Questions tagged [complex-analysis]

For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.

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3 views

Prove that fractional linear transformation $f_\theta$ is equal to composition of fractional linear transformations (circle inversion and rotation)

Prove that $f_\theta(z)=i_C\circ R_\theta \circ i_C^{-1}$ in a way that $i_C(z)= \dfrac{i\bar{z}+1}{\bar{z}+i}$ $f_\theta(z)=-\dfrac{(\cos\frac{\theta}{2})z+\sin\frac{\theta}{2}}{(-\sin\frac{\theta}{2}...
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Is $\delta\left(a+bi\right)=\delta\left(a-bi\right)$?

We can write Delta function as $$\delta(z) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{itz}\, dt=\delta\left(a+bi\right)=\frac1{2\pi}\int_{-\infty}^{+\infty}e^{-bx}\cos ax\, dx+\frac{i}{2\pi}\int_{-\...
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1answer
23 views

Fixed Points in Mobius Transformation

Let $$w(z)=\frac{e^z-1}{e^z+1}; \ z=x+iy$$ and $$w=u+iv$$ Then show that the image of the $y$-axis in the domain is $v$-axis in the co-domain. Does this contradict the result that a Mobius ...
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Are the fourier coefficents of the conjugate of an holomorphic function zeros?

Let $f(z)$ be a holomorphic function and $f^{*}(z)$ is the complex conjugate of $f(z)$. The following equation is true or not? If it is true, could you give me an relatively elementary proof? $$\int_{...
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22 views

Compact Riemann surfaces with boundary

I am familiar (to an extent) with the theory of compact Riemann surfaces without boundary, but I am interested in the theory of compact Riemann surfaces with boundary. To what extent do the following ...
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Gamelin's Complex Analyisis, Chapter XIII, Section 2, Exercise 6

Construct a meromorphic function on the complex plane whose poles are simple poles at the Gaussian integers m+ni with residue 1.
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Is this the correct way to find the length of this path $\gamma (t)$?

Let $\gamma: \mathbb{R} \mapsto \mathbb{C}$, $\gamma (t) = 12 + i(t^2-3t-3)$ be a path for $t \in \mathbb{R}$. Find the length of the path on $[0,2]$, i.e., $\ell(\gamma\vert_{[0,2]})$. I know that $$\...
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Complex integration with fractional powers

How would one compute $$ \int_{\gamma}\frac{z^{n + \frac{1}{2}}}{(z^{\frac{1}{2}} + 1)^2}\ dz, $$ for an integer $n$ and where $\gamma$ denotes the closed unit circle contour, that is using complex ...
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1answer
24 views

Why $\lim\limits_{z\to\infty}\frac{P(z)}{Q(z)}=\lim\limits_{y\to\infty}\frac{P(iy)}{Q(iy)}$?

While reading about the concept of $L$-stability I came across a result that said that for a rational (complex) function $R(z)=P(z)/Q(z)$ where $P,Q$ are polyomials we have $$\lim\limits_{z\to\infty}\...
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28 views

How do I show that a domain is not simply connected?

Let $U$ be a star-shaped domain in $\mathbb{C}$. Prove that the subset obtained from $U$ by removing a finite number of points is not simply connected. Usually when we want to show something is ...
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23 views

Residue of $\frac{\sin(1/z)}{z^2+a^2}$

Now I was able to find the residues at $z=\pm ia$, but I thought that there would be another singularity at $z=0$. However, everywhere I checked, it says that there isn't any residue at $z=0$. Why is ...
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22 views

holomorphic function on an unbounded region

Suppose $\Gamma$ is the boundary of an unbounded region $\Omega$, $f \in H(\Omega)$, $f$ is continuous on $\Omega \cup \Gamma$, and there are constants $B < \infty$ and $M < \infty$ such that $ |...
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Do Hodge $*$-operators glue?

