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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40 views

Complex series should sum to zero but it's a puzzle

If we have a finite sum defined as $$\frac{1}{N}\sum\limits_{n=N/4}^{3N/4-1} e^{-4\pi ink/N}$$ (where $k$ is an integer and $N$ is divisible by $4$), then how can we show that this sum is equal to $...
2
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0answers
46 views

Let $f:\mathbb{C} \to \mathbb{C}$ be an non constant entire function such that $f(1-z)+f(z)=1$ for all $z\in \mathbb{C}$.

Let $f:\mathbb{C} \to \mathbb{C}$ be a non constant entire function such that $f(1-z)+f(z)=1$ for all $z\in \mathbb{C}$. Then prove that $f$ is surjective. It can be solved trivially by Picard's ...
2
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0answers
36 views

Let $f(z)$ be an entire function. Prove that $f$ and $f-a$ have the same order.

Let $f$ be an entire function, the order of $f$ is defined by $$\lambda=\limsup_{r \to \infty} \frac{\log \log M(r)}{\log r},$$ where $M_{f}(r)=\max_{|z|=r} |f(z)|$. And this is equivalent to define $...
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0answers
33 views

Polynomial with roots on ellipse in complex plane [on hold]

What polynomial or complex equation produces 4 equi-spaced complex numbers Z with one real root $(1,0)$ ( for $\theta =0$) on an ellipse in the complex plane, where $$ Z =\frac{e^{i\theta}}{1-\...
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1answer
26 views

Construct a function which satisfies the following conditions

I am struggling with the following routine problem : Construct a holomorphic function f(z) with pole of order 2 at 0, an essential singularity at 1 and with residues 1 and 0 respectively. In general ...
3
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1answer
45 views

Is this a contour integral question?

I had this in my previous cats that I'm not sure whether it's really a complex analysis question, looks like a differential question with line integrals a bit $$\int_{(1,3)}^{(4,5)} (2y+x^2)\,dx + (...
3
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0answers
33 views

Comparison of entire functions with $f=cg$

Let $f,g:\mathbb C\to\mathbb C$ be holomorphic with $|f(z)|\leq|g(z)|$ for all $z\in\mathbb C$. Show that there is a $c\in\mathbb C$ with $|c|\leq 1$ such that $f=cg$. If $g\equiv 0$ then we ...
2
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0answers
56 views

Convergence of a series of complex numbers.

Let $f : \mathbb C \to \mathbb C$ be a non constant entire function. Does the series $\sum_{n=1}^ \infty \frac{1}{n} f(\frac {z}{n})$ converges at any point $z \in \mathbb C$ ? I think this will not ...
2
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1answer
42 views

How to derive this Hankel's Contour integral formula with gamma function?

This relation was put up in The Art Of Computer Programming and no derivation was offered. Please help me understand this better. $$\frac{1}{\Gamma (z)} = \frac{1}{2i\pi} \oint\frac{e^t dt}{t^z}$$ ...
1
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0answers
36 views

topology of compact convergence, closed sets

Let $H(\mathbb{D})$ be the vector space of all analytic functions on the unit disk. Then the topology induced by uniform convergence on compact subsets is metrizable. Thus the following topology ...
0
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0answers
15 views

About some good references for self study

I'm willing to start a self study of Hardy spaces, Bergman spaces and Bloch spaces. I would like to know good books on the subject. Since I'm going to study on my own, would be great to find one that ...
2
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0answers
52 views

Is this what universal covering spaces are used for?

