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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2
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1answer
50 views

The equation of a circle on a complex plane?

The equation of a circle $|z-z_0|=r$ in a complex plane has (among others) the form: $$z\overline{z}+\overline{b}z+b\overline{z}+c=0$$ where $b=-z_0 \in \mathbb{C}$. What I'd like to understand is, ...
0
votes
1answer
63 views

To prove $\overline{ f(\bar{z})}$ is analytic [duplicate]

Let $f(z)$ analytic. Prove that $\overline{ f(\bar{z})}$ is also analytic. How do I use the concept of analytic of $f(z)$ here? any help
5
votes
3answers
356 views

Existence of bounded analytic function on unbounded domain?

Given any proper open connected unbounded set $U$ in $\mathbb C$.Does there always exist a non constant bounded analytic function $ f\colon U \to \mathbb C$ ? Edit: $U$ is any arbitrary domain. ...
4
votes
1answer
66 views

Is $\frac{z-\alpha}{1-\overline{\alpha}z}$ some special function in complex analysis?

Many homework problems seem to use the following function (or something very close to it): $$F_\alpha(z)=\frac{z-\alpha}{1-\overline{\alpha}z}$$ Does it serve some special purposes in complex ...
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vote
1answer
25 views

Let $R=[p_1,q_1]\times \cdots\times[p_n,q_n]$ and show that diam $R=d(p,q)=[\sum_{k=1}^n (q_k-p_k)^2]^{1\over 2}$.

Let $p=(p_1,p_2,...,p_n)$ and $q=(q_1,q_2,...,q_n)$ be points in $\mathbb{R^n}$ with $p_k<q_k$ for each $k$. Let $R=[p_1,q_1]\times \cdots\times[p_n,q_n]$ and show that diam $R=d(p,q)=[\sum_{k=1}^n ...
3
votes
1answer
34 views

Find the value of $\frac{i}{4-\pi}\int_{|z|=4}\frac{1}{z\cos{z}}dz$

Find the value of $$\frac{i}{4-\pi}\int_{|z|=4}\frac{1}{z\cos{z}}dz$$. My attempt: The integrand has singularities at $z=0, \frac{\pi}{2}, \frac{-\pi}{2}$, so ...
1
vote
1answer
48 views

Order of an infinite sum

How to prove that, for $0<c<1$, $$ \sum\limits_{j=1}^{\infty} c^{\, j } \cdot j^{\, -(\frac{d}{2} +1 )} $$ is, for some positive constant $K$, of order $K + O( \, ( 1-c )^{\frac{d-2}{2}})$ when ...
1
vote
3answers
63 views

How does a function 'uniformly' go to zero?

I have read that Jordan's Lemma can be written a saying that: $$I=\int_{\Gamma_R} f(z)e^{iaz}\, dz$$ where $a>0$ and $\Gamma_R$ is a semi-circle in the upper-half plane will go to zero if ...
0
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4answers
125 views

Area of triangle formed by three complex numbers $-z,iz$ and $z-iz$ in the complex plane

The question bothering me is "Find the area of a triangle formed by the complex numbers $-z,iz,z-iz$ in the argand plane. I'm really sorry, but I could not understand how to proceed. However to that ...
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1answer
33 views

Why there is no biholomorphism between complex plane and unit disk? [closed]

Why there is no biholomorphism between complex plane and the unit disk?
2
votes
1answer
62 views

Conway, showing analytic function is constant

I want to show that if $G$ is a region and suppose that $ f: G \to \mathbb{C}~$ is analytic such that $f(G)$ is a subset of a circle, then $f$ is constant. in the section with this exercise, I ...
0
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1answer
25 views

Generalizing for asymmetric functions

I have the ratio of two functions $D(\rho)$ and $S(\rho)$ where $$D(\rho)= g(\rho)e^{ia\rho}-g(\rho)e^{-ia\rho} \ \ \text{and} \ \ S(\rho)= g(\rho)e^{ia\rho}+g(\rho)e^{-ia\rho}.$$ Obviously ...
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1answer
41 views

Finding Laurent series and classifying singularities

Find out the points of singularities of the following function and classify them: $f(z)=tan(\frac{1}{z}).$ my idea here is the following: the singular points are $z=\frac{2}{(2n+1)\pi}$ (isolated). ...
0
votes
1answer
49 views

