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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37 views

Verifying the complex integral: $\int_0^{\infty}\frac{\cos{ax}}{1+x^4}dx$

Verifying the integral: $$\int_0^{\infty}\dfrac{\cos{ax}}{1+x^4}dx$$ I started considering: $$\cos{x}=\dfrac{e^{ix}+e^{-ix}}{2}\implies \cos{ax}=\dfrac{e^{iax}+e^{-iax}}{2}?$$ So: ...
0
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1answer
13 views

Closedness of a complex $1$-form defined by homogeneous functions

How can I prove this 1-form is a closed one within the specific subset $A\subset \mathbb C$ ? $$\omega=\frac{f(x,y)}{xf(x,y)+yg(x,y)}dx+\frac{g(x,y)}{xf(x,y)+yg(x,y)}dy$$ Where $f, g\in C^1$ are ...
0
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0answers
36 views

Can the triangle function approximate the Gaussian curve for complex numbers?

I was thinking about approximating the Gaussian curve with a triangular curve. The graphs look like this: their respective functions are: $$ y_1(x) = t(x) = max(0, 1 - |x|)$$ $$ y_2(x) = e^{ - ...
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0answers
26 views

Roots of polynomial outside a vertical strip of $\mathbb C$

Let $P(z)$ be an arbitrary polynomial with real coefficients. I'd like to guarantee that all roots of $P$ have real parts outside the interval $(0, 1)$. Is there some simple condition on P that will ...
2
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1answer
144 views

Why are conformal mappings necessarily 1 to 1?

Say, by the Riemann Mapping Theorem, there exists a biholomorphic, conformal mapping from the upper half plane to the (open) unit disk (since the UHP is simply connected and is not the entire complex ...
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2answers
65 views

Find the Laurent series of $f(z)=\frac{1}{z(1-z)}$

I am having difficulties finding Laurent series of the above function, around these two domains: $$0<|z-1|<1$$ and $$|z-1|>1$$ The function $f(z)$ takes the form ...
2
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1answer
37 views

Harmonic complex function

Can anyone help me with this question? Show that a $C^2$ function, $f:U\longrightarrow \mathbb C$, is harmonic iff $\frac{\partial }{\partial z}\frac{\partial f}{\partial \bar z}\equiv0$. Thank you. ...
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4answers
187 views

Help with the contour for this integral using residues

$$ PV \int_0^\infty \frac{dx}{\sqrt{x}(x^2-1)} $$ A keyhole contour can't be used because we have a pole in the real positive axis, isn't it?
5
votes
1answer
107 views

Definite integral with logarithm and arctangent inside of arctangent

How to prove $$\int_0^1 \left[ \frac{2}{\pi }\arctan \left(\frac 2 \pi \arctan \frac{1}{x} + \frac{1}{\pi }\ln \frac{1 + x}{1 - x}\right) - \frac{1}{2} \right]\frac{\mathrm{d}x} x = \frac{1}{2} \ln ...
0
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1answer
19 views

Determine a meromorphic function satisfying certain conditions

Suppose $f$ is meromorphic on the Riemann sphere, and suppose also that $f(0) = 0$, $f(-1) = 2$, $f(3) = 3$, $f$ has a simple pole at $1$ with residue $1$, and $f$ has a triple pole at $2$ with ...
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1answer
30 views

Solving $\int_{0}^{+ \infty} \frac{x \cos(x)}{x^4 + 4 a^4} dx$ with residues

We also have the condition $a > 0$. My attempt was to, as usual, define $f(z) = \displaystyle\frac{z e^{iz}}{z^4 + 4 a^4}$. Then I tried to integrate $f$ over a curve $\gamma$ which goes from $0$ ...
0
votes
2answers
33 views

Proof composition of analytic functions is analytic

Title says it all I looked for a proof on this site but couldn't find one. Prove if $f$ is analytic on $D$ and $g$ is analytic on $\Omega$ containing the range of $f$ show $g(f(z)$ is analytic. ...
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2answers
55 views

Complex analysis: Prove a meromorphic function to be rational.

