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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1answer
50 views

Hamiltonian differential equation involving complex logarithm

Consider the differential equation $$\begin{pmatrix}\dot p \\ \dot q \end{pmatrix} = \frac{1}{p^2+q^2}\begin{pmatrix} p \\ q \end{pmatrix}$$ where $(p,q)^T\in \mathbb R ^2 - \{0\}$. I want to show ...
0
votes
0answers
31 views

$\lim_{\rho\to0}\int_{\gamma_{\rho}}g(z)e^{iz}dz=-\pi i Res(f,a)$ with a pole $a\in\mathbb{R}$

Let $U$ be an open neighbourhood of $\overline{\mathbb{H}}=\{z\in\mathbb{C}:\Im(z)\ge0\}$ and $g:U\rightarrow\mathbb{C}$ meromorphic with a finite number of poles in $\mathbb{H}=\{z\in\mathbb{C}:\Im(z)...
-2
votes
3answers
39 views

Express $y^2=4ax$ using complex numbers $z$ and $\bar{z}$ [closed]

Express $y^2=4ax$ using complex numbers $z=x+iy$ and $\bar{z}$ (Hint: Use appropriate substitution for x and y)
-2
votes
1answer
38 views

Prove that $u(x,y) = \ln(x^2 + y^2)^{\frac{1}{2}}$ is harmonic on $\mathbb{C}\setminus \{0\}$ [closed]

Prove that $u(x,y) = \ln(x^2 + y^2)^{\frac{1}{2}}$ is harmonic on $\mathbb{C}\ {0}$, then find a harmonic conjugate $v(x,y)$ of $u(x,y)$ so that $f(z) = u(x,y) + iv(x,y)$ is analytic on $\mathbb{C}\...
0
votes
1answer
16 views

Index with respect to a loop

Does there exist a continuous loop in the complex plane such that the index of every point not on the loop is 0?
0
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0answers
20 views

How to define a variable which is an integral involving cauchy principal value inside?

How to define a variable which is an integral involving cauchy principal value inside in any computer programming language? I want to know how to break down the procedure step by step from a ...
3
votes
1answer
109 views

Finding all $z\in \mathbb{C}$ such that the series $\sum\limits_{n=1}^{\infty} \frac{1}{1+z^n}$ converges

I am trying to find out all $z\in \mathbb{C}$ such that the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{1+z^n}$ converges. I notice that for $\left|z\right|\leq 1$, we have $\left|1+z^n\right|...
2
votes
1answer
30 views

Formula for the graph of 3 hyperbolas

I recently was doing some complex number work and found this guy: Is there a formula for a graph like this in two - dimensions? I know that the values are the same on every curve, but they are ...
-3
votes
1answer
39 views

Cauchy rieman equation. What is u?

Using cauchy rieman equation, i want to show the function is analytic. So i want to decompose from f(z) to two term (real part and imaginary part) With rectangular form or polar form. But it is so ...
0
votes
1answer
67 views

What is the benefit of representing a complex number as e^i(theta) versus e^(a+bi), what is the process of finding a solution to this example?

What is the benefit of representing a complex number as $ e^{i\theta} $ versus $ e^{a+bi} $? Am I correct in saying that these give the same information but offer convenience in different situations? ...
0
votes
1answer
66 views

Restriction of holomorphic functions [closed]

Let $U\subset \mathbb{C}$ open and $D\subset U$ an open disk of radius $r>0$ centered at $z_0\in U$. When is the restriction map $f\rightarrow f_{|D}:D\rightarrow \mathbb{C}$ of a holomorphic ...
1
vote
1answer
60 views

Zeros of $f_n(z)=1+\frac{1}{z}+\frac{1}{2!z^2}+…+\frac{1}{n!z^n}$ are in $B_{\varepsilon}(0)$

I want to prove that for every $\varepsilon >0$ there is a $N\in\mathbb{N}$ so that for every $n\ge N$ all zeros of $$f_n(z)=1+\frac{1}{z}+\frac{1}{2!z^2}+...+\frac{1}{n!z^n}$$ are in $B_{\...
0
votes
1answer
24 views

$\log \inf_{x\in \partial U} f_x(z)=\inf_{x\in \partial U} \log f_x(z)$?

