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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25 views

Laurent series of $e^{e^{\frac{1}{z}}}$ around $z=0$

Actually I need only the $res(f;0)$ where $f = e^{e^{\frac{1}{z}}}$ I thought of finding the Laurent series of $e^{e^{\frac{1}{z}}}$ around $z=0$ Any other Ideas if you have ?
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2answers
25 views

Suppose $a \in \mathbb{C}$, $|a| < 1$, and $f(z) = \dfrac{z - a}{1 - \overline{a}z}$. How to prove dependence of $|f(z)|$ on $|z|$? [duplicate]

Let $a \in \mathbb{C}$, $|a| < 1$. Also let $f(z) = \dfrac{z - a}{1 - \overline{a}z}$. I am asked to prove that $|f(z)| < 1$ if $|z| < 1$ and that $|f(z)| = 1$ if $|z| = 1$. What is a good ...
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0answers
36 views

Complex/Real Analysis mysterious quantity.

The following is a lemma to prove the Runge (only excerpt) page 629 in the link Can someone explain how they got the $$\frac{b_1 - b}{(z-b)}.$$ I believe he took $$\frac{1}{z - b}$$ and the other ...
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votes
1answer
8 views

Heine borel theorem on the complex plane

I'm trying to understand this proof of the Heine-Borel theorem on the complex plane. I'm reading Lang's Complex Analysis (page 22): I didn't understand the converse. Why there is a convergent ...
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1answer
16 views

Please could someone check my results for principal values of the complex logarithm?

I solved an exercise in my book and would greatly appreicate it if someone would check my result and tell me if it is correct:. The exercise: Find the principal values of the logarithm for the ...
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1answer
22 views

Showing this is analytic and finding its derivative $f(z)= \frac{4z+1}{z^3 - z}$

How to show the following is analytic and find it's derivative? $$f(z)= \frac{4z+1}{z^3 - z}$$ I am having trouble solving the above, since I am not sure how to break this into terms of $u,v$ for my ...
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vote
2answers
21 views

Determining whether $f(z)=\ln r + i\theta$ (with domain $\{z:r\gt , 0\lt \theta \lt 2\pi\}$) is analytic [duplicate]

Define $$f(z)=\ln r + i\theta$$ on the domain $\{z:r\gt , 0\lt \theta \lt 2\pi\}$. This domain is just a punctured disk of radius $\ln r$, correct? How does one determine whether this is ...
4
votes
1answer
28 views

Proving $\prod_k \sin \pi k / n = n / 2^{n-1}$

I am stuck trying to prove $$\prod_{k=1}^{n-1} \sin {\pi k \over n} = {n \over 2^{n-1}}$$ and I'd appreciate help. What I have done so far: $z^n - 1 = \prod_{k=1}^n (z - \xi^k)$ where $\xi = ...
1
vote
1answer
18 views

IS $f(z) = x^3 + i(1-y)^3$ analytic and where is it differentiable?

Where is $f(z) = x^3 + i(1-y)^3$ analytic and where is it differentiable? I have taken Cauchy-Riemann equations as follows: $$u(x,y) = x^3$$ $$v(x,y) =(1-y)^3$$ $$\frac{\partial u}{\partial ...
1
vote
1answer
23 views

Using de l'Hopital for complex functions?

I was wondering if de l'Hopital's rule also applies to complex functions. Some background information: This question came up as I was trying to calculate $\displaystyle \lim_{z \to 1} {z^n - 1 \over ...
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vote
1answer
11 views

Prove that there is a postive integer $n_0$ such that all the $z_n$ are nonzero for $n \le n_0$

Assume that a sequence $(z_n)$ of complex numbers converges to a nonzero limit. Then Prove that there is a postive integer $n_0$ such that all the $z_n$ are nonzero for $n \le n_0$ I know I should ...
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votes
1answer
27 views

Prove that $|\int_C f(z)dz| \le M |z_2 - z_1|$ where $M \gt 0$ such that $|f(z)|\le M; \ \forall \ z \in \Omega$

Let $z_1$ and $z_2$ be any two points in $\Omega$ and let $C$ be any oriented contour in $\Omega$ from $z_1$ to $z_2$. Also, assume that $f:\Omega \to \Bbb{C}$ is analytic on an open convex set ...
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vote
0answers
16 views

Use of Cauchy's intergral theorem (and consequences).

