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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14 views

Let $f(z)$ be an entire function. Prove that $f$ and $f-a$ have the same order.

Let $f$ be an entire function, the order of $f$ is defined by $$\lambda=\limsup_{r \to \infty} \frac{\log \log M(r)}{\log r},$$ where $M_{f}(r)=\max_{|z|=r} |f(z)|$. And this is equivalent to define $...
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0answers
26 views

Are all important function spaces vector spaces?

Are all useful (real/complex-valued) function spaces in analysis vector spaces? (Closed under "linear operations", addition and scalar multiplication.) By this I also want to include functionals and ...
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2answers
3k views

Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer.

Using residues, try the contour below with $R \rightarrow \infty$ and $$\lim_{R \rightarrow \infty } \int_0^R \frac{1}{1+r^n} dr \rightarrow \int_0^\infty \frac{1}{1+x^n} dx$$ I've ...
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5answers
149 views

“Exponential Madness”

From Euler's identity, we see that $e^{i\pi}=-1$ $\Rightarrow e^{2ik\pi}=1$ [squaring both sides]. This equation surely holds for all integers $k$. EDIT: From the second equation we get $e^{1+...
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3answers
85 views

Help, Where is wrong when I do same complex integration using two different contours

everyone! please give few hit. I want take the integral $$I=\int_{0}^{\infty}{\frac {dx}{ \sqrt{x}(1+{x}^{2})}} $$ by using the Residue Theorem. I choice two contours in complex plane with $z=r e^{i\...
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1answer
34 views

It's true that $ |\log^2(z)| \leqslant |\log(R)|^2 + |i \arg(z)|^2 $ where $z \in \mathbb{C}$

In some residue integral, when one have to prove that an integral vanish at infinity, I've found in some textbooks the inequality: $$ |\log^2(z)| \leqslant |\log(R)|^2 + |i\ \arg(z)|^2 $$ Where $z= ...
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0answers
30 views

Understanding a calculation deduced for the function $\pi^{-s/2}\Gamma(s/2)\zeta(s)$

With my current knowledges I don't know if this is a bad question, but since I am interesting in this kind of calculations I want to ask you, if I was wrong or if if my statement is obvious. From ...
2
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1answer
90 views
+50

Invariance of subharmonicity under a holomorphic map

If $f:U_1\rightarrow U_2$ is holomorphic and $u$ is subharmonic on $U_2$, then prove that $u\circ f $ is subharmonic on $U_1$. I know how to prove the same argument with $f$ being conformal. In that ...
2
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1answer
62 views
+100

Sufficient condition for an holomorphic map to be conformal

Let $U,V\subseteq\Bbb C$ be open sets, let $f:U\to\Bbb C$ be holomorphic. If we want to prove that $f$ is a conformal map $U\to V$, my teacher said that is enough to check that $f$ is locally ...
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1answer
22 views

Maximal value of real part of holomorphic function

Let $f:U \rightarrow C$ be a non-constant holomorphic function. $U$ is open, connected and $D(0,1+\epsilon) \subset U$. I'd like to show that there exists $z_0 \in \partial D(0,1)$ such that $Re(f(z))...
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1answer
50 views

Why Differential Forms on Riemann surfaces?

I am working with Rick Miranda's "Algebraic Curves and Riemann Surfaces". Right now I am in chapter four "Integration on Riemann Surfaces" and struggle with it a lot!:( It starts with the definition ...
2
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2answers
66 views

Solve this complex integral [on hold]

Solve this complex integral $$\lim_{\varepsilon \rightarrow 0} \int_{-\infty}^{\infty} \frac{d\omega}{2\pi i}\frac{e^{-i\omega x}}{\omega + i\varepsilon}$$ Where $\varepsilon > 0$ and $x$ is real....
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1answer
23 views

All power series has a point that is not regular.

Definition: Let $f = \sum_{n \geq 0} a_n z^n $ a power series and $0<R< \infty$ its convergence ratio. We say that $z_0 \in \mathbb C, |z_0| = R$ is a regular point if $\exists r > 0$ such ...
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1answer
33 views

Find the regular points.

If $f(z)$ is a power series, i.e., $f(z) = \sum_{n \geq 0} a_nz^n$, and this function is define in $B(0,R)$, where $ 0 < R < \infty <$ and $R$ is the convergence radius. We say $z_0 \in \...
2
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0answers
31 views

How to apply the Identity Theorem to this function?

Given the function $f(z)=\exp\left(z^2-\cos\left(iz\right)-4\right)$ with the domain $|z|<10$, if we try to apply the Cauchy integral formula, we'll see that f(2) "will be" $$\frac{1}{2\pi i}\int_\...
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1answer
49 views

How does one compute this heavy integral?

