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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Hermite polynomials 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
10 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
40 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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3answers
42 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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27 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
15 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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44 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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17 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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0answers
21 views

Transcendental solution to system of equations

Suppose $(A)$ $$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and there are functions ...
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2answers
16 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
18 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
49 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 $$\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^2}$$
2
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0answers
27 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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1answer
22 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
23 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
17 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
16 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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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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22 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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30 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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0answers
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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25 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
28 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
38 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 ...
2
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2answers
49 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
40 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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0answers
9 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
35 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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0answers
19 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 ...
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1answer
23 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
votes
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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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
46 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 ...
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1answer
30 views

Zeros of $z^2 \text{cos}z^2$

Is there an easy way to find the zeros of the function $z^2 \text{cos}z^2$, $z\in \mathbb{C}$ and the respective orders (multiplicities)? All I can think of is to find $f^{(1)},f^{(2)},...$ but then ...
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votes
3answers
30 views

Find couples of complex numbers

I found this exercise, given: $$u=|z|+|u|$$ and $$z=|u|+1$$ (it is a system I don't how to write it in latex from) I have to find the couples of complex numbers $u,z$ that comes from the two equation. ...
3
votes
1answer
78 views

Evaluating $\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$

Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$$ We see that the only poles are at $x=\pm bi$. Integrating over the semicircular contour implies that it is equal to $2\pi i*Res_{(+bi)}$ ...
1
vote
2answers
44 views

Integral of $((x^2+1)((x-1)^2+1))^{-1}$

Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+1)(2-2x+x^2)}dx$$ So I am going to integrate this using a semicircular contour. Is it safe to say that on the curved part, the integral vanishes? because ...
0
votes
1answer
70 views

Express mapping from $\mathbb R^2 \to \mathbb R^2$ as a function $f: \mathbb C\to \mathbb C$

Express $\underline{f}(x,y)=(4x^2,y^2)$ as a function $f: \mathbb C\to \mathbb C$. How do I go about this? I think it's something like $4x^2+y^2i$ but obviously there should only be one variable so ...
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0answers
19 views

Contour Integral of sin(z)/(z^2-z)

Find the integral $\int_{\lambda}\frac{\sin(z)}{z(z-1)}$ where $\lambda(t) = 10e^{it},t\in[0,2\pi]$ We notice that there are poles at $z = 0$ and $z=1$. So we can use residue theorem but I am ...
0
votes
1answer
31 views

Harmonic function that extends continuously up to the boundary?

I have to find a harmonic function $u$ in the open first quadrant that extends continuously up to the boundary except at the points $0$ and $1$, and that takes up the following boundary values: $u(x, ...