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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4 views

If $f$ is ananlytic in $D_r(z_0)$\ {$z_0$} and $Ref(z)>0$ for all $z\in D_r(z_0)$\ {$z_0$} then $z_0$ is a removable singularity

If $f$ is ananlytic in $D_r(z_0)$\ {$z_0$} and $Ref(z)>0$ for all $z\in D_r(z_0)$\ {$z_0$} then $z_0$ is a removable singularity. I want to prove this statement but I just cannot seem to find a ...
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16 views

Euler Characteristic $\mathbb{C} \backslash\{0,1\}$ and non-compact surfaces

I am looking at various proofs of the Little and Big Picard Theorems. I am interested in the following question: Without the Uniformization Theorem, can one calculate the Euler characteristic of ...
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11 views

Question about a example of Schwarz-Christoffel Transformation

My question is 1.What the initial point mean for the first integration, and can it be $-\infty$? 2.For the integration of $A \int_{ 1 }^{ \infty } {\frac{dt}{(t^2-1)^\frac{2}{3}}} $,why it ...
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78 views

What is a good, hi-tech textbook on complex analysis?

I am looking for an introductory textbook for Complex Analysis that is hi-tech. All the books I have looked at suffer from the same problem; they're only assuming that the reader is familiar with is ...
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17 views

Entire function with constant argument on the unit circle

What is the easiest way to show that the only entire functions with constant argument on the unit circle (i.e., $\arg(f(z))=const$ for all $z$ on the unit circle) are the constant functions?
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1answer
19 views

Application of Minimum Modulus Principle

How do I use the minimum modulus principle to characterize the family of holomorphic functions $f:\mathbb{D}\to \mathbb{D}$ which have a continuous extension to the closure of $\mathbb{D}$ and map the ...
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29 views

Misplaced complex analysis intuition on Riemann Surfaces

Next week I will be giving a lecture, based on Chapter 2.6 from Jost's book Compact Riemann Surfaces. He states the following theorem: Theorem 1 (Jost Theorem 2.6.2) Let $S$ and $\Sigma$ be Riemann ...
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1answer
34 views

Hermite functions as eigenvectors of Fourier transform

In order to find an orthogonal basis of eigenvectors of the Fourier transform operator $F:L_2(\mathbb{R})\to L_2(\mathbb{R})$, $f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x$ for ...
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3answers
12 views

Not sure how this inequality is formed - $\bigg|\int_0^{2\pi}\frac{e^{p(R+iy)}}{1+e^{R+iy}}idy\bigg| \le \frac{e^{pR}}{e^R - 1}2\pi$

I have the following inequality in my notes - $$\bigg|\int_0^{2\pi}\frac{e^{p(R+iy)}}{1+e^{R+iy}}idy\bigg| \le \frac{e^{pR}}{e^R - 1}2\pi$$ We can start as follows ...
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1answer
52 views

Evaluate $\int_0^{2\pi} \frac{d\theta}{\left(1+\beta \cos (\theta )\right)^2}$

I am trying to evaluate the integral $$\int_0^{2\pi} \frac{d\theta}{\left(1+\beta \cos \left(\theta\right)\right)^2}$$ via change of variables and applying Cauchy's Residue Theorem. Here is how I'm ...
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64 views

Similar to Cauchy inegral formula

Let $f=u+iv$ be an analytic function in disk $\mathbb{D}$ and $0<r<1$. Can you help me to prove that $$\pi{r}f'(0)=\int_{0}^{2\pi}\frac{u(re^{i\theta})}{e^{i\theta}}d\theta\;\;\;?$$ I tried ...
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1answer
55 views

Prove that there is no isomorphism between any two of the groups $ Aut(\hat{C}) $,$ Aut(H^+) $(upper half plane) and $ Aut(C) $

Referring the groups of automorphisms (holomorphic bijections) of the respective domains. An equivalent statement would be: there is no isomorphism between any two of PSL(2,C), PSL(2,R) and ...
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2answers
18 views