For ${\Bbb P}_{\Bbb C}^2 = \underset{i = 0, 1, 2}{\bigcup} {\Bbb A}_{i, \Bbb C}^2$, we have Hodge $*$-operators on each affine open ${\Bbb A}_{i,{\Bbb C}}^2$. For example $z_1 = X_0/X_2, z_2 = X_1/X_2$...
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33 views

How are multifunctions precisely defined?

From what I understand, a multifunction is a binary relation between two sets, where each input can be associated with more than one output. For instance, we could define the multifunction $x \mapsto \...
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Prove or disproven the statement listed below for my math discussion [closed]

Let m, n ∈ N such that mσ(m) = nσ(n). Then m = n. The equation ν(n) = 20 has only finitely many solutions.
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Computing Eisenstein series terms

I have to compute a few terms of the normalized Eisenstein series: $\xi = e^{2 \pi i \tau}$, where $\tau$ belongs to the upper-half plane. In particular, I have to show that: $$ E_{4} (\tau) =1+240(\...
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1answer
54 views

If $z_1,z_2,z_3,z_4$ are consecutive vertices of a quadrilateral that lie on a circle prove the following

If $z_1,z_2,z_3,z_4$ are consecutive vertices of a quadrilateral that lie on a circle prove the following: $|z_1-z_3||z_2-z_4|=|z_1-z_2||z_3-z_4|+|z_1-z_4||z_2-z_3|$. I know that you can prove this ...
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1answer
31 views

How to prove that $f(z) := \bar{z}$, for $z \in \mathbb{C}$, is bounded on a rectangle $\mathcal{R} \subseteq \mathbb{C}$?

Let $f(z):=\bar{z}$, for $z=x+iy \in \mathbb{C}$. Explain why $f(z)$ is bounded on the rectangle $$\mathcal{R} := [a,b] + i[c,d] = \{z \in \mathbb{C} : a \leq \operatorname{Re}(z) \leq b , \hspace{0....
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1answer
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Fourier transform of compact supported function is entire

Suppose $f\in L^1(\mathbb{R})$ has compact support, say $\operatorname{supp}f\subset[-r,r]$. I want to show that its Fourier transform $$ \hat{f}(z) = \int_{-a}^{a} f(t) e^{-2\pi itz}dt$$ for $z\in\...
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A new notation to describe complex arcs

Recently I've been working on quite a few (slightly harder) complex integrals. What was always bothering me is that we can not define contour arcs "easily". If we want to define a line in ...
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2answers
26 views

How to find the radius of convergence of different powers $\sum_{n=0}^{\infty} b_{n} z^{2 n+2}$

If I have a power series on the form $$\sum_{n=0}^{\infty} a_{n} z^{n}$$ A theorem tells me it has a radius of convergence given by $$R=\lim _{n \rightarrow \infty}\left|\frac{a_{n}}{a_{n+1}}\right|$$ ...
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1answer
33 views

Laurent expansion of $\frac{16}{z^2(z-3)}$

What is Laurent series of $\frac{16}{z^2(z-3)}$ at $z=3$ in the inner annulus? $\frac{16}{z^2(z-3)}=\frac{-16}{3z^2}-\frac{16}{9z}+\frac{16}{9(z-3)} = \frac{16}{9}(z-3)^{-1}+\frac{16}{27}\cdot\frac{1}...
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16 views

Differentiability of infinite series - on every bounded subset vs. all complex numbers

I'm trying to solve a problem regarding absolute convergence of a series of complex functions and finding it's derivative. Especially, I do not know how to find the derivative for all complex numbers, ...
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74 views

Does there exist a function for integral of $e^{-x^2}$?

I am wondering if the graph of $$f(x) = \int_{0}^{x}{e^{-t^2}}\,\mathrm{d}t$$ is possible to be written as some function without the Error Function, since it looks like an existing function when I ...
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18 views

Find the Mobius Transformation that satisfies the following

So this question comes from Complex Analysis by Ahlfors, Section 3.3 Problem 4. It asks to find a Mobius transformation that takes $|z|=2$ into $|z+1|=1$, $-2$ to $0$, and $0$ to $i$. I know this ...
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1answer
36 views

Finding conjugate of a complex number using some algebra

it is given that z is a complex number satisfying $z^3-iz+1=0$ I'm supposed to find argument of $ z+z \bar z + \bar z$ The only thing I could do was $(z+1)(z^2+1-z)=iz$ Any help is appreciated
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46 views

Calculate the sum of A.

