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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13 views

How to show that a complex function have a branch in a domain

I've given as homework to show that the function $$f(z)=\sqrt{\frac{z+1}{z-1}} $$ has a branch on $G = \mathbb C \backslash [-1,1] $. I'm having a hard time in finding the way to approach this kind ...
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18 views

Searching for a constant transformation in $ \mathbb C$

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with a set $B \subseteq \mathbb C $, which is bounded. Now I want to proove that $ A = f^{-1} (B)$ is NOT bounded! I know it ...
2
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1answer
48 views

An entire and one-to-one function - what can it be? [duplicate]

Show that if f is entire and one-to-one, then it must be of the form AZ+B, with A not equal to zero. My work: If we restrict our discussions to the linear fractional transformations, then we can ...
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0answers
24 views

Is restriction of $exp:\mathbb{C} \to \mathbb{C}^*$ to $A = \{ x+iy : x \in R, y \in ]1, 1+2\pi]\} \subset \mathbb{C}$ a bijection?

I have this question: Is the restriction of exp function $exp:\mathbb{C} \to \mathbb{C}^*$ to $A = \{ x+iy : x \in R, y \in ]1, 1+2\pi]\} \subset \mathbb{C}$ a bijection? Here's what I tried: ...
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1answer
16 views

Biholomorphic mapping of $\tan(z)$

I'm supposed to solve this question: Show that the function $\tan$ maps the vertical strip $-\frac{\pi}{4}<x<\frac{\pi}{4}$ biholomorphically to $\dot B(0,1)$ It is obvious that ...
8
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1answer
45 views

Is this contour continuously deformable into a circle?

As an exam question, we had to solve the integral of $\frac{1}{z}$ over the following contour: (The contour is a sequence of straights arcs joining -1, -$\frac{i}{2}$, $\frac{1}{2}$, i, ...
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2answers
39 views

How to compute the residue of $(z^2+2z+1)\sin\left(\frac{1}{1+z}\right)$

This was an example given in my notes but all it concluded was with something about an infinite principal part. How do we compute it? we have it equal to $ \left( z + 1 \right)^2 \cdot \sin \left( ...
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1answer
14 views

The Laurent series expansion of 1/((z^2)(z+1))

In an example in our notes it says: Compute the Laurent series for f(z)= 1/z^2(z+1) and determine the annulus of convergence. No more information was provided. So I did it on my own by factoring it ...
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2answers
44 views

Integral of rational function in the complex plane

Let $P$, and $Q$ be complex polynomials such that $\deg Q \ge \deg(P) + 2$ Prove that there exists $r > 0$ such that if $\gamma$ is a closed curve outside $\{z : |z| \le r\}$, then $$\int ...
8
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1answer
77 views

Prove that an analytic function, real-valued on radii $[0, 1)$ and $[0, e^{i\pi\sqrt 2})$, is constant on the open unit disk

Suppose $f$ is analytic in the open unit disc and real valued on the radii $[0, 1)$ and $[0, e^{i \pi \sqrt 2})$. Prove that $f$ is constant. I'm not sure how to solve this.
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1answer
21 views

Limit a complex contour integral

Let $z_{0}$ be a simple pole of $f$ and $\gamma_{r}$ an arc of circle centered on $z_{0}$, of the radius $r$ and angle $\alpha$, i.e., $\gamma_{r}=z_{0}+re^{it}$, with $t\in ...
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1answer
46 views

find the possible values of z

given two complex number $z,w$ such number that $|z|\le1,|w|\le1$ and $|z+iw|=|z-i\overline{w}|=2$, then find the possible values of $z$ i tryed to use triangular inequality and got that ...
4
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1answer
45 views

How to show $\int_0^1\frac{e^{e^{2\pi it}}}{e^{2\pi it}}dt=1$

I was trying to integrate the contour integral $$\int_\gamma \frac{\vert z \vert e^z}{z^2}$$ where $\gamma$ parametrizes the unit circle counterclockwise. I cannot use the Generalized Cauchy Integral ...
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1answer
52 views

Constructing an entire function

This is a question from my complex analysis final exam: Does there exists an entire function $f$ such that $f(\log k)=1/\log k$ for all $k\geq 2$, integer. My answer is a no. What do you guys think?
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2answers
55 views

How to find the residues of $\frac{1}{(z^4+4)^2}$?

How to find the residues of this function? $$\frac{1}{(z^4+4)^2}$$ So far, I found the poles: $z_1=-1-i$, $z_2 = -1+i$, $z_3=1-i$, $z_4=1+i$. I know they are of the second order. But I have ...
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1answer
21 views

Find the transformation that maps real axis to itself and imaginary axis to the circle $|w-\frac{1}{2}|=\frac{1}{2}$

Find the transformation that maps real axis to itself and imaginary axis to the circle $|w-\frac{1}{2}|=\frac{1}{2}$ What I did: $$z_{1}=0,z_{2}=i,z_{3}=\infty ...
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1answer
17 views

$\left|(1+R^2e^{2i\theta})^2\right| \geqslant (R^2-1)^2$ in complex integration

I need to prove: $$\lim_{R\to +\infty} \left|\int_0^\pi \frac{e^{iaR(\cos\theta+i\sin\theta)}}{(1+R^2e^{2i\theta})^2}iRe^{i\theta} d\theta\right| =0$$ Could someone give me some pointers? A ...
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3answers
69 views

How to use complex analysis to find the integral $\int^\pi_{−\pi} \frac 1 {1+\sin^2(\theta)} d\theta$?

