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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Piecewise continuity for contours - Definition for complex analysis

We want piecewise continuity for any contours in complex analysis. What does this refer to? I imagine it refers to the nature of referring to each line arc being continuous. E.g. We want continuity ...
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10 views

An entire function that must be a polynomial

Let $f$ be an entire function of exponential type zero such that its restriction to the real line belongs to $L^{p}$ for some $p$ > $1$. I want to prove that f is constant. My initial attempt was to ...
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7 views

extending real solution to holomorphic function

Given a rational functions $f$ and $g$ defined on an interval $]0,a[$, $a>0$, a function $h$ satisfies in $]0,a[$ the linear differential equation: ...
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1answer
16 views

Harmonic function as a real part of laurent series and log|z|

The exercise: Let $h$ be a function harmonic on $\{z\in\mathbb{C}: \rho_1 < |z| < \rho_2\}$. Using the fact that $h_x - ih_y$ is holomorphic, prove that there exist unique constants ...
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1answer
35 views

Evaluate the improper integral.

Evaluate the integral below. $ \int^{+\infty}_{-\infty} \frac{x^2}{{(x^2-8x+20)}^2} \, dx $ I feel that I know how to do this problem, but I'm getting caught up in all the calculations. I've been ...
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9 views

Intersection of a curve with a complex line

Given: $$ \left\{\begin{matrix}t =\frac{1}{n}\sqrt{n^{4}-z^{2} } & \\ z=im & \end{matrix}\right.$$ with $n<m$, positive integers (and $i$ the imaginary unit), if one wanted to ...
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1answer
20 views

Mimimum of entire function over unit disc

I'm trying to solve the following question: Let: $f:\mathbb C\to \mathbb C$ be an entire function, $|f(0)|=1$ and $f$ has exactly $n$ roots in $D(0,\frac{1}{2})$. (Not necessarily distinct) ...
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65 views

Show that holomorphic function $f: \mathbb{C} \rightarrow \mathbb{C}$ is constant

Let's $f: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function such that values $f$ are on line $y=ax+b$. Show that $f$ is constant. I think I should use Cauchy-Riemann equations but I don't ...
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1answer
18 views

Contour integration for the improper integral

How to integrate $\Large \int_{-\infty}^{\infty}\frac{x \sin (7x)}{(x^2-6x+18)} dx$ with contours? So far I have $\Large \int_{C}^{}\frac{ze^{7iz}}{(z-(3+3i)) (z-(3-3i))} dz$ With the $$ \large ...
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1answer
18 views

determine and classify all singular points of $\frac{z}{e^{\pi z}-1}*\log(z^2+4)$

determine and classify all singular points of \begin{equation*} \frac{z}{e^{\pi z}-1}\ast \log(z^2+4) \end{equation*} Obviously one sees that the singularities occur when $z = 2in$ with $n \in ...
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1answer
46 views

Qusetions on complex analysis

I was doing my homework: If $w_1, w_2\in\mathbb{C}$ satisfies $\mathrm{Im}\frac{w_1}{w_2}\neq0$ , $f$ is an entire function and there are $a,b$ such that [ \left{ ...
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1answer
21 views

Why does $f^{(n)}(x) = 0$ for all $n \geq 1$ if $|f^{(n)}(x)| \leq \frac{1}{R^n}$ and $R \to \infty$?

Claim: If for all $z \in B(a, R)$ we have $|f(z)| < 1$, then by Cauchy's estimate we have $|f^{(n)}(a)| \le \frac {n!}{R^n}$ for all $R > 0$. If $R \to \infty$, then $f^{(n)}(0) = 0$ for all ...
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22 views

Why is this function not holomorphic on a disk?

I have the complex-valued functions $$f_1\left(z\right)=Erf\left(R-z\right)$$ and $$f_2\left(z\right)=Erf\left(R-\lvert z\rvert \right)$$ Now, I'm told $f_1$ is holomorphic over the disk of radius ...
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2answers
26 views

How should I expand $\frac{1}{z(e^z-1)}$ to find the residue and order at the pole z = 0?

$\frac{1}{z(e^z-1)}$ is the function I want to expand. I tried using the expansion for $e^z$ and got $$\frac{1}{z^2+z^3/2!+z^4/3!+...}$$ Can I put this fraction into the $b_n/(z-z_0)^n$ form, or did I ...
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1answer
9 views

Harmonic function with vanishing partial derivative

Let $u:D(0,1)\to \mathbf{R}$ be harmonic on the unit disc, and suppose there exists a $z_0\in D(0,1)$ such that all partial derivatives of $u$ vanish. Show that $u$ is constant. I found this problem ...
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1answer
23 views

If $\gamma(t) = t + it^2$ and $f(z) = xy^2-iyx^2$ how do we compute $f(\gamma(t))$?

If $\gamma(t) = t + it^2$ and $f(z) = xy^2-iyx^2$ how do we compute $f(\gamma(t))$? I'm confused because there are no values of $z$ stated explicitly for $f(z).$ I can do this easily with functions ...
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1answer
41 views

When Cauchy-Riemann equations hold, WHERE does it tell you $f$ is analytic?

