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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2
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18 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 ...
0
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0answers
8 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} + ...
0
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1answer
15 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 ...
1
vote
1answer
11 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 ...
0
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0answers
13 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 ...
1
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1answer
19 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. ...
1
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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 ...
0
votes
1answer
11 views

$f$ has a zero of order $k$ at $a$ and $g$ has a pole of order $m$ at $z=0$, then $g(f(z))$ has

$f,g$ 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 ...
1
vote
0answers
29 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 ...
1
vote
1answer
21 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?
1
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0answers
19 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 ...
-1
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0answers
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.
0
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0answers
15 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 ...
0
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0answers
24 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 ...
0
votes
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 ...
1
vote
1answer
14 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 ...
1
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0answers
25 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
votes
0answers
27 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
votes
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 $1-1/k$, $k \in \mathbb N$? ...
0
votes
1answer
37 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 $r$. ...
0
votes
1answer
33 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 ...
-4
votes
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 ...
0
votes
0answers
17 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 ...
1
vote
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 ...
1
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0answers
24 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: ...
1
vote
1answer
11 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
votes
1answer
22 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
28 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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votes
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
votes
0answers
44 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 ...
0
votes
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 ...
0
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
74 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
42 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
45 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 ...
1
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0answers
17 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
27 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, ...
2
votes
1answer
20 views

Creating surjective holomorphic map from unit disc to $\mathbb{C}$?

I'm trying to formulate a surjective holomorphic map from the unit disc ($\mathbb{D}$) to $\mathbb{C}$. I know that there exists $f: \mathbb{D} \rightarrow \mathbb{H}$, which is a biholomorphism from ...
0
votes
0answers
44 views

fixed point for analytic function in a connect set [on hold]

If $\omega \subset \mathbb{C} $ be a domain and $ f:\omega \to \mathbb{C}$ is an analytic function that $\text{Im} (f) \subset \omega$ is compact set. Prove that $f$ has a uniqe fixed point.
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0answers
26 views

Show that $\sin z$ has only one series expansion

The question goes: An extension of the real function $\sin x$ into a complex analytic function is by defining $\sin z = z- z^3/3! + z^5/5!- \cdots$. Show that this is the only way6 to extend $\sin x$ ...
0
votes
0answers
39 views

$f(z)$ is $2\pi$ periodic

If $f$ is entire such that $f(x+2\pi)=f(x)$ for all $x\in \mathbb{R}$, $|f(iy)|\leq \exp(-e^{|y|})$ for all $y\in \mathbb{R}$, and $|f(z)|\leq \exp\bigl(e^{|\operatorname{Im}(z)|}\bigr) $ for all ...
-3
votes
0answers
19 views

Covering of complex manifod [on hold]

Prove that a complex manifold of dimension n is possible build a covering such that for all intersection of more of 2n open sets of covering is empty.
1
vote
1answer
31 views

Intuition behind Laurent's theorem?

Taylor series has a pretty nice intuitive explanation. If you know the position, velocity, acceleration and so on of a particle you can predict it's location at any time. Does a similar intuitive ...
0
votes
1answer
25 views

A question about the complex logarithm

So frist, define $L(z) = \log(r)+i\theta $ is the holomorphic branch of $\log(z)$ on the cut-plane $\mathbb{C} \setminus (-\infty,0]$ such that $L(1)=0$ Let$[1,i]$ denote the line segment from 1 to ...
0
votes
0answers
43 views

Showing that there exist $C \in \mathbb{C}$ such that $g(z)=C \sin(z)$ if $g$ holomorphic & $|g(z)|\leq A|\sin(z)|$ ($A\in\mathbb{R}$)? [duplicate]

I'm trying to manipulate the sine function is some complex analysis problems (I need practice) and I've encountered two slight darker points: First, I don't understand how it can be possible (I read ...
2
votes
0answers
23 views

Geometric Interpretation of Antiderivative?

Could someone please give me a geometric interpretation of the above theorem?
3
votes
1answer
29 views

Transcendence Degree of the Function field of $\mathbb{C}$

What is the transcendence degree of the field of meromorphic functions over $\mathbb{C}$? By a cardinality argument (meromorphic functions are determined by their image under a countable dense subset ...
0
votes
0answers
36 views

Geometric interpretation of Cauchy-Goursat Theorem?

This theorem seems almost magical. The algebraic derivation doesn't really provide any insight into why it works. So could someone give me a geometric interpretation of it? This: Geometrical ...
1
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0answers
20 views

Geometric interpretation of analyticity?

Suppose the real valued functions $u(x,y)$ and $v(x,y)$ are continuous and have continuous first order partial derivatives in a domain $D$. If $u$ and $v$ satisfy the Cauchy Riemann equations at ...