How can I use complex analysis to solve the following: $$\int^\pi_{−\pi} \frac 1 {1+\sin^2(\theta)} d\theta$$
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0answers
35 views

Can two analytic functions that agree on the boundary of a domain, each from a different direction, can be extending into one function?

Let $D=\{z:|z|\leq 1\}$ be the unit disc in $\mathbb{C}$. Say $f$ is analytic on $D$ and $g$ is analytic on $\overline{D^c}$, and that $f|_{\partial D}=g|_{\partial D}$. Is there necessarily an ...
3
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1answer
27 views

Is i holomorphic over the whole complex plane?

That is, is i entire? I know that it's derivative with respect to z bar is 0, so I would think that the answer is yes, although I'm not sure.
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2answers
34 views

Analytic continuation of holomorphic function along clockwise/counterclockwise path

"Write down (say, as a power series) a holomorphic function $f(z)$ on $D(1, 1)$ which satisfies $f(z)^5 = z$ and $f(1) = 1$. What is the result of analytically continuing $f$ along a path which ...
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1answer
20 views

Proving the asymptotic behavior of the prime counting function (Prop 2.1 in Ch.7 Princeton Lectures in Analysis-Complex Analysis)

This is taken from Complex Analysis by Elias M. Stein and Rami Shakarchi. $\psi(x) \text{ is Tchebychev’s ψ-function defined by}$ $$\psi(x)=\sum_{p^m\leq x} \text{log }$$ the sum is taken over the ...
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3answers
184 views

Evaluate $\int_1^\infty \frac {dx}{x^3+1}$

I would like some help with the following integral. I would like to find a contour line to evaluate $$\int_1^\infty \frac {dx}{x^3+1}$$ So one can see that on any circumference it goes to $0$, but ...
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2answers
83 views

Proving that $\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2$

Given $f$ entire show that $$ \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2 $$ I've come close to getting the exact ...
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1answer
28 views

Holomorphic function and nth derivative.

Let $K$ be a open connected subset of complex numbers and $f$ holomorphic on $K$. If $f=0$ on some open disc $D$ in $K$, then is it true that $n$th derivative of $f$ is $0$ for all points in $D$ ...
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1answer
42 views

(Though?)Expression Rearranging

I have the following expression $ 2x+3x^2+e^{5x+x^2}=7 $ which I need rearranged in a form of the type $Ye^Y=Z$ with Y a function of x and Z some constant. I have tried the substitution $y=5x+x^2$, ...
3
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1answer
25 views

an analytic function in $\Delta^n$ is bounded in $T^n$, then it is bounded in $\Delta^n$

Is true that if an analytic function in $\Delta^n$ is bounded in $T^n$, then it is bounded in $\Delta^n$? Here $\Delta^n$: polydisc and $T^n$: Torus, distinguished boundary of $\Delta^n$.
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0answers
27 views

Using residue theorem along a branch cut to evaluate the inverse Laplace transform

I am trying to find the inverse Laplace transform of $f(z)$ using the residue theorem. Can you please check to see if what am doing below is correct? I am not really sure about what I am doing. ...
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0answers
34 views

Evaluating the sum $\sum_{n=1}^{\infty}\frac{1}{n^4 + 1}$? [duplicate]

I'm trying to evaluate the sum $$\sum_{n=1}^{\infty}\frac{1}{n^4 + 1}$$ I figure that this has something to do with the Poisson summation formula, which states that $$\sum_{n \in \mathbb{Z}}f(n) = ...
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28 views

Chain rule (derivative) for for complex data

I found some difficulties in extending the chain rule for complex data. Any suggestion will be appreciated, thanks. In the complex domain, for example, we have a function ...
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1answer
25 views

Complex Analysis analytic function 1$f(z)=z$ [duplicate]

if$\text{ } f:D(0,1)\longrightarrow D(0,1)$ is analytic such that there exists $a,b\in D(0,1)$ and $\text{ }$$f(a)=a$ , $f(b)=b$ prove that $f(z)=z$ $\forall$ $z\in D(0,1)$
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22 views

Which one is correct option? [duplicate]

Let $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$. Which of the following are correct? there exists a holomorphic function $f:\mathbb{D}\rightarrow \mathbb{D}$ with $f(0)=0$ and $f'(0)=2$. there exists ...
2
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1answer
31 views

Usage of Rouche's theorem?