Uniform convergence of the series $\sum_{r=1}^{\infty} \frac{1}{(r-z)^2}$ in a neighborhood

I am asked to show that the function $f(z)=\sum_{r=1}^{\infty} \frac{1}{(r-z)^2}$ defined on $\mathbb{C} \setminus \mathbb{N}$ is holomorphic assuming that the series $\sum_{r=1}^{\infty} ...
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1answer
59 views

Exercise - analytic function

Assume that $f$ is an analytic function on $|z|<2$ $f(0)=f'(0)=0$ , $f''(0) \not= 0$, $f(1/3)=i/12$ , $|f(z)|\le3$ for $|z|<2$ then find the value of $f(2i/3)$. Thank you
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22 views

Center of real projective line or Riemann sphere

I have recently encountered the ideas of the real projective line and the Riemann sphere, and it seems to me that in any circle (representing the real projective line) or sphere, the center is a ...
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1answer
47 views

Transcendental solution to system of equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and ...
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2answers
18 views

Why does ray for trigonometric functions not need to be cut?

My question is for complex variables, I understand that ray of log Z needs to be cut starting from the origin (since log 0 does not exist) and give a domain for the theta values, so we can have ...
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0answers
19 views

Is the error function in the complex plane bounded? [on hold]

I have to show that the $ erf (\sqrt{(\lambda / 2) }r(t) x)$ is bounded where $r(t)$ is only bounded when $\lambda < 0$
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2answers
53 views

Evaluate $\displaystyle\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^2}$ using residue method [on hold]

This is a real integral but I want to evaluate it using residue integration method $$\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^2}$$
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1answer
30 views

Find a T such that T maps the real axis onto itself and the imaginary axis onto the circle $|w-\frac{1}{2}|=\frac{1}{2}$

Find a linear fractional transformation T such that T maps the real axis onto itself and the imaginary axis onto the circle $|w-\frac{1}{2}|=\frac{1}{2}$ I have no idea how to do this kind of ...
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12 views

classifying singularities of functions

(a) $$\frac{\pi}{tan \pi z}$$ (b) $$\frac{z^2-z}{1 - sin z}$$ -- for part (a), I found the singularities to be $z = n$ $\space$ $\forall n \in \mathbb{Z}$ and for part (b), $z = \frac{\pi}{2} + ...
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2answers
32 views

Showing a certain complex function is surjective

Can you help me show that $f(z)=z+e^{z}$ is surjective onto $\Bbb C$? The idea is to show that for any $z$, we can construct a closed curve $C$ around $z$ such that $z$ is contained in $f(C)$ with ...
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1answer
19 views

Prove of an addition theorem for the general binomial coefficients

Prove that: $\sum_{k=0}^n \binom{s}{k} \binom{t}{n - k} = \binom{s + t}{n}$ for all $s, t \in\Bbb C $, $n \in N\cup {0}$. That's pretty much all I'm given, and therefore, I haven't come quite far ...
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0answers
16 views

How to determine the limit of a complex function

It is easy to show that a complex function doesn't have a limit as it approaches a certain point, but is there any way to know for sure whether any given complex function has a limit as it approaches ...
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1answer
24 views

Circle in the plane of complex numbers

Let $K = \{z \in \mathbb{C}: |z−a|=r \}$ be a circle in $ℂ$. Show that, for the case that $|a|$ is not equal to r, the image of $K$ under the transformation $z$ $\to$ $\frac {1}{z}$ is a circle too. ...
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2answers
19 views

$f(w)=\frac{w}{4w^{2}-1}$, find max value of $|f(w)|$ in $|w|\geq1$

$f(w)=\frac{w}{4w^{2}-1}$, find max value of $|f(w)|$ in $|w|\geq1$ What I have done is ...
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2answers
17 views

If $f$ has a zero of order $k$ at $a$ and $g$ has a pole of order $m$ at $z=0$, what does $g(f(z))$ have at $z=a$?