The integral is $$\frac{1}{2\pi i}\int_\Gamma\frac{\exp(z^2-\cos(iz)-4)}{z-2}dz$$ where $\Gamma$ is the unit circle. Here's how I tried to parametrize it: $z=e^{i\theta}$ on $\theta\in [0, 2\pi]$, ...
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1answer
394 views

Showing Schlaefli integral satisfies Legendre equation

The integral representation of Legendre functions is $P_ν(z)=\frac{2^{-\nu}}{2\pi i} \oint_Γ\frac{(w^2−1)^\nu}{(w−z)^{\nu+1}} dw$. I'm trying to show that this satisfies Legendre's equation. When I ...
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0answers
25 views

Existence of analytic function from A to B . [duplicate]

Does there exists a non constant analytic map $f:A\to B$ . Where $A=\{z\in \mathbb C~:~ |z|\neq 0\}$ and $B=\{z \in \mathbb C ~:~ |z|>1\}$. I am unable to construct one
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1answer
28 views

Minimum modulus principle - looks like a counterexample?

The minimum modulus principle states that if $f$ is holomorphic within a bounded domain D, continuous up to the boundary of D, and non-zero at all points, then $|f (z)|$ takes its minimum value on the ...
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0answers
21 views

Minimum norm of analytic function may not be achieved on the boundary of its domain

I need to show that the minimum modulus of an analytic function may not be achieved on the boundary of its domain. I'm stuck with this question, would appreciate if someone could help me with it. I ...
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0answers
15 views

Construction of continuous/analytic maps

Let $A=\{z\in C ~:~ |z|>1\}$ , $B=\{z\in C ~:~z\neq 0\}$. Which of the following is true? 1.There exists a continuous onto map $ f:A\to B. $ 2.There exists a continuous one to one map $ f:B\to ...
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1answer
1k views

what is the argument of 0?

When $z\neq 0$, $\arg z$ is defined to be the set $\{\theta \in \mathbb{R} : z=|z|e^{i\theta}\}$. What if $z=0$? Usually does one leave the argument of $0$ undefined? Or is $\arg 0 = \mathbb{R}$?
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2answers
19 views

$|f(x)-f(y)|\leq c|x-y|^{\alpha}$ uniformly continuous, while c>0 and $\alpha\in \mathbb{Q}\cap (0,1]$

Let $f:\mathbb{C}\supset X\rightarrow\mathbb{C}$ be a function with the property that c>0 and $\alpha\in\mathbb{Q}\cap (0,1]$ exist such that for all $x,y\in X$ following holds: $|f(x)-f(y)|\leq c|x-y|...
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2answers
51 views

$\int_{|z|=1} \frac{1}{\sqrt{z}} dz$?

Can we compute the integral ? $$\int_{|z|=1} \frac{1}{\sqrt{z}} dz$$ Actual problem asks to compute: $$\int_{|z|=2} \frac{z^n}{\sqrt{z^2+1}} dz$$ To compute this I need to solve the integral: $\...
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0answers
38 views

Xi Function on Critical Strip - Mellin Transform

Story I'm trying to prove the following identity $$\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cos(xt) dt = \frac{1}{2} \pi (e^{\frac{1}{2}x} - 2e^{-\frac{1}{2}x} \psi(e^{-2x}))$$ where $$\psi(...
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1answer
20 views

existence of an automorphism F(A)=A

Let G be a bounded domain in $\mathbb{C}$ and let $A\subset G$ be finite.I got to show that for every $f\in Aut(G\backslash A)$ there is a unique $F \in Aut (G)$ with $F\vert_{G\backslash A}=F$ and ...
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0answers
19 views

convergence to a non-injective function

Let G be a simply connected domain, $G \not \neq \mathbb{C}$ and $z_0 \in G$, I got to show that for every $n \in \mathbb{N}$ there is a holomorphic and injective mapping: $f_n:G \rightarrow D_1(0)$, ...
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1answer
27 views

holomorphic injective function

Prove: Let $z_o \in G\;$. There exists a holomoprhic injective function for every $n \in \mathbb N$ $\;f_n:G\rightarrow D_1(0)$ such that $f_n(z_0)=1-\frac{1}{n}$ I don't know how to find such a ...
3
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2answers
48 views

Frullani's theorem in complex context, other examples

One has as application of Frullani's theorem in complex context that $$\int_0^\infty \frac{e^{-x\log 2}-e^{-xb}}{x}dx=\mathcal{Log} \left( \frac{1}{2\log 2}+i\frac{B}{\log 2} \right) $$ where I taken ...
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0answers
19 views

Complex conics as a Riemann surface

Consider the complex curve defined by $\{(x,y) \in \hat{\mathbb{C}}^2 | ax^2 + bxy + cy^2 + dx + ey + f = 0 \}$ for some complex numbers $a,b,c,d,e,f$ (here $\hat{\mathbb{C}}$ is the Riemann sphere). ...
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1answer
20 views

Confused with the reexpression of a Hamiltonian in eigenbasis

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring of spin chain. To compute the complexity of ...
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2answers
30 views

show that Biholomorphism operates transitively

I got to show two statements for the following domains: $G:=\mathbb{C}, \mathbb{C\backslash\{0\}}$ and $D_1(0)$ (the circle around zero with radius 1): (i) the group of all biholomorphic function,$...
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0answers
21 views

Convergence of a big sum indexed over $\mathbb{Z}^3$

For a fixed vector $r_j$ consider the function on $\mathbb{R}^3$ defined by the series $$f(r) = \sum_{\substack{n,m,k \in \mathbb{Z} \\ (n,m,k) \neq 0}} \frac{1}{n^2+m^2+k^2}e^{2\pi i(n,m,k) \cdot (r-...
1
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1answer
25 views

complex integration cauchy theorem

I need to find the integral of the following assuming a simple closed path. $f(z) = e^z - \frac{1}{z^2}$ along the lower half of the unit circle with center at the origin traversed in the clockwise ...
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1answer
67 views

Is it true that $ \sqrt{-z} = i \sqrt z $?