From the perspective of real analysis, we have: $$\int_{-1}^{1}\frac{1}{1+x^2} = \mathrm{tan}^{-1}(1)-\mathrm{tan}^{-1}(-1) = 2\mathrm{tan}^{-1}(1) = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}$$ Something ...
3
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1answer
45 views

Show that $F(z) \in H(D_1(0))$

Let $\mathbb{S}^{1} = \{ z \in \mathbb{C} : |z| = 1 \}$ and $f: \mathbb{S}^1 \longrightarrow \mathbb{C}$, $f \in \mathcal{C}^{0}(\mathbb{S}^{1})$, i.e., $f$ is continuous. Define for $z \in D_1(0) = \{...
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2answers
89 views
+100

$\log(e^z - i)$ as a holomorphic function in $\mathbb{D}$

I'm learning complex analysis, specifically holomorphic functions, and need help with the following exercise: Examine if the function $\log(e^z - i)$ can be defined as a holomorphic function in ...
3
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1answer
38 views

Conformal holomorphic mapping from disc to square

Let $f$ be a holomorphic map from the unit disc $\mathbb{D}$ to an open square $\mathbb{S}$ with its center at the origin. Given $$ f(0) = 0, \qquad f'(z) \neq 0 \quad (z \in \mathbb{D}) $$ prove that ...
1
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1answer
29 views

Proof of Reflection Principle when f(x) is imaginary

Suppose that a function f is analytic in some domain $D$ which contains a segment of the x-axis and whose lower half is the reflection of the upper half with respect to that axis then $$\overline{f(z)...
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2answers
42 views

Argument of complex numbers

If $z=re^{i\theta}$ and $w=\rho e^{i \phi} $ are two complex numbers, then $ arg(zw)=arg (z)+arg (w)$ But if $z=-1$ and $w=-1$, we get $ 0= 2\pi $ which is not correct. So why it gives us this ...
0
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1answer
35 views

Proof of non-constant analytic functions

Use the following theorem: "A function that is analytic in a domain $D$ is uniquely determined over $D$ by its values in a domain, or along a line segment, contained in D" to show that if $f(z)$ ...
1
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1answer
42 views

Lebesgue integral of $\frac{1}{\|\boldsymbol{x}-\boldsymbol{r}\|^2}$ on an infinite cylinder

Let $V\subset \mathbb{R}^3$ be a solid infinite cylinder, or cylindrical shell, and let $\boldsymbol{r}\in\mathbb{R}^3$ be any point of the space. I intuitively suppose that the Lebesgue integral $$\...
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2answers
24 views

Automorphism of Upper half plane

Let $M=\left\{\left.\displaystyle z\mapsto\frac{az+b}{cz+d}\ \right|\ \ ad-bc\not =0\right\}$,$$p:GL(2,\mathbb C)\to M, \begin{bmatrix}a & b \\ c & d \end{bmatrix}\mapsto\frac{az+b}{cz+d}.$$ ...
1
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1answer
32 views

Is a Blaschke product/rational function a covering map for a $n$-sheeted covering of $S^{1}$?

We have a Blaschke product $B(z)$ of order $n$ (you can think of it as a rational function with $n$ zeros and $n$ poles), the zeros are obviously inside $\mathbb{D}$. Why is $B(z) \colon S^{1} \to S^{...
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1answer
35 views

Help, why these are two different results of integral of $\sqrt{z}$ on unit circle depending the choice of Branch cut

everyone, I want test the effect of different choice of branch cut for contour, So I find a simple function, i.e. $\sqrt{z}$ with $z=re^{i\theta}$ on 1st Branch as $$I=\oint_{UnitCircle}{\sqrt{z}dz}$$ ...
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1answer
24 views

Finding the Laurent series of a function

I'm trying to work through the following example: Find the Laurent series of: $$ f(z) = \frac{1}{z(z-2)^3}, $$ about the singularities $z = 0$ and $z = 2$ (separately). Hence ...
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2answers
53 views

Analytic continuation of Euler's reflection formula with the Gamma function

Let $\widetilde\Gamma$ be an analytic continuation of $\Gamma$ on $\mathbb C\setminus(-\mathbb N_0)$. Show that the function $$\widetilde\Gamma(z)\widetilde\Gamma(1-z)-\frac{\pi}{\sin(\pi z)}$$ ...
1
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1answer
34 views

Show the limit exists.

