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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56 views

Complex Inequalities

Let $\mathbb{H}=\{z\in \mathbb{C} | \ \Im(z)>0\}$ and $f:\mathbb{H} \to \mathbb{H}$ analytic. Prove that for every $z_1, z_2 \in \mathbb{H}$, it must happen that $$ ...
2
votes
1answer
35 views

Help with proof that that affine plane curves in $\mathbb{C}^2$ are not compact

This is a problem from Kirwan's Complex Algebraic Curves that I'm stuck on. She gives a hint suggesting that for $C = \{(x,y)\in\mathbb{C}^2: P(x,y) = 0\}$, show that at all but finitely many points ...
3
votes
1answer
35 views

An effective way of finding the order of the zero $z=0$ of $e^{\sin z}-e^{\tan z}$

An effective way of finding the order of the zero $z=0$ of $f(z)=e^{\sin z}-e^{\tan z}$? What I tried is developing both exponentials by their Taylor series around $z=\sin z$ and $z=\cos z$, getting ...
-3
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49 views

Show smoothness of this map

Let $S^3$ be the sphere identified with the subset $\mathbb{C}^2$ as $\{(x,y) \in \mathbb{C}^2; |x|^2+|y|^2=1\}.$ Then I want to show that the map $\phi: S^3 \rightarrow \hat{\mathbb{C}}$ is ...
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1answer
84 views

Question regarding complex numbers and real numbers?

I have two questions... If we take $(-1/3)^{(-1/3)}$ it would equal $-1.44224957$ since... $$(-1/3)^{-1/3}$$ $$\frac{1}{(-1/3)^{(1/3)}}$$ $$\frac{1}{-0.6933612744}$$ $$-1.44224957\ldots$$ Yet when I ...
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2answers
52 views

Working out a plane defined by $|z-z_1|=|z-z_2|$

Well Complex Analysis is not my good friend but I am working on it, since it is part of what i should learn. I was asked to Describe geometrically the sets of point$z$ in the complex plane defined by ...
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1answer
53 views

How can I solve the integral below using complex variables?

How can I solve the integral below using complex variables? $$ \int\limits_{0}^{2\pi}\sin\frac{\theta}{2}\;\mathrm{d}\theta $$ I know how to solve the integral of $\sin θ$. I replace it by ...
3
votes
1answer
43 views

Does this product over the primes converge, and if so, to what?

I've been trying to play around with the product: $$\prod_{p \text{ prime}}\frac{1}{1-(-p)^{-1.5}}$$ Where the product runs over all the prime numbers. The product is similar in appearance to the ...
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0answers
19 views

Question on a parameter $\alpha_1$

Assume that $\alpha_1>0, \alpha_2>0, t_1 \in R, t_2 \in R$. Does the validity of the equality $\int_0^1(-1)^xx^{-\alpha_1-it_1}dx/\int_0^1(-1)^xx^{-\alpha_2-it_2}dx=0$ implies that ...
2
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0answers
36 views

The roots of a polynomial in one complex variable lie in the upper-half plane [CSIR 2015]

Let $p$ be a polynomial in 1-complex variable.Suppose all zeroes of $p$ are in the upper half plane $H=\{z \in \mathbb{C} :Im(z)> 0 \}$. Then which of the following are true? $Im \frac {p'(z)}{ ...
1
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1answer
35 views

Integral formula involving logarithms and the zeros of a holomorphic function

I have the following formula I´d like to prove: Given a holomorphic function $f:U\to \mathbb C$ such that $\overline{D_r(0)}\subset U$, $f(0)\neq 0$ and $f(z)\neq 0$ for $z\in \partial D_r(0)$, we ...
3
votes
2answers
74 views

Cauchy Riemann equations necessary and sufficient condition?

