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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27 views

Show limit does not exist

Show that this limit does not exist, z is Complex number: $\lim_{z \to -1 } \frac{1}{z^3}\sin (\frac{z}{z+1})$
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27 views

Is there a special name for polynomials related by Möbius tranformation of the variable?

If we take a general polynomial with complex coefficients: $$C_n z^n+C_{n-1}z^{n-1}+\dots+C_1z+C_0$$ We can apply a general Möbius tranformation to the variable: $$z=\frac{aw+b}{cw+d},~~~~a,b,c,d \...
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8 views

Conservation of charge in the Schrödinger equation

Let us have the Schrödinger equation $iu_t+\Delta u+|u|u=0$, where $u$ is a function, which decays rapidly at infinity. I would like to derive the conservation of charge property, i.e. $||u||_{L^2}=...
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0answers
14 views

Application of the Casorati-Weiertrass theorem

I'm learning complex analysis, specifically series applications, and need help with the following problem: Let $f(z)$ be an entire function which is not a polynomial. Show that for every $c \in \...
0
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1answer
52 views

Weak version of Cauchy's theorem in complex analysis

Let $f(z)$ be a complex valued function defined on an open subset $U$ of the complex plane. We say $f(z)$ is analytic if it can be expressed by a convergent power series on a neighborhood of every ...
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41 views

To show an entire function is zero

$f$ is entire, satisfying $$|f(z)|\le \frac1{|\text{Re}(z)|}.$$ Show that $f$ is identically zero. It suffices to show boundedness. Since $f$ is bounded on the real line, then if $f$ is ...
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16 views

How to solve complex-valued , inhomogeneous second order differential equations?

I am trying to find a general method to solve the following complex-valued , inhomogeneous second order differential equation $$ -a(x) u''(x)+b(x)u'(x)+c(x)u(x)+d(x)\bar{u}(x)=f(x),\quad x\in\mathbb{...
3
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3answers
60 views

$\int_0^\infty \frac{ x^{1/3}}{(x+a)(x+b)} dx $

$$\int_0^\infty \frac{ x^{1/3}}{(x+a)(x+b)} dx$$ where $a>b>0$ What shall I do? I have diffucty when I meet multi value function.
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1answer
27 views

If $g(z) = z^kf(\frac{1}{z})$ is bounded in some annular region, then $f(z)$ is a polynomial

I'm learning complex analysis, specifically Laurent series, and need help with the following exercise: If $f(z)$ is entire and for some $k \in \mathbb{N}$ the function $g(z) = z^kf(\frac{1}{z})$ ...
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2answers
65 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 ...
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0answers
35 views

Showing that $\prod_{k=1}^\infty \cos\left(\frac{2\pi t}{3^k}\right)$ does not vanish at infinity.

As the title says, I need to show that $f(t)=\prod_{k=1}^\infty \cos\left(\frac{2\pi t}{3^k}\right)$ does not vanish at infinity, i.e. $f\not \in C_0(\mathbb{R})$. I tried looking at the series $$ \...
2
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1answer
30 views

How to rigorously deduce the Laurent series of $\log\frac{z-p}{z-q}$?

Of course, the logarithm here is defined on the ring region $|z|>R\ge\max\{|p|,|q|\}$ as $$\log\frac{z-p}{z-q}=\int_{z_0}^z \left(\frac1{w-p}-\frac1{w-q}\right)\mathrm d w. $$ Here the integral is ...
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1answer
22 views

Maximum modulus theorem proof

I do not understand the proof for the maximum modulus theorem done with the open mapping theorem. Unfortunately my notes are a little bit cryptic. What I understand: Let $z_0$ be the maximum that is ...
0
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3answers
40 views

winding number example

Let $$\gamma:[a,b]\rightarrow \mathbb C $$ be a closed curve and$$ Int(y)=\{z\in\mathbb C-tra(\gamma): ind(z)\neq0\}, \\ Ext(y)=\{z\in\mathbb C-tra(\gamma): ind(z)=0\}, $$ where $ind(z)\;$is the ...
1
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0answers
37 views

Theta series and Jacobi theta functions

I have some difficulties with expressing the following series $\sum\limits_{b=-\infty}^{+\infty}\sum\limits_{a=-\infty}^{+\infty}q^{1 + 3 a + 3 a^2 - 3 b - 3 a b + 3 b^2}$ using standart theta ...
0
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1answer
20 views

Isolated singularity of $\frac{\sin(\pi z)}{z - 1}$

I'm learning complex analysis, specifically Laurent series and isolated singularities, and need help to understand the solution to the following exercise: Find and determine the nature of the ...
1
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1answer
483 views

Solution to equations involving Plasma dispersion function

I am trying to solve an equation involving a complex argument for the plasma dispersion function as: $z = x + \iota y$, $ x = \omega / \sqrt2 k v_{Ti} $ $ y = \nu_i /\sqrt{2} k v_{Ti} $ $S[z] = -\...
4
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1answer
59 views

Is it true that $ \sqrt{z^2-1} = i \sqrt{1-z^2} $?

I have seen a lot of times in books or on the internet that$ \sqrt{z^2-1} = i \sqrt{1-z^2} $ and I don't understand why that is correct . In general it is not true that $ \sqrt{-z}=i \sqrt {z}$ and ...
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2answers
41 views

Bound on imaginary parts of family of analytic functions

Question: Let $ F$ be the set of holomorphic maps $f$ from the unit disc into the upper half plane, such that $f(0)=i$. Show that the supremum of the imaginary parts, $\sup_{f\in F}$ Im[$f(\frac i 2)$]...
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1answer
45 views

Calculating $\int_0^{\infty } \frac{\ln (x)}{\sqrt{x} \left(a^2+x^2\right)^2} \, \mathrm{d}x$ using contour integration

I can do this integral using the keyhole contour the answer is:$$\int_0^{\infty } \frac{\ln (x)}{\sqrt{x} \left(a^2+x^2\right)^2} \, \mathrm{d}x = -\frac{\pi (-6 \ln (a)+3 \pi +4)}{8 \sqrt{2}a^{7/2}}$...
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2answers
38 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
45 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
34 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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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 ...
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0answers
32 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-\...
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 + (...
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0answers
31 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 ...
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0answers
189 views

Bounds on derivative of real positive coefficient polynomial satisfying certain properties

While thinking about this question of Clin, I wanted to consider the polynomial: $P(z) = 1+x_1z+x_2z^2+\cdots+x_nz^n$, satisfying: (I) $1\geq x_{1}\geq x_2\geq\cdots\geq x_{n}\geq0$ and $\...
2
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1answer
40 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. $$\frac{1}{\Gamma (z)} = \frac{1}{2i\pi} \oint\frac{e^t dt}{t^z}$$ ...
2
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0answers
49 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 ...
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0answers
35 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 ...
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14 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 ...
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2answers
41 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 ...
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0answers
49 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 ...
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2answers
52 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 $$\widetilde\Gamma(z)\widetilde\Gamma(1-z)-\frac{\pi}{\sin(\pi z)}$$ ...
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3answers
112 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....
3
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1answer
42 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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1answer
33 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 $f(z)$ ...
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1answer
24 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
37 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 ...
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1answer
35 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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1answer
34 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
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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2answers
23 views

Automorphism of Upper half plane

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