 views

Why does $\lim_{n\to\infty} z_n=A$ imply $\lim_{n\to\infty}\frac{1}{n}(z_1+\cdots+z_n)=A$?

I'm self-studying a bit of complex analysis, and I'm attempting to figure out the following. Suppose $\lim_{n\to\infty}z_n=A$. How can I show that $$ \lim_{n\to\infty}\frac{1}{n}(z_1+\cdots+z_n)=A. ...
0
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1answer
223 views

The sum of a series to be found by looking at a complex function.

I'm trying to solve the following problem: Find the sum of the series $\sum^{\infty}_{n=1}\frac{(-1)^n}{(2n+1)^3}$ by using the function $f(z)=|(2z+1)^3\sin\pi z|^{-1}.$ I can't see it... I do see ...
2
votes
2answers
156 views

Question from Freitag's *Complex Analysis*

Is it true that a function is analytic iff it satisfies the Cauchy-Riemann equations? I am reading Freitag's Complex Analysis and I am asked to show that ${\partial f\over \partial \bar{z}}=0$ iff $f$ ...
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1answer
69 views

Definition clarifications : on adjectives of functions

Could somebody please explain what are the differences between a differentiable function and a holomorphic function analytic function and conformal function? (Am I right to think that all analytic ...
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1answer
81 views

why not define $H^{p,q}_{\partial}(M)$?

Let $M$ be a complex manifold, $A^{p,q}(M)$ be $C^{\infty}$ $(p,q)$ form. Dolbeault cohomology $H^{p,q}_{\bar{\partial}}(M)$ is defined as the cohomology with boundary map $\bar{\partial}$, but why ...
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2answers
949 views

Bounded harmonic function is constant

Can you please help me to prove that bounded harmonic function is constant? Thanks a lot!
4
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1answer
159 views

Entire function invariant on the coordinate axes (as sets).

From old qualifying exam: Let $E$ be the union of the two coordinate axes, i.e. $E = \{z=x+iy : xy=0\}$. Describe all entire functions satisfying $f(E) \subset E$. I feel like the best approach is to ...
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5answers
4k views

How do I rigorously show $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is?

I'm doing a bit of self study, but I'm uncomfortable with a certain idea. I want to show that $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is analytic, and by analytic I mean ...
2
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1answer
146 views

If $z\bar{z}'=-1$, do $z$ and $z'$ correspond to opposite points on the Riemann sphere?

I know that if complex numbers $z$ and $z'$ correspond to opposite points on the Riemann sphere, then it must be the case that $z\bar{z}'=-1$. Is the converse true, that $z\bar{z}'=-1$ ...
2
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1answer
274 views

Existence of holomorphic n-th root and non-vanishing

Suppose $f \in H(\Omega)$, $\Omega =$ arbitrary region. Suppose $f$ has a holomorphic $n-$th root in $\Omega$ for every positive integer $n$. Then I need to show that $f(z)\neq 0$ for all $z \in ...
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2answers
359 views

On the Cayley (conformal) transform

Prove that the function $$ \begin{align} \phi (z) = i \dfrac{1 - z}{1 + z} \end{align} $$ maps the set $D = \{z \in \mathbb{C}: |z| < 1 \} $ one-to-one onto the set $U = \{ z \in ...
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1answer
3k views

How do the Laplace s-domain and the complex frequency domain differ?

In systems theory and signal processing, we often transform expressions based in the Laplace $s$-domain into the complex frequency domain with $j\omega$ (engineering notation for the angular frequency ...
5
votes
2answers
328 views

Differentiable and analytic function

I have the following function and I am trying to find if it is analytic and differentiable. I use cauchy-riemann to prove it. $$ f(x) = x^2 -x+y+i(y^2-5y-x)$$ $$u(x,y) = x^2-x+y$$ $$v(x,y) = ...
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0answers
126 views

another inequality involving complex numbers.

Let $\{z_i\}$, $i=1,2,\ldots,n$ be a set of complex numbers. Then I know that there is a set $J$ such that $$\left|\sum_{j\in J} z_j\right|\ge \frac{1}{\pi} \sum_{k=1}^n |z_k|. $$ However, how do I ...
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votes
4answers
657 views

finding roots of complex equation

I have here a complex equation: $$z^2 - (7+j)z + 24 +j7 = 0$$ How do we get the roots of this equation? I started using the quadratic formula $-b \pm \sqrt{ b^2-4ac}\over 2$, but it yielded too much ...
0
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1answer
210 views

Derivative of a bounded function

I was wondering for the bounded function $b(t)$ what statements can be made about the derivative of $f(t)=exp(b(t))$ specifically it would be nice if the derivative $f'$ were bounded.
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votes
1answer
54 views

sets in $\mathbb{C}$ questions

Show that a finite intersection of open sets in $\mathbb{C}$ is an open set in $\mathbb{C}$. Attempt: I want to show $\bigcap_{i=0}^{n}A_i$ is open. Let $z\in\bigcap^{n}A_i$ for open $A_i$ in ...
4
votes
1answer
655 views

Every Cauchy sequence in $\mathbb{C}$ is bounded

Prove that every Cauchy sequence in $\mathbb{C}$ is bounded. In $\mathbb{R}$, this is a sketch of the proof that I recall: Let {${a_k}$} be Cauchy in $\mathbb{R}$, since $1\in\mathbb{R}$, ...
13
votes
1answer
516 views

What do $dz$ and $|dz|$ mean?

I'm having a hard time understanding complex differentials. I know that when I have a field $\mathbb K$ and a $\mathbb K-$vector space $\mathbb K^n,$ then we define $dx_i\in \mathrm{Lin}(\mathbb ...
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0answers
210 views

Limit of exp(z) along a half-line

I'm still new to complex functions and not very confident yet, so I wonder if you all can check if I've understood this problem correctly: Suppose we take the limit of of $e^z$ as $|z| \to \infty$ ...
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0answers
132 views

What is wrong with these maps?