I was taught that $f(z)$ is differentiable at $z_0=x_0+y_0$ iff Cauchy Riemann equations hold at $(x_0,y_0)$. However, I was shown this example: $f(z)=\frac{\operatorname{Re}(z) \cdot ...
2
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0answers
25 views

Asymptotic values of Meromorphic map

$f(z)=\frac{e^z}{z+1}$ I know that $0$ and $\infty$ are two asymptotic values of the above function. Question:Does there exist another asymptotic values other than $0$ and $\infty$ ?
2
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1answer
33 views

Where is $f(z)=Im(z^2)$ differntiable?

Where is $f(z)=Im(z^2)$ differntiable? So what I tried $f(z)=f(x+iy)=Im((x+iy)^2)=2x^2y^2$ Then by Cauchy Riemann: $U_x=V_y \rightarrow 8y^2x=0$ and $U_y=-V_x \rightarrow 8x^2y=0$. Then, my ...
4
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1answer
42 views

Why are all open subsets not infinite in extent?

I have been looking at the definition of an open set, for a metric space. I have come across the following definition, a few times: An open set $U$ of the metric space $(X,d)$ is a set given that ...
2
votes
2answers
23 views

Property of polynomials proof

Let$$P(z)=\sum_{k=0}^n a_kz^k=a_0+a_1z+...+a_nz^n$$ be an N-th degree polynomial of a complex variable z, where the $a_k$ are complex constants. Now,$$\vert a_0\vert-\vert a_1\vert x-...-\vert ...
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0answers
10 views

Stationary Phase method with Singular test function

Consider the following integral $I(x,t) = \int_{-\infty}^{\infty}\{F(k)exp(it\psi(k)) \}dk$ with $\psi(k) = (k-k_0)(\frac{x}{t}) - (\beta(k)-\beta_0)$ where $\beta_0=\beta(k_0)$ and $F(k)= ...
3
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2answers
33 views

$G$ open connect subset of $\mathbb{C}$ and $f: G \to \Bbb C$ analytic.

$G$ open connect subset of $\mathbb{C}$ and $f: G \to \Bbb C$ analytic. Suppose $\exists a \in G$ such that $|f(a)| \leq |f(z)|$ for all $z \in G$. Prove that either $f(a) = 0$ or $f$ is ...
4
votes
1answer
40 views

Whether the fiber of a holomorphic covering of the unit disk over a non-simply-connected domain is infinite or not

Consider a holomorphic covering $f:\mathbb{D}\rightarrow \Omega$. Then for any point $a$ in the domain $\Omega$, consider the fiber $f^{-1}(a)$. If $f$ is non-constant, I know that when $\Omega$ is a ...
2
votes
0answers
31 views

Moving limit inside a contour integral

I'm trying to compute this integral as part of a larger problem I'm working on. I'm trying to solve the integral $\int_0^\infty \frac{\sin(x)}{x}dx$ and to do it I'm using the method where you ...
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39 views

Applications of Complex Analysis.

I am wondering which fields of science, and which professional positions would make regular or semi-regular use of the techniques of complex analysis.
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54 views

How can something be proved unsolvable? [duplicate]

My question specifically deals with certain real indefinite integrals such as $$\int e^{-x^2} {dx} \ \ \text{and} \ \ \int \sqrt{1+x^3} {dx}$$ Books and articles online have only ever said that these ...
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2answers
43 views

Defining set of interior points of a triangle

Is there a way, given that $z_1,z_2 \ \text{and} \ z_3$ are the vertices of a triangle in the complex plane, to characterize all point that are inside of the triangle?
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18 views

Moving the absolute value inside of an integral involving a complex function

I have the following integral to evaluate $\lvert \int_0^\frac{\pi}{4}e^{iR^2e^{i2\theta}}iRe^{i\theta}d\theta\rvert$ and I want to put the absolute value sign inside of the integral so that I can ...
2
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1answer
42 views

$f$ is entire, $f(0)=f'(0)=0$, and $D= \{|z| \lt R\}$. $\forall z \in D: |f(z)| \leq M$. Prove $\forall z \in D: |f(z)| \leq \frac{M|z|^2}{R^2} $

