0
votes
0answers
39 views

Complex Fourier Series and using the square norm

Find the complex Fourier series of $f(x)=e^{(-πx/2)}$ on $-π < x < π$ Discuss the significance of $|C_n|$ in the solution. I've tried so far Using the Complex Fourier Series: $$ %% ...
2
votes
1answer
44 views

The complex equation

In solving $|z|i +2z =1$, I seem to be constantly getting two solutions while both answer key and Wolfram claim to be only one. What am I doing wrong? Let's share the fun: $(\sqrt{x^2 +y^2}) i +2x ...
3
votes
3answers
60 views

Bound in Complex Analysis

Can someone direct me towards the right way to approach this problem? Show $$\displaystyle \left|\int_{|z|=R} \frac{Log{z}}{z^2} dz\right| \leq 2\sqrt{2}{\pi}\frac{\log{R}}{R},\; \text{ for } ...
0
votes
0answers
24 views

Biholomorphic, Hypersurface

I'm learning the Hypersurface. And my teacher has a question: Find an example such that two Hypersurfaces are biholomorphic. I think that $$A=\{(x,y)\in \Bbb C,\ \rho(x,y)= x^2+y^2-1=0\}$$ and ...
0
votes
3answers
43 views

Solving the complex polynomial

For the complex polynomial $z^3 -5z^2 +(7-2i)z +6i-3 = 0 $ $1)$ show that $2+i $ is a root. $2)$ solve the given equation. Attemp to solve: I'm not really sure how to solve this, but I ...
1
vote
2answers
103 views

Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
4
votes
5answers
226 views

How to find $\sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)}$

Let $a,b$ be two unequal integers. I have to find the sum below. $$ \sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)} $$ I should use complex analysis, but I have no clue where to start. I only now that I can ...
1
vote
2answers
69 views

Calculating $\int_0^\infty \frac{\ln x}{(x^2+9)^2} dx$

I try to calculate $$ \int_0^\infty \frac{\ln x}{(x^2+9)^2} dx $$ I use a book that tells me to replace $\ln x \ $ by $ \ \ln(|x|) + i\phi_z$ where $\phi_z$ denotes the argument of $z$, chosen between ...
5
votes
2answers
55 views

How to compute $\int_C {e^{3z}-z\over (z+1)^2z^2}$?

I am asked to compute the integral $$ \int_C {e^{3z}-z\over (z+1)^2z^2} $$ where $C$ is a circle with the center at the origin and radius ${1 \over 2}$. My approach was to separate the integral as a ...
3
votes
1answer
70 views

Solving $|z-3| \leq|z-1-i|$

I was trying to solve graphicly: $$|z-3| \leq |z-1-i|$$ I plugged x and y in proper places as real componenets of the comlex number yielding in the end $-4x+2y+7 \leq0$ this might be tackled if ...
1
vote
4answers
62 views

How to find $\frac{f(z)}{z-a}$

I hope that you can help me to find some residues. I know two ways to find the residue in a value $a \in \mathbb{C}$: Straight forward calculation: $ \int_{C(a,\epsilon)^+} f(z) dz$ Rewriting a ...
2
votes
1answer
61 views

Singularities of complex functions.

How do I determine the singularities of a function? What is a singularity? In the functions below which are the singularities? a)$$f(z)=\frac{1}{(z^4+2z)}$$ b)$$f(z)={e^{1/z}}$$
1
vote
3answers
38 views

Determining Laurent Series expansion and residues

Determining Laurent Series expansion and residues of $f(z)=\frac{z}{(z+1)(z+2)}$ around $z = -2$. What is the validity of the expanded region? What is $res(f, -2)$??
1
vote
4answers
62 views

Which way will produce the following integral?