Let $\omega=\mathrm{e}^{\mathrm{i} \frac{2 \pi}{7}}$. Calculate $$ A=\omega+\omega^{2}+\omega^{4} $$ Since $\omega=e^{i\frac{2\pi}{7}}$ it mean that $\omega$ is the 7th roots of unity. So it have ...
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1answer
41 views

How to obtain the equality between two integrals with different integrands?

I am looking into the proof of Ramanujan's approximation formula for the partition function $p(n)$ by Stein and Shakrachi. I am confused about one step towards the end of the proof just before we ...
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1answer
46 views

Book recommendation for analysis

I am an undergraduate in Physics willing to learn pure mathematics through self study. What books would you recommend for real and complex analysis which cover the essentials of the subjects in an ...
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Bound for $e^{az-e^z}$

I am trying to prove $e^{az-e^z}$ is exponentially decreasing on the strip $\{x+iy:|y|<\pi/2\}$. After some computation by definition, I got $|e^{az-e^z}|=e^{ax-e^x\cos y}$. But I got stuck here, ...
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54 views

How many zeros does the function $f(z)=z^3+z^2+4z+1$ have in the first quadrant?

How many zeros does the function $f(z)=z^3+z^2+4z+1$ have in the first quadrant? Using Rouche's theorem with the function $h(z)=1,g(z)=z^3+z^2+4z$, I've been able to show that all the roots are in $...
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1answer
66 views

$f(z)=\frac{z}{1-z} \zeta(\frac {1}{1-z})$ belongs to Hardy Space $H^{1/3}(\mathbf{D})$

The Hardy space $H^p(\mathbf{D})$ is the vector space of holomorphic functions $f$ on the open unit disk that satisfy: $$ \sup_{0< r<1}\left(\frac{1}{2\pi} \int_0^{2\pi}\left|f \left (re^{i\...
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19 views

A question regarding differentiability and the boundary of analytic functions

Consider a Jordan curve $\Gamma\subset \mathbb{C}$ and let $\Omega$ be the interior of $\Gamma$ (which is well-defined by the Jordan curve theorem). Let $f:\Gamma\cup\Omega\rightarrow\mathbb{C}$ be ...
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1answer
18 views

Conformal equivalance and modular invariance

I'm currently in an introductory complex analysis class. I'm working through the following problem: Let $\Bbb{H}$ be the upper half-plane and $B$ be left half of the fundamental region: $$ B = \{\tau \...
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1answer
36 views

Images under the transformation of inversion

Consider the transformation of inversion $Tz = 1/z$. Find the image of under $T$ of the circle $x^2+y^2=ax$ where $a \in \mathbb{R}$. I guess I am just a bit confused about these types of problems. I ...
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1answer
22 views

Visualizing conformal map from unit disk to upper half plane

I understand that $$w=\frac{z-i}{z+i}$$ maps the open upper half plane $\mathbb{H}$ to the open unit disk $\mathbb{D}$, so $$z=i\frac{1+w}{1-w}$$ maps from $\mathbb{D}$ to $\mathbb{H}$. My question is ...
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24 views

A partial converse of Cauchy's theorem

Let $V\subset \mathbb{C}$ be open and $f\colon V\to \mathbb{C}$ a continuous function, and assume $$ \int_{\gamma} f(z) \, dz =0 $$ for any closed contour $\gamma \in V.$ We need to show that $f$ has ...
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Is $f$ holomorphic in $\mathbb C$?

Let $f:\mathbb C\to \mathbb C$ such that functions $z\mapsto \sin (f(z))$ and $z\mapsto \cos (f(z))$ are holomorphic on the entire complex plane. a) Is $f$ also holomorphic in $\mathbb C$? b) If we ...
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85 views

What's wrong with my method for this integral?

