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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133 views
+50

For what values is my integral diverging or converging?

Is the following integral convergent $$\int_{\gamma}^{+\infty} \left(1-\dfrac{1}{1+sv^{-1}}\right)\left(\frac{1}{\alpha_1}v^{\frac{2}{\alpha_1}-1} \, e^{-\beta\, v^{\frac{1}{\alpha_1}} }+ ...
3
votes
0answers
52 views

Infinite sum of analytic function still analytic

Consider $$ f_n(x) = n e^{-n^6(x-n)^2} : \mathbb R \rightarrow \mathbb R$$ and the series $$ f(x) = \sum_{n=1}^{\infty} f_n(x). $$ Is $f$ analytic on $\mathbb R$? A function is analytic if for ...
-2
votes
3answers
39 views

Imaginary part of $ln(\sqrt{i})?$ [on hold]

Which of the following is the imaginary part of a possible value of $\ln(\sqrt{i})?$ (a) $\pi$ (b) $\pi/2$ (c) $\pi/4$ (d) $\pi/8$ I compute $\sqrt{i}=\dfrac{1+i}{\sqrt{2}}$, but how to proceed ...
1
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0answers
28 views

Continuous function $f:\mathbb{R} \to \mathbb{C}$ such that $\overline{f(\mathbb{R})}=K$

Let $K \subset \mathbb{C}$ be a non-empty closed set. Then there exists a continuous function $f :\mathbb{R} \to \mathbb{C}$ such that $\overline{f(\mathbb{R})}=K$? For example, let $K=\{\lambda ...
0
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0answers
16 views

Infinite product representation for the Sine Integral $\mathrm{Si}(z)$

The infinite series representation of the sine integral (http://en.wikipedia.org/wiki/Trigonometric_integral, previous m.se question: Is there any infinite series representation of the sine ...
1
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1answer
17 views

Complex var. integral: $\oint_{|z|=1} \frac{z^2\ dz}{\sin^3{z}\cos{z}}$

Integrate $\displaystyle\oint_C \dfrac{z^2\ dz}{\sin^3{z}\cos{z}}$; $C \rightarrow |z|=1$ I already know that $|z|=1$ is a circumference with $r=1$ and center at $(0,0)$. I also know there are ...
1
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0answers
51 views

Is my understanding of the argument correct?

I worked through a proof of: $$ f(z) = {1\over 2 \pi i}\int_{\partial D} {f(w) \over w -z} dw$$ where $D\subset \mathbb C$ is an open disk and $f$ is holomorphic on $D$ and continuous on ...
3
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1answer
64 views

Check my answer for find a formula for $\sum_{n=0}^{\infty} \frac{z^{n}}{4^{n+2}}$

The next question in John D'Angelo's text is exercise 4.9. I got an answer but wanted to check it because there's no solution manual: Find a formula $$ \sum_{n=0}^ {\infty} \frac{z^{n}}{4^{n+2}}. $$ ...
3
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2answers
32 views

Complex hyperbolic Trigonometry

When faced with the equation $\cos{z}=\sqrt{2}$ I want to solve for z so I break it up into a sum $z=x+iy$ and get: $\cos{z}=\cos{x}\cosh{y}-i \sin{x} \sinh{y}$ equating real and imaginary parts I ...
0
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0answers
19 views

Complex Mapping of $\mathrm{cosh}(w)=z$

Mapping in complex analysis has not been very easy for me unfortunately. I am having difficult trying to find the mapping between the z and w plane. I attempted to simply write that ...
0
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1answer
39 views

If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$.

Statement: If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$. I just wanted to verify my proof ...
0
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3answers
55 views

How to determine a map which sends $(1,-1,0)\to(\infty,0,i)$?

I know that the uninque bilinear map sending $(z_1,z_2,z_3)\to (\infty,0,1)$ is given by $$T(z)=\frac{(z-z_2)(z_3-z_1)}{(z-z_1)(z_3-z_2)}.$$ Well, could any one tell me how to determine a map which ...
-1
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1answer
53 views

Complex transformation $w=\sqrt \frac{1-iz}{z-i}$ the region $D=\{z\in \mathbb C:|z|<1\}$ [on hold]

Under the transformation $w=\sqrt \frac{1-iz}{z-i}$ the region $D=\{z\in \mathbb C:|z|<1\}$ is transformed to (a) $\{z\in \mathbb C:0<\operatorname{arg}(z)<\pi\}$ (b) $\{z\in \mathbb ...
0
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0answers
6 views

What is the region ( area) of integration in Double mellin Barnes integral?

What is the region ( area) of integration in Double mellin Barnes integral ? In H-function of two variables we are using double Mellin-Barnes contour integration on s and t planes where s and t are ...
7
votes
3answers
232 views

Does $\sin(x+iy) = x+iy$ have infinitely many solutions?

