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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how to prove that holomorphic function mapping complex onto complex is linear?

How to prove that any one to one holomorphic function mapping complex plane onto itself is linear?
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16 views

Help with simplification of an expression

I was solving the residues of $f(z)e^{zt} = e^{zt}\frac{\ln(z)}{z^2+1}$ as follows: $$\operatorname{Res}(f(z)e^{zt}, i) = \lim_{z\to i} (z-i)\frac{e^{zt}\ln(z)}{(z-i)(z+i)} = ...
5
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46 views

Function's anaytic continuation is its own derivative

This is the question we were asked at the university by our professor for complex analysis. Not as an exam, but as a challenge. I don't think he knew the answer himself. Find a nontrivial example of ...
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2answers
38 views

Taylor expansion of the Error function

The error function $\operatorname{erf}(z)$ is defined by the integral $$ \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2}\,dt,\quad t\in\mathbb R$$ Find the Taylor expansion of ...
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22 views

Need help with holomorphic functions on a domain interval removed.

I want to prove that for a region $\Omega$ with interval $I=[a,b]\subset\Omega$, if $f$ is continuous in $\Omega$ and $f\in H(\Omega-I)$, then actually $f\in H(\Omega)$. Is this problem related to ...
6
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1answer
92 views

Show that $\{e^{in}: n\in\Bbb N\}$ is Dense in the Unit Circle

This problem gave me fits when I was in grad school. Looking back at it now, it still escapes me. The problem is from Conway's Functions of One Complex Variable. I'm looking for a proof from basic ...
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1answer
17 views

Fourier transform, quadratic function

I'm trying to compute this convolution: $\frac{2 \alpha}{\alpha ^2 + 4 \pi ^2 x^2} * \frac{2 \beta}{\beta ^2 + 4 \pi ^2 x^2}$ I know that the Fourier transform of a convolution of two functions is ...
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2answers
23 views

In dual numbers, what number is represented by the following matrix?

In dual numbers, what number is represented by the following matrix? \begin{pmatrix}0 & 0 \\1 & 0 \end{pmatrix}
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1answer
28 views

Two different results with contour integration

This is probably going to be a stupid question ( I don't feel great today) but I can't get around this problem. $$I = \int_\mathbb R \frac 1 {(3x-2i)^2} dx $$ I thought that using contour ...
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1answer
30 views

$|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$

Let $p(z)= \sum_{k=0}^n a_k z^k$ , $a_n \neq 0$ , be a polynomial of degree $n$ such that $|p(z)| \leq M$ for $|z| \leq 1$. Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$ This was an exam ...
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1answer
42 views

$ \int_\gamma \frac{1}{z\sin z}dz$ where $\gamma$ is the circle $|z| = 5$

My understanding is that if this integral exists in the real sense, i.e. real Riemann-wise, then I can apply the residue theorem. If not, I may use the Cauchy Principal Value, to obtain a value. To ...
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18 views

an example of functions which is essentialy bounded but not continuous in circle

Can you give me an example of a function which is essentially bounded but not continuous in the unit circle and bounded in the open unit ball?
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1answer
30 views

How can I prove that the zeroes of $f(z)=1+1/2^z$ have no real part?

I want to prove that the zeroes of the function $f(z)=1+1/2^{z}$ have no real part. Is the following correct? $f(z)=0$ so $2^{z} = -1$ and $-1=e^{i\pi}$ so $e^{i\pi} = e^{z\ln2}$ therefore $z= ...
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0answers
12 views

Bounded functions composed with Mobius maps

Hopefully easy question here: What is the most succinct method/technique to prove the following statement?: Let $u \in L^{\infty}(\Bbb D)$. Show that $||u(\varphi_{z})||_{\infty}$ is independent of ...
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0answers
27 views

What does complex square root as defined on wikipedia look like: two questions

If you look at the third picture here, the surface representing the complex square root intersects the negative real axis at $0$. Later in the article the definition of the complex square root is ...
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3answers
27 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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24 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 ...
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1answer
69 views

An entire and one-to-one function must be of the form AZ+B, A non-zero. How to rule out higher degree polynomials in z? [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. I am editing my question, since there are duplicates on this forum to the question of why an ...
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0answers
28 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
17 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 ...
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1answer
50 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
45 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
15 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
45 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 ...
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1answer
84 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
22 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
54 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
57 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
23 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
18 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
70 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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1answer
40 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
35 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 ...
14
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3answers
192 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
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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
29 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
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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
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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30 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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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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29 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
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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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 $f$ ...
1
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1answer
40 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 ...
0
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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 $F(\theta)$ ...