$f$ is entire, $f(0)=f'(0)=0$, and $D= \{|z| \lt R\}$. $\forall z \in D: |f(z)| \leq M$. Prove $\forall z \in D: |f(z)| \leq \frac{M|z|^2}{R^2} $ What I tried so far: Define $g(z)=\frac ...
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1answer
36 views

Map conformally $D(-2,2)\setminus \overline {D (-1,\frac12)}$ to the annulus $\{1 < |z| < 2\}$

This is coming from this question: http://math.stackexchange.com/questions/1332123/moving-around-a-circle-inside-a-different-circle-conformally I will delete that question. This will be the same ...
2
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0answers
27 views

constructing Riemann surface of $\sqrt{z-1/z}$

I am trying to construct the Riemann surface of the function $\sqrt{z-1/z}$. I rewrite the function as $\sqrt{\frac{z^2 - 1}{z}}$, from which I can see that the function has branch points at 0,1,-1 ...
2
votes
2answers
74 views

Help with the integral $\int_{0}^{\infty}\frac{y^{2}e^{y}}{e^{sy}+e^{-sy}-2}dy$

I want to do the integral : $$I(s)=\int_{0}^{\infty}\frac{y^{2}e^{y}}{e^{sy}+e^{-sy}-2} \, \mathrm{d}y$$ $s$ being a complex parameter. I tried expanding the dominator of the integrand, but this way ...
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votes
2answers
47 views

Alternative definition of Hardy spaces

Classically, Hardy spaces $H^p$on the disk are introduced as set of functions analytic on $\mathbb{D} = \{z \in \mathbb{C}: |z|<1\}$, which has bounded $H^p$ norm: $$ \|f\|_{H^p} = \sup_{0\leq r ...
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0answers
21 views

Proving Kramers-Kronig relationships from Fourier sine / cosine transform

Given a real function $A(t)$, the Fourier-sine and –cosine transforms are defined as $$ A’(\omega) = \omega \int_0^{\infty}A(\tau) sin(\omega \tau) \mathrm{d}\tau $$ $$ A’’(\omega) = \omega ...
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1answer
43 views

Showing that a Fourier transform is holomorphic

Let $f\in \mathcal{S}(\mathbb{R})$ be a Schwartz function. I would like to show that the Fourier function $$F(z)=\int_{\mathbb{R}}f(t)e^{itz}\, dt$$ is an entire function. Here is my approach: ...
4
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0answers
33 views

Asymptotic behavior of many derivatives

To compute the residue of a pole of very high order $M$ at $z=0$, one needs to compute $\frac{d^M}{dz^M} g(z)$ Suppose that $g(z)$ is a reasonable but not trivial function, that itself may depend on ...
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2answers
55 views

Tough Polynomial Root Problem

Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either ...
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1answer
26 views

Proof of Schwarz Lemma

I'm analysing proof of Schwarz lemma that is presented here on page 4: http://www.dm.unipi.it/~abate/libri/libriric/files/IterationThTautMan1-1.pdf I don't understand, why author claims that ...
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2answers
23 views

Injectivity of the extension of a injective function

I have two questions about injectivity of holomorphic functions: I know that if $f$ is a holomorphic function on a disk $U$ that is injective, then $f'(z)\not = 0$ for all $z\in U$. Does this result ...
2
votes
1answer
114 views

Is it feasible for a sophomore in high school (15 years old) to learn complex analysis? [closed]

I've been reading up on complex analysis and it seems an incredibly fascinating subject to me and one I'd like to learn more about. However, most of the books I've come across are for graduates, which ...
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2answers
58 views

Value of $\int_{|z+1|=2} \frac{z^2}{4-z^2}dz$

$\int_{|z+1|=2} \frac{z^2}{4-z^2}dz=-2\pi i$. Am I correct, I used cauchy integral formula
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Proof of Sufficiency of Cauchy-Riemann equations

I understand that the Cauchy-Riemann equations $$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$$ and $$\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$ are necessary for a ...
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1answer
18 views

Finding the residue, $z=n\pi$, and $e^{n\pi}$

I have reached the following point in a residue calculation and am now unsure what to do: $$Res_{z= n\pi}=\lim_{z\to n\pi}\{(z-n\pi)\frac{ e^z}{\sin(z) } \}$$ $$=\lim_{z\to ...
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52 views

Integration of certain real functions using Euler's Formula.