How to prove that $\sin(x+iy) = x+iy$ has infinitely many solutions? I know how to prove that $\sin(x) = x$ has only one solution, but I do not know how to extend this to complex analysis.
-2
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1answer
37 views

Divergence and convergence of the integral. [on hold]

I have the following integral, $$I=\int_a^b |x|^{-p} dx$$ where $a<b$ are finite real numbers and $p\leq 0$ is a non-negative real number. If we start solving it we will come up with the ...
1
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3answers
58 views

Fourier series of a periodic odd function

Given $\ f(\theta)=\theta(\pi-\theta)$ is a $2\pi$-periodic odd function on $[0,\pi]$. Compute the Fourier coefficients of $f$, and show that $\ f(\theta)=\frac{8}{\pi} \sum_{\text{$k$ odd} \ \geq 1} ...
10
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4answers
145 views

How to compute $\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$

Could you explain to me, with details, how to compute this integral, find its principal value? $$\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$$ $f(z) =\frac{\sqrt{z}}{z^2-1} = \frac{z}{z^{1/2} ...
3
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1answer
52 views

Moving the integral $Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds$ past Re(s) = 1.

Given the integral $$Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds,$$ I know that the integrand is holomorphic except for simple poles at ...
1
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1answer
32 views

Proving Fundamental Theorem of Algebra using Maximum Principle

I'm trying to prove FTA by using the maximum principle. Here's what I did, Let $P$ be a polynomial of degree at least $1$ and assume that $P$ has no zeros. Define $$f(z):=\frac{1}{P(z)}.$$ Then ...
1
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1answer
20 views

Extending a harmonic function satisfying a growth condition at an isolated singularity

Consider a harmonic function u on the punctured disc $\Delta(\rho)^*:= \{ z\in \mathbb C: 0 <|z|<\rho\}$ with $\lim\limits_{z \rightarrow 0} z*u(z) = 0$. Prove that $u$ can be written in the ...
0
votes
3answers
40 views

Geometric proof of complex number equation

Use geometric reasoning to find a value for $θ \in [−\pi, 0]$ satisfying $|e^{iθ} − 1| =\sqrt2$. So far I have converted to exponential form as $|\cos \theta + i\sin \theta -1|=\sqrt2$. I'm having ...
2
votes
2answers
107 views

Question about entire function [duplicate]

Suppose $f$ and $g$ are entire functions,and $|f(z)|\leq|g(z)|$ for every z.What conclusion can you draw? My conjecture : $f$ and $g$ are constant but I don't know how to deal with it. I will ...
8
votes
2answers
151 views

Expressing a complex function in terms of z

Use the Cauchy-Riemann equations to determine all differentiable functions that satisfy $Re(f(z))=xy$ I think I know how to do this problem. If we let $z=x+iy$, then $f(z)=u(x,y)+iv(x,y)$. We ...
0
votes
1answer
15 views

Showing that $\Re z \le |\Re z| \le |z|$ and $\Im z \le |\Im z| \le |z|$

What I'm wanting to show is that $$\Re (z) \le |\Re (z)| \le |z|$$ and also $$\Im(z)\le |\Im(z)| \le |z|$$ So what I've done so far is to consider $$z=x+iy$$ Using the above $z$ I also said that $$\Re ...
1
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2answers
51 views

Is this identify valid?

$$\sin(t) \dot{}e^{if(t)} = \sin(t+f(t))$$ I've never seen this identify before but it follows directly from the relation between complex exponentials and the trigonometric functions.
0
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1answer
22 views

Proving that a complex function is not differentiable anywhere

Show from the definition of the derivative that $f(z) = Re(z)$ is not complex differentiable at any point. Easy with the Cauchy-Riemann equations, but I need to do it a different way. Here's my ...
0
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0answers
16 views

Steepest descent from saddle point

I have the function $w(z)=\frac{1}{3}z^3+z$ where $z=x+iy$, i.e. a complex number. I am asked to find the saddle points of this function and then show the paths of steepest descent are ...
3
votes
1answer
51 views

False equations with Euler's Identity [duplicate]

What's wrong with the following equations? $$1 = 1^{-i} = (e^{2πi})^{-i} = e^{-i2πi} = e^{2π}$$ My guess would be the third equation, but I can't really tell why... in the first equation, we use the ...
5
votes
0answers
43 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
0
votes
1answer
25 views

Universal covers of lattice complements.

Background: I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ ...
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votes
1answer
22 views
1
vote
0answers
40 views

Show that if $\lvert a \rvert \neq 1$, then the equation $\overline{z}^2 = az^2+bz+c$ has only a discrete number of solutions.

I knew the proof for this at some point, but I'm having trouble piecing it back together. At least, I think the proof I'm thinking of was for this result, or a result which implied this result. The ...
1
vote
2answers
38 views

What is unique about the Möbius transform?