Let $U$ be an open set of $\mathbb{C}$. Fix $x\in \partial U$. Define $f_x(z)=|z-x|$. Is the following true? $\log \inf_{x\in \partial U} f_x(z)=\inf_{x\in \partial U} \log f_x(z)$. My try: $\...
0
votes
0answers
7 views

Complex Valued Autocorrelation in an Autoregressive Process.

Suppose we have an autoregressive process $$y_{n+1} = \alpha y_n + \epsilon_n$$ Here $y_n,\alpha, \epsilon \in \mathbb{C}$. Suppose that $\alpha = \exp(\lambda)$ for complex $\lambda$. If $\Re(\...
3
votes
3answers
59 views

If $x= m-m^2-2$ then find $x^4+3x^3+2x^2-11x+6$ where m is a cube root of unity

If $$x= m-m^2-2$$ then find $$x^4+3x^3+2x^2-11x+6$$ where $m$ is a cube root of unity. My try: Since $ m+ m^2+1=0$ the value of $x$ is $-1$. Let $f(x)=x^4+3x^3+2x^2-11x+6$ then $ f(-1)=5$ So the ...
1
vote
0answers
39 views

Is there only one complex structure on complex plane $\mathbb{C}$? [duplicate]

There is a trivial complex structure on $\mathbb{C}$. Do we have other complex structures on complex plane $\mathbb{C}$? If not, how to prove it?
3
votes
2answers
70 views

Solution of $e^{-z}+z=\lambda$

Let $\lambda\in\mathbb{C}$ with $\Re(\lambda)>1$ be given. I want to show, that $e^{-z}+z=\lambda$ has exactly one solution in $U=\{z\in\mathbb{C}:\Re(z)>0\}$. I think that exercise can be ...
22
votes
5answers
1k views

Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$

I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all $n\in\mathbb{N}^+$ in general: $$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$ ...
5
votes
3answers
1k views

Use the Residue Theorem to evaluate the integral:

$$\int_{0}^{∞} \frac{\sqrt{x}}{x^2+2x+5} dx$$ I'm thinking of using the "keyhole" contour, but I'm not sure how to proceed from there. Please help! Thanks!
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0answers
27 views

interchange boundary with image of holomoprhic functions

Let $f:D\rightarrow \mathbb{C}$ be an holomorphic non costant function; there are some conditions on $f$ or on the domain $D\subset \mathbb{C}$ in such a way that $f(\partial D)=\partial f(D)$ ? ...
1
vote
0answers
54 views

Multiple choice excercise with more than one answer correct

Q. Consider the function $$F(z)=\int_{1}^{2} \frac {1}{(x-z)^2}dx, {\text {Im}(z) \gt 0}$$ Then there is a meromorphic function function $G(z)$ on $\Bbb C$ that agrees with $F(z)$ when ${\text {Im}(...
0
votes
0answers
19 views

Different radius of convergence for ratio test and Hadamard's formula

I'm pretty sure I'm missing something very basic... But I have the following question: Determine the radius of convergence of $\sum \alpha_n z^n$ with $\alpha_n=\frac{1}{n+1}$. Now, with the ratio ...
1
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0answers
13 views

A false identity involving $2^{\frac{1}{\zeta(s)}}$ for $\Re s>1$, from these particular values of the Riemann Zeta function and its alternating

Yesterday when I was exploring symbolic calculations $\dagger$ about specializations in $z=\frac{1}{n}$ with $n>1$ an integer, of $$\zeta(z)=(1-2^{1-z})^{-1}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^z}:=...
0
votes
0answers
30 views

Mobius transformation from upper plane onto the unit disc

Let $H =\{ z = x +iy \in \mathbb C : y>0 \}$ and $D = \{ z \mathbb C : |z| <1\}$ be the open unit disc . Suppose $f$ is a mobius transformation which maps conformaly onto $D$. Suppose that $f(2i)...
1
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3answers
134 views

Evaluate $\int_{-\infty}^{\infty} \frac{(1-\cos { y } )}{\mid{y}\mid^{1+\alpha}}dy$ [on hold]

How do I evaluate the following integral? $$\int_{-\infty}^{\infty} \frac{(1-\cos { y } )}{\mid{y}\mid^{1+\alpha}}dy=\frac{\pi}{\Gamma(1+\alpha)\sin(\frac{\pi\alpha}{2})}$$ Thank you in advance. ...
0
votes
0answers
13 views