$f,g$ be meromorphic on $\mathbb{C}$, $f$ has a zero of order $k$ at $a$ and $g$ has a pole of order $m$ at $z=0$, then $g(f(z))$ has a zero of order $km$ at $z=a$ a pole of order $km$ at $z=a$ a ...
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0answers
33 views

How to conclude from here that $f$ is a constant function or not?

This is an extension of a previous problem I had asked before $f:\mathbb C\rightarrow \mathbb C$ be analytic with the property that $f (z)=i$ when $z=1+\frac{k}{n}+i \forall k \in \mathbb N$.how to ...
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1answer
22 views

How to conclude from here that $f$ is a constant function or not?

let $f:\mathbb C\rightarrow \mathbb C$ be an analytic function with the property that $|f(z)| \in \mathbb Z \forall z\in \mathbb C$. How to conclude from here that $f$ is a constant function or not?
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0answers
23 views

Biholomorphic in two Complex Variables

I am looking for an example of a biholomorphism $f:U\to \mathbb{C}^2$ where $U$ is an open proper subset of $\mathbb{C}^2$. Please provide any references/books/papers that you know of where I can look ...
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0answers
31 views

$f^{-1}(w)$ is finite $\forall w \in \Bbb C \Rightarrow f$ is a polynomial. [on hold]

Let $f$ be an entire function s.t i) $f^{-1}(w)$ is finite $\forall w \in \Bbb C \Rightarrow f$ is a polynomial. ii) If $\exists N\in \Bbb N$ s.t $|f(z)|<|z|^N$ then $f$ is a polynomial.
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17 views

Computing integral

Let $E(s)=\dfrac{a_0+a_1s+a_2 s^2}{b_0+b_1 s+b_2 s^2+s^3}$ and $a_0\ne 0;a_1a_2-a_0\ne 0$.Compute $$A=\dfrac{1}{2\pi}\int\limits_{-\infty}^{\infty}|E(j\omega)|^2 d\omega$$ where $j^2=-1$. The book 's ...
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0answers
28 views

Proving that $f(z)\neq \frac{z}{z+1}$ in $D_1(0)$

Suppose $f$ is analytic in $D_r(0)$ for some $r>1$. I want to prove that $f(z)\neq \frac{z}{z+1}$ in $D_1(0)$. This is how I tried to prove this. Assume $f(z)= \frac{z}{z+1}$ in $D_1(0)$. Now ...
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1answer
22 views

Complex Analysis-Taylor Series

I have the following problem Show that if $f$ is an analytic function in the unite disc $\Bbb D $ such that $f(-z)=f(z)$ for each $ z \in \Bbb D $ then there is an analytic function $h$ such that ...
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1answer
15 views

Analytic map from slit plane to disc [on hold]

In Conway's complex analysis chapter 3 there is an exercise: map $C-\{z:-1\leq z\leq 1\}$ onto the open unit disc by an analytic function f. Who know how to do it? I want some detail. Thank you very ...
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0answers
29 views

Find the Fourier transform of $u(x) = \frac{x \cos(2x)}{(1+x^2)^2}$

Find the Fourier transform of $$u(x) = \frac{x \cos(2x)}{(1+x^2)^2}$$ My work Okay so we want $$\int_\mathbb R \frac{e^{-ixt}x\cos(2x)}{(1+x^2)^2}dx$$ Of course we want to apply the residue ...
3
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1answer
43 views

Decomposition of analytic functions

Given two open overlapping sets $\Omega_1$, $\Omega_2$ and an analytic function $f$ on $\Omega_1\cap\Omega_2$, how does one prove that there are analytic functions $g_1$ on $\Omega_1$ and $g_2$ on ...
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2answers
50 views

On the constuction of a series of meromorphic functions that converges to a meromorphic function with prescribed poles and residues.