For $|z|\neq 1$,show that the following limit exists: $$f(z)=\lim_{n\to\infty}\frac{(z^n -1)}{(z^n+1)}$$ Is it possible to define f(z) when $|z|\neq 1$ in such a way as to make $f$ continuous? ...
0
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1answer
21 views

Contour Integral involving hyperbolic functions

I would like to evaluate: $\displaystyle\oint_C \frac{e^{4z}-1}{\cosh(z)-2\sinh(z)}\,\mathrm dz$ where $C$ is a unit circle in the complex plane and $z=x+iy$. I did not find any singular ...
2
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1answer
33 views

Does this function Contradict this colloary?

Colloary: If $G$ is a domain and $f: G \to \mathbb{C}$ is analytic and not identically zero, then the zeros of $f$ are isolated. If the domain $G$ is closed and bounded, then the zeros are finite in ...
2
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3answers
121 views

Show the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle

So I need a little help with the following: Considering separately the cases of real and complex roots show that the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle (i.e....
0
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3answers
44 views

How to determine Laurent series associated to $f(z)$ [closed]

The function is $$f(z)= \frac{1}{(e^z -1)},$$ $z$ belong to $\mathbb{C}$ and $0<|z|<1$. I need a general expression in term of a sum from 0 to infinity
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0answers
17 views

Prove that these are the automorphisms of the Poincare upper half plane

If $P = \{Im(z) > 0\} \subset \mathbb{C}$, prove that $f:P \rightarrow P$ is an automorphism iff $f(z) = \frac{az+b}{cz+d}$, for $a,b,c,d \in \mathbb{R}$, with $ad-bc > 0$. I had thought about ...
0
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1answer
31 views

Does $f(z)$ has an essential singularity at $z=z_0$?

Let $f(z)=g(z)h(z)$ such that $g$ has an essential singularity in $z=z_0$ and $h$ is holomorphic in a neighbourhood of $z_0$ then $f$ has an essential singularity in $z=z_0$? Im trying to see this ...
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0answers
53 views

Elementary fact about holomorphic function

I found some assertion in this article (bottom of third page) of François Trèves which in a way states that taking a bounded simply connected domain $D$, $H$ the open upper half plane, a holomorphic ...
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0answers
38 views

contour integral z/conjugate(z)

I am trying to calculate: $$\int_C \frac{z}{\bar{z}}dz$$ where C is the upper semicircles of the circles centred in (0,0) of radii 1,2 joined at their intersections with the real axis by the real ...
6
votes
1answer
106 views

Evaluate $\int \frac {\sin(x)}{x^2 + 4x + 5}dx$

Question: Evaluate $$ \int \frac{\sin(x)}{x^2 + 4x + 5} dx=\int \frac {\sin(x)}{(x + 2)^2 + 1}dx $$ By using the change of variable $y = x + 2$ we have that $dy = dx$ then $$I = \int \frac{\...
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1answer
47 views

Residue of a trig function multiplied by a polynomial

can somebody help me to find the residue for: I tried to make two series centered at $(z - k\pi)$ for $\sin(z)$ and $1- \cos(2z)$ but I don't know what to do with the $(z+\pi)^2$....and obviously, i ...
1
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1answer
35 views

Write $\,-4i\,$ in polar form

Write $\,-4i\,$ in polar form ${re}^{i\theta}$, with $r$, $\theta\in \mathbb R$, and $\,r\geq0,\;0\leq\theta<2\pi$. I let $\,z=-4i\,$ first, then get $\,r=\sqrt{0+{4^2}}=4$. However, $\,\tan\theta\...
0
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1answer
20 views

square of polynomial still harmonic? [closed]

Let $P(z)=\sum_{i=0}^n a_i z^i$ be a polynomials on $\mathbb{C}[z]$ such that $a_i$ are real numbers. $|P(z)|^2$ is a harmonic function ?
2
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4answers
172 views

“Exponential Madness” (Gauss's challenge)