Local representation of antiholomorphic map

Let $f:\mathbb{C} \to \mathbb{C}$ be an antiholomorphic map, $f(0)=0$. How can I show that there exists a holomorphic function $z(w)$, $z(0)=0$, defined in a neighborhood of $0$, such that $f(z(w))=z(\...
1
vote
2answers
102 views

Closed form for $\int_0^1 d u \, \frac{1}{u + \lambda} \ln \left(\frac{1 + u}{1 - u} \right)$

The parameter $\lambda$ is complex and it's not on the real axis. There are some similar cases: Help me evaluate $\int_0^1 \frac{\log(x+1)}{1+x^2} dx$ Evaluate $\int_0^1 \frac{\ln(1+bx)}{1+x} dx $ ...
0
votes
2answers
86 views

Riemann zeta function functional equation proof explanation

In Riemann zeta function functional equation proof I arrived to a following equation $$\frac{\Gamma\left(\frac{s}2\right) \zeta(s)}{\pi^{\frac{s}2}}=\sum_{n=1}^\infty \int_0^\infty x^{\frac{s}2-1}e^{-...
0
votes
1answer
26 views

Finding complex limit when conjugate is involved

Any ideas on how to solve this? $$\lim_{Z \to Zi} (Z^2- \bar Z)$$ I would just plug in $ Zi $ but if it were not for the conjugate which isn't something you can just derive from a complex number. I'...
18
votes
5answers
661 views

Geometrical Interpretation of Cauchy Riemann equations?

Differentiation has an obvious geometric interpretation, and the Cauchy Riemann equations are closely linked with differentiation. Do the Cauchy Riemann equations have a geometric interpretation?
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0answers
30 views

Complex fixed points on the bifurcation diagrams

I'm working with bifurcation diagrams, an extesion that is being made of them is the determination of complex fixed points in addition to the real fixed points, my question is: what information ...
4
votes
2answers
64 views

Non constant analytic function from $\{z\in\mathbb{C}:z\neq 0\}$ to $\{z\in\mathbb{C}:|z|>1\}.$

Does there is non constant analytic function from $\{z\in\mathbb{C}:z\neq 0\}$ to $\{z\in\mathbb{C}:|z|>1\}?$ According to me there is no such non constant analytic function because if there is ...
0
votes
0answers
48 views

Biholomorphic function such that $\phi(1+i)$ and $\phi'(1+i)=1/\sqrt{2}$

Let $S=\{z \in \mathbb{C}: 0 \lt Arg(z) \lt \pi/2\}$ and $\mathbb{D}=\{{z\in \mathbb{C}:|z|\lt 1}\}$ Find a biholomorphic function $\phi: S \rightarrow \mathbb{D}$ such that $\phi(1+i)=0$ and $\...
2
votes
2answers
47 views

one one continuous function from $\{z\in\mathbb{C}:|z|>1\}$ to $\{z\in\mathbb{C}:z\neq 0\}$

Does there is one one continuous function from $\{z\in\mathbb{C}:|z|>1\}$ to $\{z\in\mathbb{C}:z\neq 0\}?$ I tried many examples but did't found. Is there any concept about existence or non-...
0
votes
2answers
66 views

Prove or Disprove: A holomorphic function with the given property exists

Does there exist a holomorphic function $f:U_{1+\epsilon}(0)\rightarrow \mathbb{C}$ for all $n\in \mathbb{N}_{\ge 2}$ such that $$ f\left(\log\left(1+\frac{1}{n}\right)\right) = \left(\frac{1}{n^...
1
vote
0answers
31 views

Laurent-Series on an annulus

I solved an exercise and I want to know if it is correct. I'm trying to find the Laurent-Series for $$g(w)=\frac{w}{1+w^2}$$ On the annulus $D_{1,2}(-i)$ What I did so far: We have 2 poles in $w=i$ ...
0
votes
1answer
30 views

How do i prove that $Res(f(z)e^\frac{1}{z};0)=\sum_{n=0}^\infty \frac{a_n}{(n+1)!}$ with $f(z)=\sum_{n=0}^\infty a_nz^n$

$f(z)=\sum_{n=0}^\infty a_nz^n$ (around $0$) I need to prove that $Res(f(z)e^\frac{1}{z};0)=\sum_{n=0}^\infty \frac{a_n}{(n+1)!}$ I know that $Res(z^ne^\frac{1}{z},0)=\frac{1}{(n+1)!}$ but I don'...
5
votes
1answer
65 views