Which way $\gamma$ will produce the following integral? $$\int\limits_{\gamma}\frac{3+i}{z^5 - z}dz = 0$$
0
votes
2answers
33 views

Calculate the integral using the Cauchy integral formula

$f(z) = \frac{z}{z^2 -1}$ Calculate $\int_\gamma f(z) dz$ where $\gamma(t) = i + e^{it}$ for $t\in[0,2\pi]$ using the Cauchy integral formula.
0
votes
3answers
81 views

Complex Roots and calculations

roots of the equation $z^6 =1-\sqrt3 i $ are $$z_1,z_2,z_3,z_4,z_5,z_6 $$ calculate:$$|z_1|^3 +|z_2|^3+|z_3|^3+|z_4|^3+|z_5|^3+|z_6|^3$$ also calculate: $$z_1^6 +z_2^6+z_3^6+z_4^6+z_5^6+z_6^6$$ ...
0
votes
3answers
40 views

Complex solutions of polynomial question

$2z^3-6z^2+mz+n = 0$ $m, n$ are real and $1+\sqrt{ 2} i$ is a solution. Find $m$ and $n$. Attempt to solve : Giving the known theorem $1-\sqrt{2}i$ is also a solution, so we can substitute each time ...
2
votes
0answers
22 views

A question on normal family

Let $f$ be an entire function. Suppose that the sequence $F_0=\{f(0),f'(0),f''(0),\cdots,f^{(k)}(0),\cdots \}$ is bounded. Show that $F_z=\{f(z),f'(z),f''(z),\cdots,f^{(k)}(z),\cdots \}$ is a normal ...
0
votes
1answer
43 views

Prove that These Families of Level Curves are Orthogonal

From p. 79 in Brown's and Churchill's "Complex Variable and Application": Let the function $f(z) = u(x, y)+iv(x, y)$ be analytic in a domain $D$, and consider the family of level curves $u(x, y) = ...
1
vote
0answers
22 views

Curves composition with holomorphic function

Statement $(i)$ Let $\gamma:\mathbb R \to \mathbb C$ a $C^1$ curve. Let $v={\gamma}'(t_0)$ the complex number that one obtains from translating to the origin the tangent vector to $\gamma$ at ...
0
votes
1answer
34 views

Complex conjugate root theorem question

From the Complex conjugate root theorem we get that if a polynomial in one varaible with real coefficients has as solution $a + bi$ , than $a-bi$ must also be a solution...however, what happens if ...
-2
votes
2answers
40 views

Fourier transform of t*(sent/pi*t)^2

Here's the function (I need it's fourier transform).
0
votes
4answers
65 views

complex numbers quadratic equation question

how to solve $z^2 +3|z| = 0 , z$ complex ? treating the complex number as $a+bi $ or anything similar didnt help much...also solving like simple algebric equations also didnt prove effective and ...
0
votes
1answer
17 views

Uniform convergence of sequence of complex functions and integral over a curve

I am trying to solve a problem, its statement is: Let $\Omega \subset \mathbb C$ an open set and $f_n,f:\Omega \to \mathbb C$. Show that if $f_n \rightrightarrows f$ on a curve $\int_{\gamma} \subset ...
0
votes
1answer
21 views

Rouche's theorem for unit disk

I have a function $F(z)=2z^4+5.41z^3+10.24z^2+4.83z+1.414$. And I need to prove how many roots does it have in unit circle, on unit circle and out of unit circle. By Rouche's theorem I was able to ...
0
votes
1answer
39 views

How to integral : (1-i) / cosh z on C

Let C be the triangular contour connecting 3i, -1-i, 1-i. How to interal (1-i) / cosh z on C ? Don't use the residue theorem..
1
vote
1answer
52 views

Contour Integral of $\int_0^{\infty} \frac{1}{x^4+1} dx $ - Missing a factor of 2

I'm supposed to evaluate: $$ \int_0^{\infty} \frac{1}{x^4+1} dx $$ Consider $$ \oint \frac{1}{z^4+1} dz = \oint \frac{1}{(z - \frac{1-i}{\sqrt 2})(z + \frac{1-i}{\sqrt 2})(z - \frac{1+i}{\sqrt ...
0
votes
0answers
41 views

Complex Integration: $\int_0^{\infty} \frac{\sin x}{x(k^2x^2 +1)} dx $

I'm supposed to evaluate: $$ \int_0^{\infty} \frac{\sin x}{x(k^2x^2 +1)} dx $$ Attempt Consider $ \int_0^{\infty} \frac{e^{iz}}{x(k^2x^2 +1)} dz $ Simple poles at $z = \pm \frac{i}{k} $, simple ...
0
votes
2answers
85 views