In my textbook, it is said that $z+1\over z-1$ maps the left half plane to the unit disk. So since it is its self-inverse, (right?) the unit disc should be mapped to the left half plane. But on ...
9
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1answer
632 views

If a rational function is real on the unit circle, what does that say about its roots and poles?

While doing a bit of self study, I ran across a situation whose wording confused me. Suppose $R(z)$ is some rational function which is real on the circle $|z|=1$ in the complex plane. The question ...
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vote
3answers
227 views

properties of sets in $\mathbb{C}$

(1) Show, by example, that an infinite intersection of open sets in $\mathbb{C}$ need not be an open set in $\mathbb{C}$. Consider $\bigcap_{i}^{\infty}A_i \subset \mathbb{C}$ for $A_i$ open. ...
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votes
2answers
81 views

Finding a third point

I have learnt that if we are given 3 points in the extended complex plane and their corresponding image points, we have a unique Möbius map that can perform the mapping. Suppose I have 2 orthogonally ...
3
votes
2answers
89 views

Could someone please explain these notes on Möbius maps to me?

Suppose $M(z)={az+b\over cz+d}$ is a Möbius map. Then $M'(z)={ad-bc\over (cz+d)^2}$, which is $\neq 0$ for $z\neq -{d\over c}$ or $\infty$. So we can say that $M$ is conformal at these points, so far ...
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1answer
106 views

The zero set of sums of polynomials

As I am new to this forum, please correct me if this post is not appropriate. In that case I apologize. Let $P(z)$ and $Q(z)$ be polynomials with coefficients in $\mathbb{C}$, furthermore let $Z(P)$ ...
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3answers
1k views

Complex differentiation

Is differentiation in the complex plane the same as that in the reals? In particular do the normal differentiation rules apply in the complex case such that I can just treat a complex map as a real ...
4
votes
1answer
138 views

The $n$ complex $n$th roots of a complex number $z$

Suppose $z$ is a nonzero complex number, so $z=re^{i\theta}$. Show that there are only $n$ distinct complex $n$-th roots, given by $r^{1/n}e^{i(2\pi k+\theta)/n}$ for $0\leq k\leq n-1$. My proof: ...
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3answers
96 views

Evaluate a complex set

Can you please help me finding an exact description of the set: $$ E_{R}=\{\cos{z} | z \in \mathbb{C}, |z|>R\} $$ For any $0<R \in \mathbb{R}$. My feeling is the $E_R = \mathbb{C}$, for any ...
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2answers
394 views

Why are $u(z)$ and $u(\bar{z})$ simultaneously harmonic?

I'm trying to learn a bit of complex analysis, and this idea has got me stuck. I would like to show that, for $u$ a function of a complex variable $z$, that $u(z)$ and $u(\bar{z})$ are ...
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2answers
377 views

Area of Validity of Writing an Exponential Integral as Sum of IntegralSinus and -Cosinus

I'm confused by the two online references shown below. To me, they give different areas of validity of writing an exponential integral as sum of integralsinus and -cosinus. On this Wiki page, I find ...
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3answers
102 views

finding bound for the integral

I am trying to get bound for the following integral $$ \int_0^{\infty}\frac{1}{|x|^r}dx, \mbox{for } 1\leq r< \infty $$ In particular, the bound of the form $\frac{constant}{r}$. Sorry, we can ...
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0answers
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Finding general harmonic polynomial of form $ax^3+bx^2y+cxy^2+dy^3$.

I'm trying to find the most general harmonic polynomial of form $ax^3+bx^2y+cxy^2+dy^3$. I write this polynomial as $u(x,y)$. I calculate $$ \frac{\partial^2 u}{\partial x^2}=6ax+2by,\qquad ...
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2answers
335 views

How to evaluate this integral?

How would I prove that$$\int_{0}^\infty \frac{\cos (3x)}{x^2+4}dx= \frac{\pi}{4e^6}$$ I changed it to $$\int_{0}^\infty \frac{\cos (3z)}{(z+2i)(z-2i)}dz$$, and so the two singularities are $2i$ and ...
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2answers
75 views

Simplifying this exponential equation

I am wondering how does $$\frac{{{e^{zk}}}} {{{z^2} + 1}} = \frac{1} {{2i}}\left( {\frac{{{e^{zk}}}} {{z - i}} - \frac{{{e^{zk}}}} {{z + i}}} \right)?$$ I can see that $z^2 + 1 = (z + i)(z − ...
4
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1answer
92 views

Fundamental domain for the group of transformations generated by $\tau \mapsto \tau + 2$ and $\tau \mapsto -1/\tau$

Define the following fractional linear transformations (acting on elements of $\mathbb C$): $T_2:\tau \mapsto \tau + 2$ $S: \tau \mapsto -1/\tau$ Let $G$ be the group of transformations generated ...
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2answers
130 views

Why is the there a $-i$ in this partial derivative?

Suppose $f(z)=u(z)+iv(z)$ is a complex function of a complex variable $z=x+iy$. In the book I'm reading, it states for real values $h$, the imaginary part $y$ is kept constant, so the derivative ...
2
votes
4answers
436 views

Calculating this residue (residue theorem)

Given $$\int_{\gamma}\frac{1}{(z-a)(z-\frac{1}{a})}dz,$$ and $0<a<1$, where $\gamma(t)=e^{it}$ and $0\le t \le 2\pi$ I am trying to find the residue of$$f(z)=\frac{1}{(z-a)(z-\frac{1}{a})}$$ ...