Let $f$ be an analytical function, with $|f(z)|\leq\displaystyle\frac{1}{1-|z|}$ for $|z|<1$. I have to prove that : ...
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votes
0answers
14 views

Find the set of points on which the maps of $e^z$ and $\log(z-1)$ are expanding and contracting.

I understand that $e^z$ is has a domain $\Omega$ such that $\Omega = \Bbb {C}$ and is analytic on the whole complex plane, but I have never been tasked with understanding the map of a function that is ...
1
vote
1answer
32 views

$f$ holomorphic, calculate $f(1+i)$ with two informations about $f$

Let $f(x+iy)=u(x,y)+iv(x,y)$ be a holomorphic function, knowing that : 1) $Im(f'(x+iy))=6x(2y-1)$ 2) $f(0)=3-2i$ Find $f(1+i)$. There is nothing in my notes, but I have read online that ...
4
votes
0answers
19 views

A metric such that every Blaschke factor is an isometry

Let $\mathbb{D} := \{z \in \mathbb{C} : |z| < 1\}$. Define $d : \mathbb{D}\times \mathbb{D} \rightarrow \mathbb{R}$ to be $$ d(z,w) := \left| \frac{z-w}{1-\overline{w}z} \right| $$ I am supposed to ...
0
votes
2answers
29 views

Find image of complex set:

Find image of set: $$ \{ z \in C : 0 \le Im (z), 0 \le Re(z) \}$$ and $$f(z)=\frac{i-z}{i+z}$$ I caclulate $ w=\frac{i-z}{i+z} $ and then $z=\frac{i(1-w)}{w+1}$ and don't know what to do next... I ...
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0answers
16 views

Analytic nowhere - C-R equations satisfied on line $(0,y)$

Analytic nowhere - C-R equations satisfied on line $(0,y)$ I have the C-R equations satisfied on the above line, but I imagine it is still analytic nowhere, since there is no open neighborhood where ...
4
votes
1answer
18 views

Fourier Series for a conformal map on unit disk

Given that a conformal map on the disk $\mathbb{D}$ will always have the form $f(z)=\lambda \displaystyle\frac{z-w}{1-\overline{w}z}$ for some $\lambda\in \partial \mathbb{D}$ and some $w\in ...
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votes
1answer
21 views

Solving an equation with complex numbers

I want to use complex numbers to solve the following problem: $x^2 = 95 - 168i$. I am sure there are a few ways of doing this but the way I want to do it is to let $x = a + bi$ and then solve for $a$ ...
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1answer
24 views

Image of Möbius transformation

What's the image of the first quadrant $Rez\ge0$ and $Imz\ge0$ under transformation $f(z)=(i-z)/(i+z)$? I know that real axis is mapped to the unit circle, $f(0+i*0)=1$ and $f(\infty)=-1 $.
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0answers
36 views

Showing $f(z) = e^ye^{ix}$ is defined on all $\Bbb C$

Now I asked how to determine if a function was defined on all of $\Bbb C$ in my previous question, but since that function was a polynomial, the answer was: Yes because its a polynomial, what if it ...
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votes
0answers
22 views

Geometric solution for $\frac{1-R}{| 1 - R e^{j\theta} |} = k$

Given: $\frac{1-R}{| 1 - R e^{j\theta} |} = k$ How to solve for R? (Suppose R is the only unknown quantity -- the task is to rearrange with R as subject). I encountered this problem in an academic ...
2
votes
1answer
27 views

Showing $f(z)=2xy+i(x^2+y^2)$ is defined on all of $\Bbb C$

What does it mean to be defined on all of $\Bbb C$? That is has no points at infinity? How do I show the below is defined on all of $\Bbb C$? $$f(z)=2xy+i(x^2+y^2)$$ Is it something to do with ...
2
votes
1answer
54 views

$\lim_{z\to 0} \frac{z}{\overline{z}}\text { does not exist }$

How can I make this rigorous? $$\lim_{z\to 0} \frac{z}{\overline{z}}\text { does not exist }$$ Proof: $$\lim_{z\to0}\frac{x+iy}{x-iy} \text{ taking } y\ne 0, x\to 0 \implies ...
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0answers
25 views

Prove that for every $g$ meromorphic there exists an entire function $f$ such that $f(z)\neq g(z)$ for all $z$ in $\mathbb C $

Prove that for every $g$ meromorphic there exists an entire function $f$ such that $f(z)\neq g(z)$ for all $z$ in $\mathbb C $. This problem is in pg 137 in Classical Topics in Complex function ...
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1answer
18 views