How to show that $\int_{-\infty}^\infty\frac{t}{(a^2+t^2)(b^2+t^2)(e^{2\pi t}-1)}dt=\frac{1}{2ab(a+b)}+\frac{1}{b^2-a^2}\sum_{a<k\leq b}\frac{1}{k}$

I'm stuck on this problem. Here $a,b\in\mathbb{N}$ with $b>a$. I have already shown that $$-\lim_{\varepsilon\searrow 0}\int_{|t|>\varepsilon}\frac{\coth(\pi ...
1
vote
1answer
33 views

Show that $\sum_{k=1}^N\frac{1}{(k+a)(k+b)}=\frac{1}{b-a}\sum_{a<k\leq b}\frac{1}{k}-\frac{1}{b-a}\sum_{a<k\leq b}\frac{1}{k+N}$

I am quite stuck on this problem and I don't know how to proceed. The question states: Let $a,b,N\in\mathbb{N}$, $b>a$, $N\geq b-a$. Show that ...
1
vote
1answer
49 views

Complex integration Question - Contour Method [duplicate]

I'm asked to find: $$\int_{-\infty}^\infty \frac{\ln(x^2+1)}{1+x^2} dx $$ Attempt Considering $$ \oint \frac{\ln(z^2+1)}{(z+i)(z-i)} dz $$ So first I find the branch points of the function. This ...
2
votes
0answers
101 views

Prove that periodic analytic function can be written as $\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$

This question involves the following homework problem: PROBLEM Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form ...
2
votes
3answers
38 views

Find all value of $z$ for which each equation holds.

$$(a) \sin z=\cosh 4$$ $$(b) \cos z=2$$ $$(c) \sin z=i\sinh 1$$ $$(d) \cosh z=1$$ my answer (a) $\sin z= \sin x \cosh y+\cos x \sinh y=\cosh 4$ so, $x=2n\pi+{\pi\over2}$ and $y=4$ (b) since ...
2
votes
1answer
41 views

Evaluating the definite integral of a trig function via complex analysis methods.

Show that $$ \int_0^{\pi} \frac{1}{1 + \sin^2\theta} {\delta}\theta= \frac{\pi}{\sqrt{2}} $$ Normally if the limits of integration were from $0$ to $2\pi$ what I would, and have been taught ...
0
votes
1answer
40 views

show analytic function such $f(z)={\operatorname{Log}(z+5)\over z^2+3z+2}$

Show that $f(z)=\dfrac{\operatorname{Log}(z+5)}{z^2+3z+2}$ is analytic everywhere except at the point $-1,-2$ and on the ray $\{(x,y):x\le -5,y=0\}$. i think that separate denominator and ...
1
vote
1answer
42 views

Limit of a complex function (resulting in divide by zero situations)

I have the following limit which I'm attempting (and failing) to evaluate: $\lim_{x \to i} (z-i)\frac{e^{imz}}{(z^2+1)^2}$ Evaluating directly, we get $\frac{0}{0}$, so can rewrite using L'Hopital's ...
1
vote
0answers
19 views

show that there exists $n_0$ such that for $n\geq n_0$ $f_n(z)$ has $k$ roots.

The situations is as follows. We have sequence of holomorphic functions $\{f(n)\}$ which is defined on a open $U$ in $\overline{B(0,1)}$. Suppose now that this sequence converges uniformly to a ...
1
vote
0answers
33 views

Show that $f(z)\leq \left|\frac{z-a}{1-\overline{a}z}\right|$

Given is that $|a|<1$ and the transformation $$T\colon z\mapsto\frac{z-a}{1-\overline{a}z}$$ This maps $B(0,1)$ onto $B(0,1)$. Now suppose for some $\epsilon >0$ the function $f$ is holomorphic ...
6
votes
1answer
93 views

zeros of a polynomial

Given $P(z)=z^6+6z+10$, find how many roots are in each quadrant I have already seen that $P(z)$ has six different roots, and that none of them are real or of the form $ki$, $k\in \Bbb R$. Since ...
3
votes
1answer
44 views