From Euler's identity, we see that $e^{i\pi}=-1$ $\Rightarrow e^{2ik\pi}=1$ [squaring both sides]. This equation surely holds for all integers $k$. EDIT: From the second equation we get $e^{1+...
1
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1answer
41 views

It's true that $ |\log^2(z)| \leqslant |\log(R)|^2 + |i \arg(z)|^2 $ where $z \in \mathbb{C}$

In some residue integral, when one have to prove that an integral vanish at infinity, I've found in some textbooks the inequality: $$ |\log^2(z)| \leqslant |\log(R)|^2 + |i\ \arg(z)|^2 $$ Where $z= ...
2
votes
3answers
104 views

Help, Where is wrong when I do same complex integration using two different contours

everyone! please give few hit. I want take the integral $$I=\int_{0}^{\infty}{\frac {dx}{ \sqrt{x}(1+{x}^{2})}} $$ by using the Residue Theorem. I choice two contours in complex plane with $z=r e^{i\...
0
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0answers
32 views

Understanding a calculation deduced for the function $\pi^{-s/2}\Gamma(s/2)\zeta(s)$

With my current knowledges I don't know if this is a bad question, but since I am interesting in this kind of calculations I want to ask you, if I was wrong or if if my statement is obvious. From ...
0
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1answer
25 views

Maximal value of real part of holomorphic function

Let $f:U \rightarrow C$ be a non-constant holomorphic function. $U$ is open, connected and $D(0,1+\epsilon) \subset U$. I'd like to show that there exists $z_0 \in \partial D(0,1)$ such that $Re(f(z))...
2
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1answer
38 views

All power series has a point that is not regular.

Definition: Let $f = \sum_{n \geq 0} a_n z^n $ a power series and $0<R< \infty$ its convergence ratio. We say that $z_0 \in \mathbb C, |z_0| = R$ is a regular point if $\exists r > 0$ such ...
3
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1answer
58 views

Why Differential Forms on Riemann surfaces?

I am working with Rick Miranda's "Algebraic Curves and Riemann Surfaces". Right now I am in chapter four "Integration on Riemann Surfaces" and struggle with it a lot!:( It starts with the definition ...
2
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0answers
37 views

How to apply the Identity Theorem to this function?

Given the function $f(z)=\exp\left(z^2-\cos\left(iz\right)-4\right)$ with the domain $|z|<10$, if we try to apply the Cauchy integral formula, we'll see that f(2) "will be" $$\frac{1}{2\pi i}\int_\...
0
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1answer
54 views

How does one compute this heavy integral?

The integral is $$\frac{1}{2\pi i}\int_\Gamma\frac{\exp(z^2-\cos(iz)-4)}{z-2}dz$$ where $\Gamma$ is the unit circle. Here's how I tried to parametrize it: $z=e^{i\theta}$ on $\theta\in [0, 2\pi]$, ...
0
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0answers
25 views

Existence of analytic function from A to B . [duplicate]

Does there exists a non constant analytic map $f:A\to B$ . Where $A=\{z\in \mathbb C~:~ |z|\neq 0\}$ and $B=\{z \in \mathbb C ~:~ |z|>1\}$. I am unable to construct one
0
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1answer
28 views

Minimum modulus principle - looks like a counterexample?

The minimum modulus principle states that if $f$ is holomorphic within a bounded domain D, continuous up to the boundary of D, and non-zero at all points, then $|f (z)|$ takes its minimum value on the ...
0
votes
0answers
27 views

Minimum norm of analytic function may not be achieved on the boundary of its domain

I need to show that the minimum modulus of an analytic function may not be achieved on the boundary of its domain. I'm stuck with this question, would appreciate if someone could help me with it. I ...
0
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0answers
17 views

Construction of continuous/analytic maps

Let $A=\{z\in C ~:~ |z|>1\}$ , $B=\{z\in C ~:~z\neq 0\}$. Which of the following is true? 1.There exists a continuous onto map $ f:A\to B. $ 2.There exists a continuous one to one map $ f:B\to ...