So I've been asked 'Is $f$ analytic anywhere? Everywhere? Justify your answer.' My function is $f(z)=z^2-iz+iz^2$ which I've expressed as $f=u+iv$ so I can use Cauchy-Riemann equations to test to ...
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16 views

Parametrising the boundary of an open ball

I have been asked to evaluate line integrals of complex functions, namely of the form $$\int_{\gamma} f(z) dz$$ where $f: U \rightarrow \mathbb{C}$ is an analytic function and $\gamma : [t_{0}, t_{1}] ...
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2answers
39 views

Image under Möbius transformation

I would like to find the image of $$ {z \in C: |z|<1, Im{z}>0 } $$ under the complex map $$ w(z) = \frac{2z-i}{iz+2} $$. Well, since $w(2i)=\infty$ the interval $[-1,1]$ and $ {z \in C: ...
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34 views

Complex $\Gamma$-function is holomorphic?

How can I show that $\Gamma$ function is holomorphic ? I have to show it by dominated convergence theorem or by Morera's Theorem For $\Re(z)>0$ the $\Gamma$-function is defined as ...
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2answers
24 views

Proving a norm on the space of differentiable functions

I consider the space $C^1[a, b]$ of (complex) functions that are at least once differentiable on $[a, b]$. I want to show that $$||f||_{C^1} := ||f||_\infty + ||f'||_\infty$$ defines a norm on ...
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36 views

How to integrate $\int_{\gamma}\frac{dz}{z}$

How to integrate $\int_{\gamma}\frac{dz}{z}$ If $\ A=1+i, B=1-i, C=-1-i, D=-1+i$ and $\gamma$ is the path $[A,B,C,D,A]$ If I write $\frac1z=\frac{1}{x+iy}=\frac{x-iy}{x^2+y^2}$ The first segment ...
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30 views

zeros of Riemann zeta function.

My question is, is $z = 0$ is zero of Riemann zeta function? by putting z $= 0$ in Riemann functional equation $$\zeta(z) = [2(2\pi)^{z-1}]\times\zeta(1-z)\times\Gamma(1-z)\times\sin(\frac{\pi ...
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1answer
20 views

Show that a function maps a region into the unit circle

Show that the following function maps the indicated region into the unit circle. $$f(z) = e^{i \Psi}\frac{z^{\frac{\pi}{\alpha}} - a}{z^{\frac{\pi}{\alpha}} - \bar{a}}$$ where $\Psi \in \mathbb{R}$, ...
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0answers
11 views

Start to Proof of bernoulli polynomials and sums

I need help starting this proof: For all integers k,l,m>=0 and not all equal to 0, (3.7) It says that that comparing the above equation (3.7) with the one discussed earlier in the paper(3.6) (shown ...
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1answer
22 views

Lens Conformal Map

Please help me find a conformal map of the set $ A = \left \{\; z: \; |z-1| < \sqrt{2} \; and \; |z+1| < \sqrt{2} \; \right \}$ one-to-one onto the open first quadrant. First, I noticed that ...
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1answer
45 views

Proving $\frac{\sin\pi z}{\pi z}=\prod_{n=1}^{\infty}\Big(1-\frac{z^{2}}{n^{2}}\Big)$

I apoligize if this has been answered already; the quick searches I've done have proven fruitless. I'm given that $\displaystyle\pi\cot(\pi z)=\frac{1}{z}+\sum_{1}^{\infty}\frac{2z}{z^{2}-n^{2}}$ ...
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0answers
18 views

Find the maximum of a |cos(z)|

How do you find the maximum of the complex function $|\cos{z}|$ on $[0,2\pi]\times[0,2\pi]$. I believe I'm to use the maximum modulus principle, since the function is entire. I'm just having problems ...
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1answer
23 views

Analytic function with a zero of multiplicity 1

Suppose that $f(z)$ is analytic and has exactly one zero, at $a$, inside the circle $\gamma$, and that it has multiplicity $1$. Show $$ a = \frac{1}{2i\pi} \int_\gamma \frac{zf^{\prime}(z)}{f(z)}dz. ...
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1answer
22 views

Subtle Differences between Analiticity and differentiability at a point

I am stuck on the second part of this problem .Prove that the function $f(z)=|z|^4$ is differentiable , but not analytic at $z=0$. I can do the differentiability since $f(z)=(x^2+y^2)^2$ and hence at ...
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1answer
16 views

Question about a Möbius transformation/Conformal map

I have a question about a conformal mapping. The map $f(z)=\frac{1+z}{1-z}$ takes the unit disk to the right half plane. Composing this map with $z^2$ gives $f(z)=(\frac{1+z}{1-z})^2$, which I think ...
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1answer
14 views

Laurent series, function representation

Write the Laurent series for the function $f(z)=\frac{1}{1+z}$ $1<|z|<\infty$ I did $$\frac{1}{1-z}=\sum_{i=0}^\infty z^n\rightarrow \frac{1}{1+z}=\sum_{i=0}^\infty (-1)^nz^n$$ Is it right? ...
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0answers
25 views

Cauchy integral formula for n=1

I am working on proving cauchy integral formula (C.I.F) for $n=1$: $$f(z_0)=\frac{1}{2\pi i} \int \frac{f(z)}{z-z_0}dz.$$ Is there another method rather than taking $z= z_0+re^{i \theta}$, putting it ...
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3answers
58 views

Factor $z^4 +1$ into linear factors

$z$ is a complex number, how do I factor $z^4 +1$ into linear factors? Do I write z in terms of $x+yi$ so that $z^4+1=(x+yi)^4+1?$
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1answer
63 views

How exactly does $\frac{\partial f}{\partial \bar{z}}$ work?