I'm trying to find the number of zeros for the function $f(z) = z + 2 - e^z$ in the half plane $\{\mathscr{R}z < 0\}$. I know I'm supposed to use Rouche's theorem, which states that if both $f$ ...
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1answer
38 views

Meromorphic function with bounded order of zeros and poles

The following problem has been bothering me for a long time; Let $X$ be a compact Riemann surface of genus $g$. Is there a non-zero meromorphic function on $X$ with all of its poles and zeros have ...
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33 views

Evaluate $h(z)=\frac {k}{2\pi} \int_CF(\theta)e^{ikz\cos \theta}\,d\theta$

Suppose this integral $$h(z)=\frac {k}{2\pi} \int_CF(\theta)e^{ikz\cos \theta}\,d\theta$$ $$0\le\theta\le\pi$$ $$|z|\le l$$ We are in complex $\theta$ plane. Assume we have knowledge of $F(\theta)$ ...
2
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1answer
18 views

Complex Integral with constant Function

Show $$\dfrac{1}{2\pi i} \oint_{C}\dfrac{f'(z)}{f(z)-f_{0}}dz=N$$ Where $N$ is the number of points "$z$" where $f(z)=f_{0}$(a constant) inside of $C$; $f'(z)$ and $f(z)$ are analytic inside and on ...
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2answers
87 views

Calculate $I_m = \int_{-\infty}^\infty \frac{dx}{1+x+x^2+\cdots+x^{2m}}$ using complex variables

I have come as far as deducing that the denominator can be written as a geometric series. Hence, for $m=2$, \begin{align*} \int_{-\infty}^\infty \frac{1-x}{1-x^5} dx &= 2 \pi i ( B_1 + B_2 ) - ...
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2answers
35 views

to find radius of convergence of power series.

I have a power series given as: $f(z) =1 + z+ \frac{z^2}{2^2} +\frac{z^3}{3!} + \frac{z^4}{2^4} \frac{z^2}{2^2}+ \frac{z^5}{5!}+ \ldots$ I have to find radius of convergence of above series. My ...
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1answer
28 views

Branch cut and principal value

I do not understand the principal value and it is relation to branch cut. Please tell me about principal value with some examples, then explain the branch cut concept. For instance, what is the ...
2
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1answer
40 views

Proving that a function admits a primitive in a specific set?

I'm trying to show that $$f(z) = \frac{z}{(z^2 - 1)(z^2 - 4)}$$ admits a primitive in the set $\{|z| > 4 \}$ I know that the only singularities of $f(z)$ are poles that occur at the points $z = ...
2
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2answers
47 views

Best way to evaluate integral with contour integration?

I'm trying to evaluate the integral: $$\int_{-\infty}^{\infty}\frac{\sin^2{x}}{x^2}dx$$ with contour integration and am not sure if the basic idea of what I'm doing is correct. I know that $$\sin{x} ...
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0answers
41 views

List of simple, common functions with an incomplete domain or range on $\mathbb{C}$

This may seem like a strange question, but it's an interest of mine and I would appreciate the help of the community in addition to brainstorming on my own. As the question states, I'm looking for ...
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1answer
31 views

Holomorphic function satisfies estimate

Determine whether there exist functions $f$ which are holomorphic in a neighborhood of 0 and satisfy $$n^{-5/2}<|f(1/n)|<2n^{-5/2}$$ for $n\geq 1$. What method should you use?
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2answers
25 views

Find the Order of the Zero of the Function [on hold]

Determine the order of the zero of the function $z=0$ given a) $e^{\sin(z)}-e^{z}$ b) $(\cos(z)-1)^{3}\sin(z)$ Please, can anyone help me, what should I do?
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0answers
26 views

Expressing principal value of integral as real/imaginary

How is it that we can express $$ \mathrm{p.v.}\int_{-\infty} ^{\infty} \frac{\cos 3x}{x^2+4}=\Re \ \mathrm{p.v.}\int_{-\infty} ^{\infty} \frac{e^{3xi}}{x^2+4} $$ while we cannot for $$ ...
1
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1answer
90 views

Help to solve complex equation related to the Gamma function

I would need some help to solve the next complex equation for $y\in\mathbb {R}$, which I already know to be real-valued: $$ \frac {1} {2i}\left ((2\pi)^{\text {iy}}\text {}\text {Sin}\left (\frac ...
1
vote
1answer
52 views

how to solve complex integration problem

While working on complex integration problem I got stuck at the following problem: $\int \frac{|dz|}{|z-2|^2}$ where $|z| = 1$ is the domain. The only idea that I am getting is that we can use the ...
0
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0answers
40 views

Complex Function in the unit disc

If $f$ is a complex valued function which takes the unit disc $U$ to itself and $f(\frac43)=\frac43$ while $f'(\frac23)=\frac43$, how can we find $f$ if it exists?
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0answers
15 views

Inverse transformation of continous transformation is bounded

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with $B \subseteq \mathbb C $ bounded. Now I want to proove that $ A = f^{-1} (B)$ is bounded! How can I proove that this ...
1
vote
0answers
47 views

Prove that $\overline{f(z)}=f(\overline z)$ [duplicate]

Let $f:\Bbb C \to \Bbb C$ be a entire function sutch that $f(\Bbb R) \subseteq \Bbb R$, prove that $$\overline{f(z)}=f(\overline z)$$ In the hint of question said, cosider $g:\Bbb C \to \Bbb C$, ...