So I have to solve $$\int_0^1\frac{1}{x^{2/3}(1-x)^{1/3}}dx.$$ To do this I made a branch cut from $z=0$ to $z=1$ and took the bone-shaped contour that straddles the real axis, going clockwise. Now on ...
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51 views

All continuous solutions of $f(z)=f\left(\frac{1}{z}\right)$

After going through this article, I thought of the use of cyclic functions (as described in the article). I solved the following: $$f:\, \mathbb{C}\setminus \{0\}\mapsto\mathbb{C},\, f(z)=2f\left(\...
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10 views

Magnus Effect - Coupled linear inhomogeneous ODE with variable coefficients

I am trying to derive the equations of motion for an object under influence of the Magnus Force (ball spinning in air). This gives me the following coupled ODE: $$ \begin{align} \begin{bmatrix} ...
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23 views

Conformal equivalence between annulus and the upper half plane

So, I'm trying to do the following problem: "Let $A$ be the region inside the disk $D(1,1)$ and outside the disk $\overline{D(\frac{1}{2}, \frac{1}{2}})$. Construct the conformal equivalence ...
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Can anyone prove the laplace equation solution is $z=x+yi$ and $\overline{z}=x-yi$ rigorously by fully computing the second order partial derivatives? [closed]

Can anyone prove the laplace equation solution is $z=x+yi$ and $\overline{z}=x-yi$ rigorously by fully computing the second order partial derivatives?
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24 views

Square root of complex logarithm

Question: Let the square root be computed using the branch $L_{\pi/2}$ of the logarithm. Where is $(z^2-i)^\frac{1}{2}$ not analytic? My solution: The function is not analytic when $\sqrt{z^2-i}=yi, y ...
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55 views

Show that the map $w(z)=\frac{1}{1-z^2}$ is a Conformal Mapping..

Show that the mapping $$w(z)=\frac{1}{1-z^2}$$ is a Conformal Mapping. My try- $$w(z)=\frac{1}{1-z^2}$$ $$ w'(z)=\frac{2z}{(1-z^2)^2}$$ $$ w'(z)\neq 0 \ \forall\ z\neq 0$$ So $w(z)$ is a Conformal ...
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2answers
57 views

Real part of an analytic function

I'd like to know how to take the real part (of the power series representation) of an analytic function with the goal of showing the (real part of the analytic) function is real analytic. I know this ...
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29 views

Existence of global logarithm on manifolds

Consider we have a smooth map $f$ from a smooth manifold $M$ to $S^1$. When can we find a global logarithm accordingly, i.e. a smooth map $g: M \mapsto \mathbb{R}$ s.t. $f=e^{ig}$? Any topological ...
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1answer
35 views

Powers of primitive roots.

Let $w= \cos\frac{2k\pi}{n} + i \sin\frac{2k\pi}{n}$ be a primitive $n^{\text{th}}$ root of unity, ie, $w^n=1$ and $w^m \neq 1$ for $m \leq n$. Then the powers $$1, w, w^2, \ldots, w^{n-1}$$ are all ...
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30 views

If a harmonic function $u\colon G \to \mathbb{R}$ has a zero in $z_0\in G$, than there is another zero in every neighbourhood of $z_0$

I have a proof but I am not sure if it is right. Let $\epsilon > 0$ with $U_{\epsilon}(z_0 \in \mathbb{R}^2)\subset G$. Let $p<\epsilon$. From the mean value property of $u$, it follows: $0=u(...
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1answer
37 views

Prove$\sum_{|\alpha|<1, f(\alpha)=0}\log \frac{1}{|\alpha|}= \sum_{\Re(\rho)>1/2}\log\ |\frac{\rho}{1-\rho}|$ [closed]

Let $$ f(z)= (s-1)\zeta(s) $$ where $s=\frac{1}{1-z}$ and $\zeta$ denotes the Riemann Zeta function. Prove that,$$\sum_{|\alpha|<1, f(\alpha)=0}\log \frac{1}{|\alpha|}= \sum_{\Re(\rho)>1/2}\log\ ...

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