How can I constuct a series of meromorphic functions on $D_1(0)$ that converges locally uniformly to a meromorphic function with simple poles with residue $1$ at the points $1-1/k$, $k \in \mathbb N$? ...
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1answer
42 views

complex plane questions

Find where the points of the complex plane are if, a) |pi - arg z| < pi/4 b) |Re z| < 1 c) Im {(z+1)/(z+i)} = 0 d) z = z1 + t(cosx + isinx), 0<=x<=pi/4 where z1 = 1+2i and t=2 Please ...
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1answer
13 views

A linear fractional transformation and mapping of concentric circles

Q: A fractional linear transformation maps the annulus $r < \|z \| <1$ (where $r > 0$) onto the domain bounded by the two circles $\|z- \frac{1}{4} \|=\frac{1}{4}$ and $\|z \|=1$. Find $r$. ...
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1answer
36 views

Intuition/Understanding of “Infinite” Countour Integrals

I'm trying to clarify some thoughts on contour integration. If I have an integral $\int_{c-i\infty}^{c+i\infty} f(z) dz$, where $f(z)$ has finitely many poles in the complex plane...is this ...
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1answer
49 views

complex analysis questions [on hold]

evaluate $\Bigl[\frac{1-i\sqrt{3}}{1+i}\Bigr]^{21}$ write down the number $(1+\cos x+i\sin x)^{2n}$ where $0<x<\pi$ in polar form solve the equation $z^6+1=i\sqrt{3}$ write down the expression ...
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1answer
26 views

Proving all conformal mappings from the upper half-plane to the unit disc take a certain form

Bit confused about the above solution... why is the inverse of $\phi$, $w = \frac{i-z}{i+z}$, substituted into $$e^{i\mu}\frac{\alpha - w}{1 - \overline{\alpha}w}?$$ What exactly is the solution ...
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1answer
26 views

Where the power series is convergent

Where $f(z)=\sum_{n=1}^{\infty}\frac{(2i)^n}{n}z^n$ is convergent? I checked that the radius of convergence is equal to $\frac{1}{2}$. Now, since we know that the series ...
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0answers
26 views

A function on the punctured complex plane which turns out to be constant

Let $f: \mathbb{C}- \{0\} \rightarrow \mathbb{C}$ be a holomorphic function on the punctured complex plane, and suppose that $f(2z)=f(x)$ for all $z \neq 0$. Prove that $f$ is constant. Proof: ...
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1answer
12 views

Behavior of Points on Conformal Mapping Boundary

Carathéodory's theorem states that given a conformal mapping $f: J \to D$ from a Jordan region to the unit disc, we can extend this to a homeomorphism from the Jordan curve bounding $J$ to the unit ...
0
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1answer
24 views

Laurent Series Expansion for $f(z)=\dfrac{z+2}{(z+1)(z-2)}$ in $\{1<|z|<2\}$ and $\{2<|z|<\infty\}$

I'm trying to get the Laurent Series expansion of the function stated in the title in the stated regions. My approach is as follows: We can first break up $f(z)$ using partial fractions ...
2
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1answer
31 views

Using complex analysis to find the Inverse Laplace transform

I have been reviewing for my comprehensive graduation exam where I have been solving the Inverse Laplace transform via complex analysis. Consider $$ H(s) = \frac{s^2 - s + 1}{(s + 1)^2} $$ Then we ...
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0answers
22 views

Computing integration [on hold]

1) Let $E(s)=\displaystyle \frac{1+2s}{2+9s}$. Compute $\displaystyle A=\int_{-j\infty}^{j\infty}E(s)E(-s)ds $ 2) Let $E(s)=\displaystyle\frac{1+2s+8s^2}{2+9s+10s^2}$. Compute $\displaystyle ...
4
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0answers
47 views

exponential function, lie group homomorphism

Let $f: \mathbb{R} \to \mathbb{C}^*$ be a continuous map satisfying for all $x, y \in \mathbb{R}$: $f(x + y) = f(x)f(y)$. $f(x) = 1$ for all $t = 2\pi n, n \in \mathbb{Z}$. Show that there exists ...