$\mathbb{R}$-linear function could be uniquely written as $f(z)=\alpha z + \beta \bar{z}$, with $\alpha$, $\beta$ $\in \mathbb{C}$

With the identification $\mathbb{C} \equiv \mathbb{R^2}$ (usual isomorphism between $\mathbb{C}$ and $\mathbb{R^2}$, note by $g : \mathbb{R^2} \to \mathbb{C}$), show that each $\mathbb{R}$-linear ...
1
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1answer
32 views

Can we see this integral as the line integral of a 1-form

In Stein and Shakarchi's complex analysis, the following definition is given on pg. 21 Let $z:[a, b]\to \mathbf C$ be a parameterization of smooth curve $\gamma$ in $\mathbf C$ and $f$ be a ...
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2answers
35 views

line integral on a closed curve

I'm reading Conway's complex analysis book and I'm trying to solve the integral $\int_{\gamma}f$, where $f(z)=|z|^2$ and the curve $\gamma$ is the closed polygon $[1,1+i,i,0,1]$. I didn't understand ...
0
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1answer
29 views

Two convergent power series are the same if they equal on an infinite set of points having 0 as a limit point.

I'm having difficulty following the proof of the theorem below. First of all, how do we know that h(z) is a power series having a non-zero radius of convergence from the fact that f(z) is. And, ...
0
votes
2answers
31 views

Proof for Convergence of complex series

Hi i am looking for someone for clarification regarding a step in a proof that lies between Real and Complex Analysis. I have the following: For a complex sequence $(z_n)$, $z_n\to (z)$ iff ...
0
votes
1answer
30 views

Given one solution to $z^6 = z_0$, how do I find the others? [closed]

Suppose that $w \in \mathbb{C}$ is a solution to the equation $z^6 = z_0$, for some fixed $z_0 \in \mathbb{C}$. Find six numbers $\zeta_0, \ldots, \zeta_5 \in \mathbb{C}$ with the property that the ...
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1answer
36 views

(Ahlfors, p198) Why is it clear we can write $G(z-1)=ze^{\gamma(z)}G(z)$ when deriving the Gamma function?

In Complex Analysis by Ahlfors (p198), the author starts with the functional $$ G(z) = \prod_1^\infty \left( 1 + \frac{z}{n} \right) e^{-z/n} $$ and goes on to state that we may obviously write $$ ...
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2answers
60 views

what is $\int_{|z|=3}\frac{1}{(z-2)^n(z-4)^m}dz$ for $m,n\in\mathbb{N}$?

I saw many examples how to calculate integrals with the residue theorem. But now I'm stuck with this integral: $$\int_{|z|=3}\frac{1}{(z-2)^n(z-4)^m}dz,$$where $m,n\in\mathbb{N}$ and $z=2$ and $z=4$ ...
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0answers
39 views

Theorem in complex analysis?

What are the theorems that I supposed use it to prove that : If $\mathcal f $ is entire and $\mathcal f(z)=2f(z)$ then $ \exists $ $\lambda $ such that $\mathcal f(z)=\lambda\mathcal z$
0
votes
1answer
24 views

Big-O Landau equation for a complex number

Let $u=y+iz$ be a complex number. For the Big-O Landau symbol, do we have that $$ O(\lvert u\rvert^4)=O(y^4+z^4)? $$ I am not sure. Anyhow it is $\lvert u\rvert^4=y^4+2y^2z^2+z^4$. So I think we ...
2
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0answers
37 views

Prove that $\Re \oint_\gamma \overline{ f(\zeta)}f'(\zeta)d\zeta=0$ [duplicate]

Suppose that $f$ is homolorphic in a open subset $\Omega$ of $\mathbb{C}$. Why $$\Re \oint_\gamma \overline{ f(\zeta)}f'(\zeta)d\zeta=0\ ?$$ with $\gamma$ a closed curve in $\Omega$ This is my ...
0
votes
2answers
62 views

Questions about delta function.