... is it the only map to accomplish a transformation in 2D and keep certain characteristics invariant? Which? What else makes it special to be studied so much?
2
votes
2answers
55 views

Show $\lvert\lambda_1a_1 + \lambda_2a_2 + \cdots + \lambda_na_n\rvert < 1$ when $\lvert a_i\rvert < 1$ and $\lambda_i\geq 0$

If $\lvert a_i\rvert < 1$, $\lambda_i\geq 0$ for $i = 1,\ldots,n$ and $\lambda_1 + \lambda_2 + \cdots + \lambda_n = 1$, show that $$ \lvert\lambda_1a_1 + \lambda_2a_2 + \cdots + ...
0
votes
0answers
19 views

To obtain an explicit expression of f1(z) from the Laurent decomposition of f(z)

How do you do to obtain an explicit expression for f1(z) where the function is f(z)=tan z in the annulus {3<|z|<4}? Let f(z)=f0(z) + f1(z) be the laurent decomposition of f(z), so that f0(z) is ...
-1
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0answers
28 views

Radius of Convergence of Complex Series [on hold]

Please help me with this question, I've tried using D'Alembert Ratio but I don't understand when theres the complex z involved, thanks. Find the radius of convergence of: $$\begin{align}a)\ \ \ \ ...
1
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0answers
27 views

Zeta functions and their poles

consider a zeta function as follows $f(x):=\sum\limits_{m=1}^{\infty}\sum\limits_{n=0}^{\infty}\frac{1}{\left(a\cdot m+n+\frac{1}{2}\right)^{2x}}$, for $a>0$ and $\Re(x)>1$. How can I construct ...
1
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0answers
21 views

lacunary series with Jensen's formula

Let $\gamma_{n}\in \mathbb R$ such that $\gamma=\liminf_{|n|\rightarrow \infty} \frac{\lambda_{n}}{n}>0$. We suppose that $\sum_{n}|c_{n}|<\infty$ and $f(t)=\sum_{n\in\mathbb ...
0
votes
0answers
23 views

Is there an analytic bounded function on $\omega \subset \mathbb C\setminus]-\infty;0]$ such that $|f(x)|\leq e^{-x^{1/2}}$

Is there an analytic and bounded function on $\Omega \subset \mathbb C\setminus]-\infty;0]$ such that $|f(x)|\leq e^{-x^{1/2}}$; $x\geq 0$ ?
2
votes
2answers
29 views

Analyticity: Uniform Limit

Problem Consider a uniformly bounded sequence over the real line: $$f_n:\mathbb{R}\to\mathbb{C}:\quad|f_n(x)|\leq L$$ Suppose they have analytic continuations with common domain: ...
0
votes
2answers
36 views

Is the set of continuous functions from $[0,1] \rightarrow \mathbb{R}$ closed in the same set from $[0,1]$ to $\mathbb{C}$?

Let $X$ be the set of continuous functions from $[0,1]$ to $\mathbb{C}$, equipped with the norm $\|f\| = \int\limits_0^1 |f(x)|dx$, and let $S$ be the subspace of those functions into $\mathbb{R}$. I ...
2
votes
1answer
65 views

Cauchy integral formula: can it be proved like this?

Consider the Cauchy theorem: Let $D\subset \mathbb C$ be a domain such that $\partial D$ is smooth and $\overline{D}$ is compact. Let $f$ be holomorphic on $D$ and continuous on the closure. Then ...
3
votes
1answer
80 views

Find all holomorphic diffeomorphisms $f:\mathbb{CP}^1\to\mathbb{CP}^1$

The complex projective line $\mathbb{CP}^1$ is the complex manifold defined by the quotient of $\mathbb{C}^2-\{(0,0)\}$ by the relation $z\sim w$ if $z=\lambda w$ for $\lambda\in\mathbb{C}-\{0\}$. I ...
4
votes
1answer
51 views

If all the roots of a polynomial P(z) have negative real parts, prove that all the roots of P'(z) also have negative real parts

If all the roots of a polynomial $P(z)$ have negative real parts, prove that all the roots of the derivative $P'(z)$ also have negative real parts. Could anyone provide a proof for this please?
0
votes
1answer
32 views

Number of zeros of holomorphic function

Let $f(z)=z^{10}+10ze^{z+1}-9$. How to find number of zeros of $f$ in a unit disk ? Probably, I should use Rouche theorem, but I don't know how.
0
votes
0answers
27 views

Analytic bounded in the half right plane is identically zero provided that $\limsup_{x \rightarrow \infty}$ $\frac{\log|f(xe^{ni})|}{x}\leq-n$

I have to prove that an analytic bounded function in the right half complex plane $\mathbb C_{+}$ is the null function, that is $f=0$ if we suppose that f verifies : $\limsup_{x \rightarrow \infty}$ ...
1
vote
2answers
35 views

Polar form equations on the unit circle

If $l \in [0, 2 π)$, $k, n \in N$, proof the following equations: $$\mid{e^{i k l/n} - e^{i (k-1) l/n}}\mid = \mid e^{i l/n} - 1\mid$$ and: $$\lim_{n \to \infty} \sum_{k = 1}^n \mid e^{i k l/n} - ...
1
vote
1answer
42 views

Roots less than 1 if at least one coefficient is greater than one

I have this doubt. If you have this equation with $\alpha_i \in \mathbb R$ $$P(z)=1-\alpha_{1}z-\alpha_{2}z^{2}- \cdots - \alpha_{p}z^{p}=0$$ I believe that if there exist an $\alpha$ greater or equal ...
0
votes
1answer
14 views

sketching and domain question

Sketch the region in the complex plane given by $|z+i|<2$, with Im$z<0$, and decide whether this region is a domain or not. So correct me if I am wrong but would the combined region be ...