$\lim_{z\to\infty} \frac{(az+b)^2}{(cz+d)^2}=\frac{a^2}{c^2} \text{ if }c\ne0$

$$\lim_{z\to\infty} \frac{(az+b)^2}{(cz+d)^2}=\frac{a^2}{c^2} \text{ if }c\ne0$$ Now I am not sure how to prove this. Can I ignore the pesky square and do this? $$\lim_{z\to\infty} ...
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2answers
31 views

How find $\max _{z: \ |z|=1} \ f \left( z \right)$ for $f \left( z \right) = |z^3 - z +2|$

Let $f : C \mapsto R $, $f \left( z \right) = |z^3 - z +2|$. How find $\max _{z: \ |z|=1} \ f \left( z \right)$ ?
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0answers
23 views

Explain the geometrical interpretation a pair of harmonic function conjugated each other.

Explain the geometrical interpretation a pair of harmonic function conjugated each other. Could you help me? I am wondering how to draw it but unfortunately my abstract imagination can't cope with ...
1
vote
3answers
67 views

How to finish proof of $ {1 \over 2}+ \sum_{k=1}^n \cos(k\varphi ) = {\sin({n+1 \over 2}\varphi)\over 2 \sin {\varphi \over 2}}$

I'm trying to prove the identity $$ {1 \over 2}+ \sum_{k=1}^n \cos(k\varphi ) = {\sin({n+1 \over 2}\varphi)\over 2 \sin {\varphi \over 2}}$$ What I've done so far: From geometric series ...
2
votes
2answers
19 views

Solutions of an exponential function

Find all solutions of $e^{z} = -1+i$ These are the things for what I did: 1) Let $z=x+iy$ 2) $e^{iy} = e^{z} = √2(e^{i(3π/4+2kπ)})$ 3) Equate moduli and arguments to see that: ...
3
votes
1answer
21 views

What's the $z$-derivative of $|g|^2$ for $g(z)$ analytic?

Let $g\colon \mathbb{C} \to \mathbb{C}$ be holomorphic in a domain. What's $$\frac{\partial}{\partial z} (g \bar{g}).$$ I would think that since $\bar{g}$ is independent of $z$ (it's only dependent on ...
4
votes
1answer
53 views

How can I maintain notes while self studying Maths?

Thank you for stopping by this thread. I'm an engineering student rekindling an interest in Maths. I just love studying Maths in my free time (and sometimes it trespasses into my non free time). I ...
0
votes
1answer
36 views

Factorization $\cos(z) - \sin(z)$

How do I find the product expansion of $\cos z - \sin z$ We have $\cos z = \sin z$ iff $z = \pi/4 + k \pi$ where $k$ is an integer. The sequence $\sum (r/(|\pi/4 + k \pi|)^2$ converges For some ...
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vote
1answer
19 views

Limit of sequence of analytic functions

If $\Omega_1$ and $\Omega_2$ are two nonempty disjoint open subsets ${\bf C}$ and $\{f_n\}$ is a sequence of analytic functions from $\Omega_1 \to \Omega_2$ which converges pointwise to a function $f ...
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0answers
32 views

$\frac{1}{2i\pi} \int{\frac{1}{(z^{2}-a^{2})^{1/2}}dz}$

Can you please integrate $$\frac{1}{2i\pi} \int{\frac{1}{(z^{2}-a^{2})^{1/2}}dz}$$ over a circle of radius $R$ centered at the origin enclosing the points $z=+a$ and $-a$ where $a>0$ (Principal ...
2
votes
1answer
41 views

Integral with residues $\int_0^\infty \tfrac{\sin^2(x)}{1+x^4}dx$

I am trying to calculate $\displaystyle\int_0^\infty \dfrac{\sin^2(x)}{1+x^4}dx$ using method of residues. I have already seen this post, "Integrating $\int_{-\infty}^\infty \frac{1}{1 + x^4}dx$ ...
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1answer
32 views

$\int_{0}^{\infty}{\frac{x^{1/2}\log {x}}{1+x^2}dx}$

integrate in a keyhole contour and show that $$ \int_{0}^{\infty}{\frac{x^{1/2}\log {x}}{1+x^2}dx}=\pi^2/\sqrt(8)$$ and $$ \int_{0}^{\infty}{\frac{x^{1/2}}{1+x^2}dx}=\pi/\sqrt(2)$$ We use the ...
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0answers
17 views