Let $z \neq 1$ be a complex number. Then \begin{align} \frac{1}{1-z} = \sum_{n=0}^{\infty} z^n. \end{align} We have \begin{align} \frac{z^{-1}}{1-z^{-1}} = \sum_{n=1}^{\infty} z^{-n}. \end{align} ...
0
votes
1answer
34 views

Find all the harmonics functions constants on the rays

I'm stuck with this exercise, I don't know how characterize the harmonic functions of the exercise. I'd appreciate your help. Thank you. Let $G=\mathbb C\setminus\{(-\infty,0]\}$. Find all the ...
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2answers
64 views

Calculating $\int_{|z|=2}\frac{e^{1/z^2}}{1-z}dz$

Find the value of $\displaystyle\int_{|z|=2}\frac{e^{1/z^2}}{1-z}dz$ The first idea I had was to apply Cauchy's integral formula. But can this be done? $e^{1/z^2}$ is not holomorphic on the ...
4
votes
1answer
68 views

Show $f_n = f \circ f \circ \dots \circ f \longrightarrow 0$ uniformly on compact sets

I am seeking help on a complex analysis qualifying exam problem. Let $D$ be a bounded open connected subset of $\mathbf{C}$ containing $0$ and let $f \colon D \to D$ be an analytic function ...
3
votes
2answers
73 views

Fourier transform of a Lévy density $\frac{1}{\sqrt{2\pi }}\int_{0}^{\infty} e^{ikx-\frac{1}{2x}}x^{-\frac{3}{2}}dx$

A Lévy density is defined as $$q(x;1/2,1)=\frac{1}{\sqrt{2\pi }}e^{-\frac{1}{2x}}x^{-\frac{3}{2}}$$ for $x>0$ I am looking for it's Fourier transform: $$g(k;1/2,1)=\frac{1}{\sqrt{2\pi ...
1
vote
3answers
45 views

Laurent expansion of $1/(1+z^n)$ for $n \in \mathbb{N}$.

I've seen in many texts and answers on this website that the residue of $$1/(1+z^n)$$ can be computed easily since it has a simple pole at $z=e^{i \pi / n}$. That is all well and good but nothing ...
0
votes
1answer
36 views

Show that $|f(z)|\le M\frac{\prod_{k=1}^n|z-z_k|}{\prod_{k=1}^n|z+\overline{z_k}|}$ on the right half plane

Let $f$ be analytic on $H=\{z:\Re(z)\ge 0\}$. Suppose that: There exists $M>0$ such that if $\Re(z)=0$ then $|f(z)|\le M$. There are $z_1,...,z_n$ such that $\Re(z_k)>0$ and ...
1
vote
2answers
41 views

Determine the image of the unit circle by this transformation.

Compute $u$ and $v$ ($z=x+iy$, $w = u + iv$) if $w=\frac{2z-1}{2-z}$. Determine the image of the unit circle by this transformation. I succeed to to the first part of the question. I obtained : ...
5
votes
3answers
78 views

Is the integral $\int e^{2 \pi i z^2} dz$ uniformly bounded for any interval of $\mathbb{R}$?

I was wondering if there exists a constant $C$ such that $| \int_I e^{2 \pi i z^2} dz | \leq C $ for any interval $I$ of $\mathbb{R}$? Here I want $C$ to be independent of the choice of the interval ...
2
votes
1answer
72 views

Is there an holomorphic function? If that function exists, is it unique?

I am solving a problem that asked me if exist an holomorphic function which satisfies only two condition. both equally: $$ f\left(\frac{1}{\alpha n} \right)=0\ \ \ \ and\ \ \ f\left( \frac{1}{\alpha ...
2
votes
1answer
48 views

Complex power series and radius of convergence

Let $c$ be a non-zero complex number, and consider the power series \begin{equation} S(z)=\frac{z-c}{c}-\frac{(z-c)^2}{2c^2}+\frac{(z-c)^3}{3c^3}-\ldots. \end{equation} By using the Ratio Test, or ...
3
votes
2answers
55 views

Finding Laurent's series of a function

I am trying express the function $$f(z)=\frac{z^3+2}{(z-1)(z-2)}$$ like a Laurent's series in each ring centering in $0$, but I do not now how could I express it, in first I said that ...
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0answers
35 views

Proof: the resolvent operator is holomorphic.

