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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60 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
28 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
19 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
71 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: $$\int^\pi_{−\pi} \frac 1 {1+\sin^2(\theta)} d\theta$$
2
votes
1answer
45 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.
4
votes
2answers
37 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 ...
1
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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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4answers
213 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
84 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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votes
1answer
31 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 $D$ ...
4
votes
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
votes
1answer
27 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
31 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
36 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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0answers
32 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 ...
0
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1answer
26 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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0answers
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
votes
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
41 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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0answers
36 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
votes
1answer
19 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 ...
4
votes
2answers
90 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 ) - ...
0
votes
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 ...
2
votes
1answer
32 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
votes
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
votes
2answers
48 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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votes
0answers
42 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 ...
1
vote
1answer
32 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
vote
1answer
123 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
53 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 ...
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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
16 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 ...
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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 $g:\Bbb C \to \Bbb C$, ...
4
votes
1answer
48 views

Calculate $\int_{|z|=1}\frac{dz}{\sin z}$

I have to evaluate $\int_{|z|=1}\frac{dz}{\sin z}$. Any tips?
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1answer
39 views

Schwarz Lemma/Conformal mapping problem

Let $F:\mathbb{H}\rightarrow \mathbb{D}$ be holomorphic, where $\mathbb{H}$ is the upper half plane and $\mathbb{D}$ is the unit disc. Show that if $F(i)=0$, then $$|F(z)|\leq ...
2
votes
1answer
48 views

Complex Analysis Integrals

I'm unsure how to apply what I've learned in complex analysis to the following question types: $$ \int_{-\pi}^\pi \frac 1 {1 + \sin^2(\theta)}\,d\theta $$ and $$ \int_{-\pi}^\pi \frac ...
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0answers
35 views

Uniform and absolute convergence of $\frac{1}{n^2+z^2}$

Let $z \in \mathbb{C}.$ I am asked to prove that $\sum\limits_{n=0}^{\infty} \frac{1}{n^2+z^2}$ converges on the set $\mathbb{C} \backslash \{ni : n\in \mathbb{Z}\} $. And also to prove that the ...
3
votes
1answer
38 views

What conditions are necessary on $a,b,c,d$ so that the Mobius transformation $w=\frac{az-b}{cz-d}$ has only one fixed point?

Question: What conditions are necessary on $a,b,c,d$ so that the Mobius transformation $w=\frac{az-b}{cz-d}$ has only one fixed point? Attempt: We examine $$ z=\frac{az-b}{cz-d}$$ to find that ...
3
votes
1answer
44 views

Proving analytic continuation, choosing suitable branch cuts,

Consider the function $$f(z)=\log[(z^2+1)^{1/2}],\quad z>0$$ where the branch is chosen so that $(z^2+1)^{1/2}>0$ for $z>0$ and the log denotes the principal branch. Let $R$ be the union of ...
1
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1answer
36 views

Show that $f(z):=\sum a_n (z-z_0)^n$ is continuous whenever $z$ is in disk of convergence.

Consider a power series $\sum a_n(z-z_0)^n$, and assume it has radius of convergence $r$. Then we know that $\forall z\in(z_0 -r,z_0 +r)$, this power series converges absolutely by root test. Thus we ...
2
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0answers
27 views

Calculate a complex integral using residues

Let $f(z)= \frac{2(e^\frac{1}{z}-1)(\sin^2z)}{z^3}$. Calculate $\int\limits_{\partial B_+(O,1)} f(z)\operatorname{d}z$ Could someone confirm my solution? Solution? I try to calculate the ...
3
votes
3answers
51 views

Let $z_1$, $z_2$ and $z_3$ be complex vertices of an equilateral triangle. Show $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$.

Prompt: Let $z_1$, $z_2$ and $z_3$ represent vertices of an equilateral triangle in the complex plane. Show $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$. Question: I hope the following ...
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votes
0answers
28 views

For which values of $n$ does $x^n+y^n=i$ has a zeros in $\mathbb{R}$? [closed]

$ x $ and $y$ are real numbers and $i$ : is unit imaginary part . 1-for which values of $n$ does $x^n+y^n=i$ has a zeros in $\mathbb{R}$ ? 2-what are the possible geometrics forms of $x^n+y^n=i$ ...
1
vote
1answer
26 views

interpolation properties of analytic paths

Assume we are given $n$ points in $\mathbb{C}^k$ can we find an analytic path $\phi:[0,1]\to \mathbb{C}^k$ passing through these $n$ points?
2
votes
1answer
27 views

Compute $\int_0^\infty \frac{dx}{x^5+1}$ using a contour in the upper half complex plane that encloses one of the roots of $z^5+1=0$

Question: Compute $\int_0^\infty \frac{dx}{x^5+1}$ using a contour in the upper half complex plane that encloses one of the roots of $z^5+1=0$. Hint: The contour should consist of the ...
3
votes
2answers
50 views

Proving that for all complex $z$, $\lim_{x\to0}\frac{1-\cos^{z}x}{x^2}=\frac{z}{2}.$

What do I need to study beforehand in order to prove it (not necessarily in only one way)? I found this sperimentally, at the moment we're beginning derivatives at school. By induction, I succeeded in ...
3
votes
1answer
54 views

The Computation of a special kind of Laurent Series

Let $a\in\mathbb{C}$ and $k\in\mathbb{N}$, we wish to compute the Laurent Series for the function $$ f(z)=\frac{1}{(z-a)^k} $$ about $z=0$ (NOT $z=a$). So there should be two Laurent Series which are ...