Is it correct to write $ \sqrt{-z} = i \sqrt z $ , for every complex $z$? I think it's not true but I have seen it in some books . The reason I think it's not correct is for example if $z=i$ then $\...
1
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2answers
39 views

An example of a power series that has a radius of convergence of 3

The problem states "Give an example of a power series $\sum^{\infty}_{n=0}$a$_{n}$z$^{n}$ that has a radius of convergence of 3 and that represents an analytic function having no zeroes. I'm sorry if ...
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1answer
27 views

Complex bilinear transformation.

Let $H=\{z=z+iy\in\mathbb{C}:y>0\}$ be the upper half plane and $D=\{z\in\mathbb{C}:|z|<1\}$ be the open unit disc. Suppose that $f$ is a Mobius transformation, which maps $H$ conformally onto $...
2
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0answers
54 views

Double integral in complex variables form. [on hold]

Rewrite $\displaystyle \iint f(x,y) dx \, dy$ in complex variable form of $\displaystyle \iint g(z, \bar{z}) dz \, d\bar{z}$? Where $z=x+iy$, $\bar{z}=x-iy$ and $x$ changes from $0$ to $a$ and $y$ ...
2
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2answers
100 views

Find limit of $\lim_{z\rightarrow 0}\left(\frac{\sin z}{z}\right)^{1/z^{2}}$ [on hold]

where $z$ is a complex number. Please help me to solve it. I have no idea how to solve this but I have little bit knowledge of limits. my textbook's answer is $e^{-1/6}$. I am very confused at this. ...
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1answer
42 views

Complex Analysis - Integration [on hold]

I am trying to evaluate the integral for $f(z) =\frac{z} {(z^2-1)}$ , along the path $|z-\pi|=1.$ It is a simple closed path and positively oriented. Any help would be greatly appreciated.
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0answers
25 views

Plotting $\sum_{n\geq 1}\frac{1}{n}z^{n}$ for $|z|<1$ (the natural boundary): Coding

As part of my tutorial, I would like to show the plots of a)$-\sum_{n\geq 1}\frac{1}{n}z^{n}$ and b)$Log(1-z)=log(|z|)+iArg(z)$, to drive home the point of analytic continuation. Plotting the Log is ...
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0answers
32 views

Exciting applications of the Riemann-Roch-theorem for Riemann-surfaces

This semester I took a lecture on Riemann surfaces. The professor proved the Riemann-Roch-theorem (stated below). As an application of it, he proved elementary results, we did earlier in the course ...
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1answer
18 views

Solve the Cauchy-Riemann Equations for $u_x$ and $u_y$

I know the Cauchy Riemann Equations in Polar Coordinates are as follows: $$u_r= u_xcos\theta + u_ysin\theta$$ $$ u_\theta= -u_xrsin\theta + u_yrcos\theta$$ I need to solve the following equations ...
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1answer
35 views

Calculate the residues of this complex function

Calculate the residues of this complex function $$\frac{1}{z^2\sin(z)}$$ I can notice that we have singularities at $z=n\pi$, where $n=0,1,2,3,\dots$ But, how to find the residues?
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1answer
24 views

Radius of convergence of power series of $f(z) = \frac{z^{3}-1}{z^{2}+3z-4}$ at $0.$

The function $f(z) = \frac{z^{3}-1}{z^{2}+3z-4}$ has a power series expansion in a neighborhood of the origin. What is its radius of convergence. I believe I have to use the ratio test and show that ...
0
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0answers
49 views

Imaginary, holomorphic and bounded function, zero everywhere at $(i+n^\beta)$. Blaschke condition?

Is there a function $h\ne 0$, holomorphic and bounded on $\mathbb{H_+=\{z \in \mathbb{C}: Im\,z > 0}\}$ zero exactly at at all points $(i+n^\beta)$, $0 \lt \beta \lt +\infty$, $n\gt 0$ ? I have ...
1
vote
2answers
50 views

Automorphism of unit disk without zero

Let $S$ be the unit disk without $0$. Find all $f \in Auto(S)$ I got the following idea. By Riemann 0 is a removable singularity. Since for $g\in Auto(D)$ where $D$ is the unit disk. $g(z)= e^{i{\...
0
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0answers
28 views

Determine the nature and situation of the singularities of this function [on hold]

Determine the nature and situation of the singularities of this function $f(z) = \frac{1}{z(e^z -1)}$ and show their residues.