I come across a problem about complex analysis: Show that a meromorphic function on the complex plane, which achieves any complex number no more than fixed given times, must be rational. The only ...
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1answer
63 views

The “argument” of a quaternion

My question is pretty simple. I've been trying to read a pretty introductory text on Clifford algebras, and I encountered how they define the "argument" of a quaternion as an ordered quadruple ...
3
votes
3answers
38 views

Where do these Mobius transformations map the coordinate half-planes?

They are $$\frac{z-1}{z+1}, \frac{z+1}{z-1},\frac{z-i}{z+i},\frac{z+i}{z-i}.$$ All four look virtually identical, so I would like to know how to best distinguish between them. For example, the ...
6
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1answer
87 views

Convergence of $\prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)$

I'm looking at some notes from my previous complex variables course and I need help verifying some things about the convergence of $$ \prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right) $$ on compact ...
3
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1answer
37 views

How to show that $\int_{S}\frac{1}{z}\frac{1}{\cos(2\pi i a)-\cos(2\pi z)}dz\rightarrow 0$ as the sides of the square $S$ go to $\infty$.

I have a question where I am asked to show that the following sum is $$\sum_{k=-\infty}^\infty\frac{1}{a^2+k^2}=\frac{\pi}{a}\frac{e^{2\pi a}-e^{-2\pi a}}{e^{2\pi a}+e^{-2\pi a}-2}$$ by integrating ...
1
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3answers
87 views

Solve integrals using residue theorem? [on hold]

$$\int_{0}^{\pi}\frac{d\theta }{2+\cos\theta}$$ $$\int_{0}^{\infty}\frac{x }{(1+x)^6} dx$$ My problem is that I don't know how to start solving these integrals, or how to convert them into usual ...
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1answer
29 views

Separate real and imaginary part of $\arccos(z)$

Beginning with $$i \cos \left[ \frac{1}{n} \arccos \left( \frac{i}{\epsilon} \right) + \frac{m \pi}{n} \right]$$ where $m,n \in \mathbf{Z}$, $\epsilon >0$, $\epsilon \in \mathbf{R}$ and $i$ is ...
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0answers
43 views

Using Rouché prove that $a_0\geq a_1 \geq … \geq a_n > 0$ the polynomial $P(z)=a_n z^n + … + a_1 z + a_0$ has no roots $|z|<1$

As asked in the title I am trying to prove this using Rouché's theorem. But I am having a hard time trying to find a second polynomial $f(z)$ such that $$|P(z)-f(z)| < |f(z)|$$ on the unit ...
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0answers
18 views

Total derivative of a complex function using Wirtinger derivatives

The mathworld.wolfram page on Cauchy-Riemann Equations states that given a complex function $f(z) = f(x,y) = u(x,y) + iy(x,y)$ has the derivative: $$\frac {df} {dz} = \frac {\partial f} {\partial x} ...
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3answers
58 views

Complex numbers $z$ satisfying $|z−a|+|z+a| = 2|b|\Leftrightarrow |a|\le |b|$.

could you help me to prove the next statement? Show that there are complex numbers $z$ satisfying $$|z−a|+|z+a| = 2|b| $$ if and only if $|a| \le |b|$. I did the first implication using the ...
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1answer
23 views

convergence radius and sum of a series

1) Given $S(z)$ find the radius of convergence and sum. $$S(z)= \sum_{n \geq 1} \frac{(4z-2)^n}{n}$$ Then: $$S(z)= \sum_{n \geq 1} \frac{(4z-2)^n}{n} = \sum_{n \geq 1} ...
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2answers
37 views

Solve complex exponential equation

I need to solve an expression of this kind (solve for $x$): $e^{\pi i x} -e^{-\pi ix} = 2yi$ Both $x$ and $y$ are real numbers, $y$ is given. I have no clue on how to solve it analytically. All I ...
2
votes
1answer
56 views

Using the complex logarithm as a conformal mapping,

I want to map the upper half plane, y>0, conformally onto the semi-infinite strip u>0, $-\pi < v < \pi$ in the w-plane. I then studied the complex logarithm, and noticed that the principal ...
0
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1answer
21 views

Proper definition of concyclic?