I'm currently learning about complex analysis, and I keep coming across expressions involving $\frac{\partial f}{\partial \bar{z}}$. But I don't understand what this means. For example people might ...
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0answers
16 views

Understanding an example involving Rouches Theorem from Marsden-Hoffman Basic Complex.

I am trying to understanding two of the steps in an example involving Rouches Theorem from Marsden Hoffman Basic Complex Analysis. Specifically, its example 6.2.13. $$Let\; ...
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1answer
21 views

Laurent series , function representation

Write the Laurent series around zero for the entire function $f(z)=z^2e^{3z}$ I'm a little confused on how to represent the complex functions by series, as I did in the calculation of real functions, ...
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1answer
29 views

Prove whether the following subset is open, closed, or neither

We've sketched the subset, and we now need to prove whether it's closed, open or neither. We have no idea where to start so any help is appreciated, thanks. $S\subset \mathbb C$
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2answers
19 views

Bijective continous from R to closed half interval

Can we have bijective continous function from Set of real number R to closed half interval 0 to infinity.
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1answer
54 views

Why are the Cauchy-Riemann equations in polar form 'obvious'?

In my book on complex analysis I'm asked to prove the Cauchy-Riemann equations in polar form, which I did. However, at the end of the question the author asks why these relations are 'almost obvious'. ...
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3answers
28 views

Suppose $a,b \in \mathbb C$ (with non-zero imaginary parts) satisfy that $a\cdot b \in \mathbb R$. Can I then conclude that $b = \bar a$?

Suppose $a,b \in \mathbb C$ (with non-zero imaginary parts) satisfy that $a\cdot b \in \mathbb R$. Can I then conclude that $b = \bar a$? Can I then conclude that $b = \bar a$, i.e. $b$ is the ...
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1answer
17 views

Entire function with a condition.

Find all entire function on $\mathbb C$, such that $|f (z)|\le 100 \log|z|$ for each $z$ with $|z|\ge 2$ and $f (i)=2i$.
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1answer
51 views

For which $z \in \Bbb{C}$ does this series converge?

How can I determine for which $z\in \Bbb{C}$ the following series converges? $$\sum_{n=0}^{\infty} \frac{z^n}{1+z^{2n}}$$ I've tried the root and ratio test with no success.
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0answers
36 views

If $f$ is harmonic on $G$ and $f\big|_U=0,$ then $f\equiv0$

Let $G\subset\mathbb C$ be a connected open set, and let $f:G\to\mathbb R$ be harmonic. If there is an nonempty open set $U\subset G$ s.t. $f\big|_U=0$ then $f\equiv0.$ In the proof $N:=\{z\in ...
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1answer
39 views

Zeroes of a complex function

Let $\mathcal{O} = \{z: f$ has zero of order $\infty$$\}$. We claim that $\mathcal{O}$ is open. To prove this we note that, if $f$ has a zero of infinite order at $z_{0}$, then all the Taylor ...
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0answers
21 views

Exterior of the square $ABCD$ is a connected

If $\ A=1+i, B=1-i, C=-1-i, D=-1+i$ and $\gamma$ be the path $[A,B,C,D,A]$, that i the contour of the square $\ ABCD$, show that the exterior of the square $\ ABCD$ is a connected subset of the ...
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1answer
35 views

a maximum modulus problem

Let $0<r<R$ and $A=\{z:r\leq |z| \leq R\}$. Prove that there is a positive number $\epsilon >0$ such that for each polynomial $p$, $$\sup \{|p(z)-z^{-1}|:z\in A\}\geq \epsilon $$ I ...
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1answer
23 views

Quotient of rational functions is meromorphic

I feel like this should be a very straight forward problem but I am having difficulties with the definition. I'm trying to prove that $f: \mathbb{C}P^1 \to \mathbb{C}P^1$ given by $f(z) = g(z)/h(z)$ ...
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1answer
14 views

Argument of the function $2 \frac{\sin(x)}{x} + \frac{\sin(x/2)}{x/2} e^{-i x/2} $

Plot the argument (phase ) of the Complex function $$2 \frac{\sin(x)}{x} + \frac{\sin(x/2)}{x/2} e^{-i x/2}$$ This can be written as $$\frac{1}{ix} (e^{ix} + 1 - 2 e^{-ix})$$ Wolfram shows ...
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1answer
39 views

Group theory and Complex Analysis

Actually this time I am having interest in group Theory and complex analysis. So I want to study the topics which relate both. So please suggest me which topic or book should I read to explore the ...