Proving that a Function is Analytic Given that it is Equal to the Complex Conjugate of an analytic Function

So I'm working a problem that states: A function $f$ is analytic in an open set $U$. Define $g$ by $g(z)=\overline{f(\overline{z})}$ (just because the notation can be hard to read, this is the the ...
1
vote
0answers
38 views

Calculate all the local automorphisms

The Kohn - Nirenberg domain $\Omega_{KN}$ defined by $$\Omega_{KN}=\left\{(z,w)\in \Bbb C^2:\text{Re}\ w+|zw|^2+|z|^8+\dfrac{15}{7}|z|^2\text{Re}\ z^6<0\right\}$$ How to compute all the ...
2
votes
1answer
54 views

Integration of complex function with respect to complex variable

I was given as homework to calculate the complex integral limit $$\lim_{T\rightarrow \infty} \frac {1}{2\pi i}\int_{c-iT}^{c+iT}\frac {x^s}{s^{k+1}}ds $$ where $c>0$ and $k\geq1$ is an integer. ...
1
vote
0answers
47 views

Existence of non-constant bounded analytic function on $\mathbb{C}\setminus \mathbb{Z}$

Show that there is no non-constant bounded analytic function on $\mathbb{C}\setminus \mathbb{Z}$. In this homework problem i tried to proceed in the following way: If possible, let $f$ be a ...
2
votes
2answers
39 views

Find all holomorphic functions on $\mathbb{C}$, except for some singularities, such that $|f(z)|\leq C(|z|^{3/2}+|z-1|^{-3/2}), z\in\mathbb{C}-\{1\}$

First I wrote the Laurent series of $f(z)$ around $z=1$: $$ f(z)=\sum_{n=-\infty}^{-1}c_n(z-1)^n+\sum_{n=0}^{\infty}c_n(z-1)^n. $$ Now if $|z|$ becomes very large, the first sum with the negatives ...
4
votes
2answers
122 views

Integrating $e^{e^{ix}}$

Evaluate $\int_0^{2\pi}e^{e^{ix}}dx$. Attempt: $e^{ix}=\cos{x}+i\sin{x}$, so we can write $$e^{e^{ix}}=e^{\cos{x}}e^{i\sin{x}}$$ and then use the same identity to get ...
0
votes
0answers
38 views

A problem in complex analysis from Stein's book [duplicate]

I'm trying to solve an exercise from Stein's Complex Analysis: Let $t>0$ be given and fixed, and define $F(z)$ by $F(z)=\prod _{n=1}^\infty (1-e^{-2\pi nt}e^{2\pi iz})$ Show that $F(z)$ is of ...
0
votes
0answers
34 views

Vanishing Partial Derivatives of a Harmonic Function

If $u$ is a harmonic function such that all of its partial derivatives vanish at some point $z$, show that $u$ is constant.
1
vote
2answers
55 views

Determining if any general funtion u(x,y) makes f(z)=u(x,y)+iv(x,y) analytical

I have a question about Complex Analytical functions. I have some homework that asks: let $f(z) = u(x,y) + iv(x,y)$. Indicate the following functions for which u(x,y) may be analytic: $6(x^2-y^2)$ | ...
1
vote
1answer
72 views

Laurent expansion of $\frac{1}{1+z^2}$ in $A:= \{z \in \mathbb{C} : | z - z_0 | \gt |z_0 + i|, Im(z_0) \gt 0\}$

I need to find the Laurent expansion of $f(z) := \dfrac{1}{1+z^2}$ in the set $A:= \{z \in \mathbb{C} : | z - z_0 | \gt |z_0 + i|, Im(z_0) \gt 0\}$. I've drawn a picture of this: I know that if $r ...
1
vote
1answer
31 views

Singularities of a complex function

I have a function of the form $$ \frac{\sqrt{z}}{\sqrt{1+z^2}} $$ I want to find the singularities of it. Obviously there is a branch point at $z=0$ and $z=\infty$ because of $\sqrt{z}$, I'm all OK ...