Holomorphic Hermitian metrics

Let $E\to M$ be a complex vector bundle. A hermitian metric $h$ on $E$ is a hermitian inner product on each fiber $E_{p},\, p\in M$. Suppose that $M$ is also a complex manifold and that $E$ is ...
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vote
1answer
34 views

Keyhole Integration $\int_{0}^{\infty}{\frac{x^{k-1}}{(x+1)^2}dx}$

Can you please integrate $$ \int_{0}^{\infty}{\frac{x^{k-1}}{(x+1)^2}dx}$$ using the keyhole integration.I tried to integrate like in $$ \int_{0}^{\infty}{\frac{\log{z}}{(z+1)^2}dx}$$ but I couldn't ...
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vote
3answers
23 views

How to prove this complex binomial power series identity?

I am trying to prove the following: For $k \in \mathbb N$ and complex $z$ such that $|z|<1$: $$ {1 \over (1-z)^{k +1}} = \sum_{n \ge 0} {n+k \choose k} z^n$$ But I can't do it. My first idea was ...
0
votes
2answers
47 views

Integral of $\frac{\cos(2x) − \cos(x)}{x^2}$

Will you please solve $$ \int_{0}^{\infty}{\frac{\cos 2x− \cos x}{x^2}dx}$$ using indented contour. I tried like in $$\frac{\sin x}{x}$$ but couldn't figure out.
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votes
1answer
10 views

Evaluating this contour integral

Let $R$ be the rectangle with vertices at $-1$, $1$, $1+2i$, $-1+2i$. Compute $$\int_{\partial R} \frac{(z^2 +i)\sin(z)}{z^2+1}dz$$where the boundary of $R$ is traversed counterclockwise. Here is ...
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votes
3answers
49 views

Prove the identity $\frac{d}{dt} |z(t)|= |z(t)|Re \frac{z'(t)}{z(t)}$

Let $t \mapsto z(t)$ be a complex-valued function of the real variable $t$. Assume further that $z'(t) \not= 0$ for every $t$. Prove the identity $\frac{d}{dt} |z(t)|= |z(t)|Re \frac{z'(t)}{z(t)}$. By ...
3
votes
3answers
56 views

Using residues to evaluate the integral $\int_{-\pi}^{\pi} \frac{\cos(n\theta)}{1-2a\cos(\theta)+a^2}d\theta$, $|a|<1$

Calculate the integral for $\left|a\right|<1$ $$\int_{-\pi}^{\pi} \dfrac{\cos(n\theta)}{1-2a\cos(\theta)+a^2}d\theta$$ I'm supposed to evaluate this using method of residues, but the parameter a ...
0
votes
1answer
10 views

Do these intervals have the same image under the $e^z$ transformation?

Is the image of $D_1=\{z \in C: 0 \lt Re(z) \lt \infty, 0 \lt Im(z) \lt \pi\}$ the same as the image of $D_2=\{z \in C: 0 \lt Re(z) \lt \infty, \alpha \lt Im(z) \lt \beta \}$ as $0 \lt \alpha \lt ...
0
votes
1answer
19 views

Determine the order of entire function $G(z)=\sum_{n=0}^\infty \frac{z^n}{(n!)^\alpha}$.

Let $$G(z)=\sum_{n=0}^\infty \frac{z^n}{(n!)^\alpha}$$ for $\alpha>0$. Prove that it's an entire function and determine its order. Any suggestions please?
2
votes
1answer
31 views

Find all holomorphic functions with the following property

Let $D$ be the unit disc. Find all holomorphic functions $f:D\to D$ such that $f(\frac14)=\frac14$, and $f'(\frac14)=\frac7{15}$. I guess that we should use Schwarz lemma. And I guess that the only ...
2
votes
1answer
24 views

Complex differential operators

Consider the differential operators $\dfrac{\partial}{\partial z}$ and $\dfrac{\partial}{\partial \bar{z} }$ defined by $\frac{\partial}{\partial z} = \frac {1}{2} (\frac{\partial}{\partial x} - ...
4
votes
2answers
31 views

Prove that if $f$ is holomorphic so that $f'(z)=\alpha f(z)$ then $f(z)=ce^{\alpha z}$

Prove that if $f$ is holomorphic so that $f'(z)=\alpha f(z)$, $\alpha$ being a constant, for every $z \neq 0$ then $f(z)=ce^{\alpha z}$, $c \in \mathbb C$. So what I tried doing is defining ...