Complement of a simply connected set is simply connected

I saw the following surprising statement in Wikipedia: When $D\subseteq\Bbb C$ is a simply connected compact set, then its complement $E=D^c$ is a simply connected domain in the Riemann sphere ...
1
vote
0answers
23 views

Convergence of $\sum_{n=1}^\infty\frac{\psi(n)}{e^n}\sin ns$ on an horizontal closed strip

Let $\psi(x)=\sum_{k\leq x}\Lambda(k)$ the Second Chebyshev function, and $\epsilon>0$. I would like to ask Question. Can you prove or disprove that the series $$\sum_{n=1}^\infty\frac{\psi(n)}{...
0
votes
1answer
40 views

Radius of convergence of $f(z)?$

Let $p(x)$ be a polynomial of the real variable $x$ of degree $k\geq 1$ .Consider the power series $$f(z)=\sum_{n=0}^{\infty}p(n)z^n$$ where z is a complex variable .Then the radius of convergence of $...
-1
votes
1answer
65 views

Image drawing complex analysis [closed]

$w=u+iv,z=x+iy$ are complex numbers and we have $w=z^2-2z$. Determine the image in the $w$-plane of the unit circle $x^2+y^2=1$. I have tried to answer this here Question and Answer. I have problems ...
0
votes
1answer
44 views

Analytic function $f$ satisfying $f=f \circ g$

In this question, Is there an analytic function with $f(z)=f(e^{iz})$?, it was settled that there exists no non-constant analytic function $f$ such that $f=f \circ g$, where $g(z)=e^{iz}$. Below is an ...
2
votes
0answers
40 views

Does there exist a non constant entire function $f: \mathbb C \to \mathbb C$ which is square integrable? [duplicate]

Does there exist a non constant entire function $f: \mathbb C \to \mathbb C$ which is square integrable i.e. $\int_{\mathbb C} \vert f(z)\vert^2 dz< \infty$ ? I think that answer to above ...
0
votes
0answers
35 views

Singular expansion of an implicit function

In the book of Flajolet and Sedgewick (this context is not so important, though), the following argumentation is used: Let $y(z)$ be a function given implicitly by $y - \phi(z,y) = 0$, where $\phi$ ...
0
votes
0answers
40 views

Imaginary part of a function

Please, can somebody help me with this problem? Let $\Delta=\{z \in \mathbb{C}:|z|=1\}$ and $S=\{z \in \mathbb{C}:Imz>0\}$; let $f:\Delta \to S$ be an olomorphic function, with $Im(f(z))>0, \...
1
vote
1answer
126 views

Let $f=u+iv$ is an entire function. If $u+v$ is bounded then is $f$ constant or not? $(u=Ref(z)$ and $v=Imf(z))$

$u+v$ is bounded $\Rightarrow$ $(u+v)^2=u^2+v^2+2uv$ is bounded $\Rightarrow$ $|f|^2+2uv$ is bounded. What can I do further to reach the conclusion? or this is not the way to reach there? Thank you.
12
votes
1answer
113 views

The divergent sum of alternating factorials

So I came across this exposition of a paper by Euler here where Euler is trying to sum the divergent sum: $$s = 1 - 1 + 2! - 3! + 4! \dots = \sum_{k\geq 0}(-1)^k k!.$$ There are a couple of questions ...
1
vote
1answer
73 views

Show map is norm-preserving and determine Eigenvalues [duplicate]

Can someone of you give me a solution for this? Let $N\in \mathbb N$. a) We define the map $\mathfrak F:(\mathbb C^N, ||\cdot||_2)\to(\mathbb C^N, ||\cdot||_2)$ by $$(\mathfrak F(x))_k := \frac{...
7
votes
1answer
509 views

Strengthened version of the Casorati-Weierstrass Theorem

Suppose the $f$ has an essential singularity at $z=a$.Prove that if $c\in \mathbb{C}$,and $\varepsilon >0$ are given,then for each $\delta >0$ there is a number $b$,$|c-b|<\varepsilon$,such ...
42
votes
4answers
1k views

How to prove that $\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$?

How can one prove this identity? $$\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$$ There is a formula for $\zeta$ values at even integers, but ...