Let $z_1,z_2...,z_n$ be points in the complex plane, then if there exists $Z$ such that $$\vert Z-z_k\vert=a\in\{\text{Real Numbers}\}$$for all $k\in \{1, 2, 3...,n\}$, then $z_1,z_2...,z_n$ are ...
2
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2answers
37 views

Why does this sum equal zero?

Le}t $\gamma$ be a piece-wise, smooth, closed curve. Let $[t_{j+1}, t_{j}]$ be an interval on the curve. Prove, $$\int_{\gamma} z^m dz=0$$ In the proof it states $$\int_{t_{j}}^{t_{j+1}} ...
0
votes
1answer
22 views

Meromorphic function with constant magnitude on closed unit disk

I was reading the solution of a problem in the book "Berkeley problems in mathematics", and last part of the proof is unclear to me. It somehow proves using Maximum principle that the rational ...
2
votes
3answers
52 views

Find the sum of the roots given no multiple roots.

Find the sum of the roots, real and non-real, of the equation $$ x^{2001} + \left( \frac{1}{2} - x \right)^{2001} = 0 $$ given that there are no multiple roots. I am in a weird situation here. ...
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2answers
101 views

Showing properties of a subset of $\mathbb{C}$

Let $\omega(k)=\alpha_{n}k^{n}+\alpha_{n-1}k^{n-1}+\dots+\alpha_{0}$ be a polynomial of degree $n$ on $\mathbb{C}$. Define $D=\{k\in\mathbb{C}:\text{Re}(\omega(k))<0 \}$. How do i show that ...
1
vote
1answer
38 views

Finding Laurent series of a function $f(z)=\frac{3z^2-6z+1}{(z-1)(3z-1)}$ [on hold]

How do i transform this function into Laurent series $$f(z)=\frac{3z^2-6z+1}{(z-1)(3z-1)}$$ where $ \frac{1}{3} < |z| < 1 $.
0
votes
1answer
19 views

Show that the inclusion relation of Hardy spaces is proper

The definition of Hardy spaces for the unit disk is here. It is clear that for $0<p<q\le\infty$, $H^q\subseteq H^p$, by Hölder's inequality. I'm asked to show that the inclusion relation is ...
0
votes
2answers
31 views

Could someone explain a step in this proof please

Let $u(z)$ be a continuous function in the disc $|z-z_0|<r$ and let $\gamma_{\epsilon}$ be the circle $|z-z_0|=\epsilon$. Prove, $$\lim_{\epsilon \to 0} \frac{1}{2\pi i} \int_{\gamma_{\epsilon}} ...
6
votes
0answers
116 views

why this three complex are equal (2014 Pan African olympaid problem) [on hold]

EDIT(This following is 2014 Pan African olympiad problem) Let $$H(p,q)=\dfrac{\omega p}{\omega-1+a(\omega p-q)},a>0$$ where $\omega^3=1,\omega\neq 1$,if ...
0
votes
1answer
35 views

Residue of $\frac{e^{iz}}{z^2+4z+5}$ [on hold]

I need to find the residue of $\dfrac{e^{iz}}{z^2+4z+5}$ at its singular points. How do I do that?
0
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0answers
36 views

Show the integration with a complex variable

I want to show that there exists inverse Laplace transform, $f(t)$ of the function $F(\lambda)$. In other word, given $F(\lambda)$, existence of function $f(t)$ such that $$ ...
1
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0answers
27 views

A holomorphic line integral?

Let $f$ be a holomorphic function on a connected open set $U$ which contains the unit disc $D$, and suppose $|f(z)| \geq 1$ for $|z| = 1$. For $w \in D$, the argument principle tells us that ...
0
votes
1answer
37 views

Square root of a function

Let $D$ be a circular annulus in $\Bbb C$ with center at $0$. Put $v(z)=z$, for every $z\in D$. Show that $v$ has no square root measurable function. I think if define function $h$ such that ...
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30 views

When is a line integral of a holomorphic function holomorphic?