I've heard about using Euler's formula $$e^{ix}=\cos(x)+i\sin(x)$$ to transform rational functions of sine and cosine into computable indefinite integrals. However, upon attempting to apply this ...
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1answer
46 views

Fractional part expansion of $\frac{1}{(e^{x}-1)^{2}}$

What's the fractional part expansion of $$\frac{1}{(e^{x}-1)^{2}}$$ I know that: $$\frac{1}{e^{x}-1}=\sum_{n=-\infty}^{\infty}\frac{1}{x+2\pi i n}-\frac{1}{2}$$
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1answer
106 views

Cosh and Sinh analogs

We know that $$\cosh{x}+\sinh{x}=e^x$$ and that his can be expressed as $$\frac{e^x+e^{-x}}{2}+\frac{e^x-e^{-x}}{2}=\frac{(e^x+e^x)+(e^{-x}-e^{-x})}{2}=e^x$$ and this works out nicely because the ...
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3answers
81 views

Show that $\int^\infty_0 \frac{x^a}{1+x^3}\frac{dx}{x} = \frac{\pi}{3 \sin(\pi a/3)}$ for $0<a<3$ (Mellin transform)

I am stuck on proving that this is true, I am really close but there seems to be an error in my calculation and I am unable to find where I am making a mistake... Relevant equation $$\int^\infty_0 ...
2
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1answer
49 views

Estimating $\log|\zeta(\sigma+it)|$ for $\sigma$ sufficiently large

In a paper I am reading, I've come across the estimate $$ 2\pi\sum_{\substack{\sigma<\beta<\sigma_0\\ T<\gamma\leq ...
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vote
2answers
31 views

Show that $f(z)=x^3+ i(1-y)^3$ is differentiable only at $z=i$.

Here's the exact phrasing of the question: Show that when $f(z)=x^3+i(1-y)^3$, where $z=x+iy$. it is legitimate to write: $$f'(z)=u_x+iv_x=3x^2$$ only when $z=i$ Here's my best attempt We have ...
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0answers
20 views

Continuity of complex functions with sequences

In real functions, a function $f$ is continuous in $c$, if for every sequence $x_n \rightarrow c$, $f(x_n) \rightarrow f(c)$. Does this condition hold as well for complex variable function? Thank ...
2
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3answers
55 views

Complex integration with trigonometric and logarithm

Show that $\int_0^{2\pi}\log\sin^22\theta dx=4\int_0^\pi\log\sin \theta d\theta=-4\pi \log2$ I did $$\int_0^{2\pi}\log\sin^22\theta d\theta=4\int_0^{\frac{\pi}{4}}\log\sin^22\theta d\theta$$ ...
3
votes
1answer
82 views

exponential type of entire function?

I'm looking at an entire function of the form $$ f(\lambda):=p(\lambda)e^{-\lambda}+q(\lambda)\;, $$ where $p$ and $q$ are polynomials and $\lambda\in\mathbb{C}$. I need to establish that $f$ is an ...
0
votes
1answer
9 views

Degree theorem for Runge's approximating rational functions

Suppose that $f$ is analytic on an open set $D\subset\mathbb{C}$, and one uses Runge's theorem to obtain a sequence of rational functions $\{r_n\}$ which approach $f$ uniformly on compact subsets of ...
2
votes
1answer
51 views

Complex sum using Laurent series?

By considering $f(z)=exp(z-\frac{1}{z})$ show that $$ \frac{1}{2\pi}\int_{0}^{2\pi}cos(n\theta-2sin\theta)d\theta=\sum_0^{\infty}\frac{(-1)^k}{k!(n+k)!}\ \forall n\ge1$$ f is holomorphic in ...