I tried to prove that the resolvent operator $$\rho(A) \to \mathbb C,\space \lambda \mapsto R_{\lambda}(A):=(\lambda id_X -A)^{-1} $$ is holomorphic, where here $A$ is a bounded linear operator from ...
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vote
3answers
60 views

How to find Laurent expansion

I have been presented with the function $g(z) = \frac{2z}{z^2 + z^3}$ and asked to find the Laurent expansion around the point $z=0$. I split the function into partial fractions to obtain $g(z) = ...
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vote
1answer
57 views

Bounded holomorphic function

Lef $f\in\mathcal{H}(\mathbb{D}):f(0)=0$ with $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$. If there exists $n\in\mathbb{N}$ and $c>0$ such that: $$|f(z)|\leq\frac{c}{(1-|z|)^n},\qquad \text{ for all ...
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1answer
19 views

Find the line which makes angle $\frac{\pi}{4}$ with $(2-i)z+(2+i)\overline z+3=0$ which passes through point $(-1,4)$.

Find the line which makes angle $\frac{\pi}{4}$ with $(2-i)z+(2+i)\overline z+3=0$ which passes through point $(-1,4)$. Complex slope of the given line is $$ w_1=\frac{-3-4i}{5}$$ The complex slope of ...
1
vote
1answer
46 views

$2^z$ behavior when changing real and imaginary components of $z$

I'm reading The Music of the Primes by du Sautoy and I've come across a section that I'm having difficulty understanding: Euler fed imaginary numbers into the function $2^x$. To his surprise, out ...
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0answers
34 views

Convolution of complex functions (Laplace Domain)

Convolution of functions in the time domain is equivalent to multiplication in the frequency domain. However, I am interested in multiplication of functions in the time domain, which is convolution in ...
2
votes
3answers
57 views

If $f(i)=2i$, then what will be $f(1)$? [duplicate]

Let $f$ be an entire function on $\Bbb{C}$ such that $|f(z)|\le 100 \ln |z|$ for $|z|\ge 2$ and if $f(i)=2i$, then what is $f(1)$? I think $f$ will be constant i.e. $f(1)=2i$, but I'm unable to ...
3
votes
1answer
87 views

How to calculate $\int_{-\infty}^{\infty}\frac{x^2}{\cosh(x)}\mathrm{d}x$ [duplicate]

I know the poles are $z=i\pi/2+i n\pi$ and therefor I got an rectangular contour for the integration which wasn't so useful. I also know with change of variables I can get to ...
2
votes
1answer
142 views

Finding a solution to $\sum\limits_{n=1}^{n=k} \frac{1}{n^s}=0$

Finding ONE solution to: $$\sum\limits_{n=1}^{n=k} \frac{1}{n^s}=0$$ can apparently be done by iterating the following formula: $$\Large s(m+1)=\frac{\log \left(-\frac{1}{\sum _{n=1}^{k-1} ...
0
votes
1answer
38 views

Integral along the boundary is zero, if function has a compact support?

I've just missed a point on complex analysis lectures. Namely, we did an integral representation formula using Green's formula: Suppose we have function $f$ continuously differentiable on open set ...
1
vote
0answers
43 views

Property of Weierstrass sigma function

In theorem 1.2.3 of Schertz' Complex Multiplication says that For any $\omega \in \mathcal{L}$, a fixed lattice, we have the property: $$ \sigma(z + \omega) = \psi(\omega)e^{\eta(\omega)(z + ...
3
votes
3answers
109 views

How to show that $\int_0^{\infty} dx \frac{\log{x}}{1+x^2}$ is zero using complex analysis

I want to show this using contour integration, the appropriate contour is a keyhole I think.
0
votes
0answers
39 views

Is there anything wrong with the following work on the Argument Principle?

The Argument Principle states that : $$\oint_C {d\over dz}(log (f(z))) \, dz = 2\pi i(N-P)$$ Let $g(z)={d\over dz}\log(f(z))$ If $f: C \to C$ is a continuous function on a directed smooth curve, ...
2
votes
2answers
174 views

how to calculate $\int_{0}^{\infty}\frac{x}{\sqrt{e^x-1}}\mathrm{d}x$

I was trying to solve another integral when then I reached this, I've no idea of how to select the contour for the integration.