Let $U$ be a connected open set, $z_0 \in U$, and $g$ holomorphic on $U$. I know that if $U$ is simply connected, then $$G(z) := \int_{z_0}^z g(w)dw$$ is holomorphic on $U$, and $G'(z) = g(z)$. Here ...
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0answers
36 views

On the definition of modular forms

In many books, I see people defining modular forms to be holomorphic/meromorphic functions in the upper half plane such that it is invariant under the $|_k$ action of the group ...
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votes
3answers
74 views

find all $v(x,y)$ so that $f(x+iy)=u(x,y)+iv(x,y)$ is entire

I'm practicing to write down solutions clearly and thoroughly. Is this a proper answer to this exercise? How can it be improved? Let $u(x,y)=x^3-3xy^2$, find all $v(x,y)$ so that ...
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0answers
54 views

Find $\int_0^{2\pi} \frac{d\theta}{2\pi\cos^{2n}(\theta)} \ n=1,2,3,\dots$ via Residue Theorem

So the question is as follows: Use the Residue Theorem to calculate $$\int_0^{2\pi} \frac{1}{2\pi\cos^{2n}(\theta)} d\theta \quad\quad n=1,2,3,\dots.$$ Now I believe the first step would be to use the ...
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0answers
27 views

If $f=u+iv:D\to \Bbb C$ is analytic on a domain D, is then the curves $u(x,y)=c_1$ and $v(x,y)=c_2$ intersect orthogonally?

If $f=u+iv:D\to \Bbb C$ is analytic on a domain D (an open connected subset of $\Bbb C$), is then the curves $u(x,y)=c_1$ and $v(x,y)=c_2$ intersect orthogonally, for any constants $c_1$ and $c_2$? ...
6
votes
1answer
61 views

Does there exists an entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line?

Does there exists a nonconstant entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line? Clearly,$ f(z)=sin(z)$ is an example of an entire function which is ...
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0answers
34 views

Integral curves of complex function components

How to generally find integral curves $F(x,y,c) = 0$ from $u(x, y), v(x, y)$ when $$ x'(t) = u (x, y),\; y'(t) = v (x, y) $$ where $u, v$ are real and imaginary parts functions of a complex ...
3
votes
2answers
37 views

plot graph of function $f(z)=\frac{1+z}{1-z}$

I am not able to plot graph of function $f(z)=\frac{1+z}{1-z}$. can anyone tell me how to do this without using any software?
3
votes
2answers
68 views

Proving that the cross ratio is a Möbius transformation

I'm trying to show that given three distinct points $z_1,z_2,z_3\in\mathbb C$, the rational function $$ f(z) = \frac{(z-z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)} = \frac{(z_2 - z_3)z + (z_1z_3 - ...
1
vote
1answer
21 views

Extending functions on proper subsets of $\mathbb C$ to functions on proper subsets of $S^2$.

There are a number of nice results about extending holomorphic and meromorphic functions from the complex plane $\mathbb C$ to the Riemann sphere $S^2$. See for instance Does entire function extend ...
4
votes
2answers
49 views

Complex integral with exponential and tangent

Suppose that $k \in \mathbb{R}.$ Evaluate as a function of $k$ the integral $$I(k) : = \int_{-\pi/2}^{\pi/2} e^{i \ k \ \mathrm{tan}(\phi)} d \phi.$$ Any suggestions on how to approach this problem? ...
3
votes
1answer
37 views

Why this map is a mobius transformation

Question: Let $D_2=\bar D(2,1)$ and $D_{-2}=\bar D(-2,1)$ be the closed disks of radius $1$ centered at $z=2$ and $z=-2$ in the complex plane, respectively. Set $X= \mathbb C-\{D_2 \cup D_{-2} \}$, ...