Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.
0
votes
2answers
33 views
Real and Imaginary Parts of $\frac{\cos(z)}{(1-e^{ix})}$
Find
$$\mathrm{Re}\bigg(\frac{\cos(x+iy)}{(1-e^{ix})}\bigg)$$
and
$$\mathrm{Im}\bigg(\frac{\cos(x+iy)}{(1-e^{ix})}\bigg)$$
Please help I've been trying for some time now...
1
vote
1answer
26 views
A method for solving cubic equation
So I'm reading Beardon's Algebra and Geometry, and in chapter on complex numbers, author gives the following method for solving cubic equation:
Suppose we want to solve cubic equation $p_1(z)=0$, ...
0
votes
2answers
61 views
What is the principal 12th root of one?
Let $w$ be the principal 12th root of 1. What is $w$, and what are the real and complex parts of the following:
$w w^∗$ (* = complex conjugate)
$w^9$
2
votes
2answers
21 views
Small inequality on unit open disc
For $|u|,|z|<1$, $u,z$ complex numbers, how to show the inequality:
$|\frac{u-z}{1-\bar uz}|<1$?
1
vote
2answers
24 views
nasty exponentials
While trying to find the fourier transform of $\Large \frac{1}{1 + x^4} $, using the definition and the residue theorem has required me to evaluate nasty looking expressions like
$$\large \rm ...
1
vote
2answers
38 views
determining residue for the purposes of calculating an integral
Determine the integral
$$ \int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2}$$
using residues. This is from Section 79, Brown and Churchill's Complex Variables and Applications.
In order to do this. We ...
1
vote
1answer
21 views
similarity : $z'=(1-i)z+1+i$ with the curve of $e^x-1-x$.
Let $S$ be the similarity defined by : $S(z)=(1-i)z+1+i$, for a complex number $z$ in the complex plane.
What is the image of the curve : $y=e^x-x-1$ by the similarity $S$.
My work : Let $z=x+iy$ ...
0
votes
1answer
18 views
convergence of complex series
Set that Re $z_n>=0$,$\forall$ n $\in$ N,Proof that if $\sum z_n$ and $\sum {z_n}^2$ are both convergent,then $\sum |z_n|^2$ is also convergent.
Well I've no idea how to tackle it.
1
vote
2answers
29 views
Does $\lim_{n\to\infty}\sum\limits_{k=1}^n|e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi i(k-1)}{n}}|$ exist?
Does $\lim_{n\to\infty}\sum\limits_{k=1}^n|e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi i(k-1)}{n}}|$ exist? If yes, what is its value?
0
votes
2answers
74 views
Simplification of product of complex numbers
I look for a closed formula to the expression
$$\prod_{k=1}^{n-1}\left(e^{\frac{2ik\pi}{n}}-1\right)$$
Any suggestion is welcome. Thanks.
-1
votes
0answers
58 views
{University Complex Analysis] contour and Laurant series [closed]
I am really lost on these problems. Please help.
$(1)$ Evaluate $$\int_\Gamma \bar z^2 dz$$ where $\Gamma$ is the following contour from $z=0$ to $z=1+i$.
$(a)$ A simple line segment
$(b)$ The ...
-1
votes
0answers
34 views
question about complex analysis [closed]
Sketch the lines defined by the following equations:
$(a)$ $\text{Re}(z^2) = r$,
$(b)$ $|z^2-1| = r$,
$(c)$ $|z + 1| + |z - 1| = r$,
where $r > 0$ is some positive, real number.
-3
votes
1answer
65 views
$\mathbb{Z}[\sqrt{-23}]$: A uniquely written set?
I suspect that
$\mathbb{Z}[\sqrt{-23}] \implies \forall~z=\sqrt{23b+a}~e^{i\arctan{\frac{23b}{a}}},~\text{where $z$ is uniquely written}~\forall~z\in \mathbb{Z}[\sqrt{-23}]$
2
votes
4answers
99 views
When does $az + b\bar{z} + c = 0$ represent a line?
$a,b,c$ and $z$ are all complex numbers. My idea was to show that it passes through the point $\infty$ in the extended complex plane, but I'm not quite sure how to execute that.
Update:
It says in ...
0
votes
0answers
8 views
Argument of a fraction of complex numbers times the exponential function
How do I find the argument a function that looks like this:
$G(s)=\frac{K(1+\frac{s}{z_0})}{s^p(1+\frac{s}{p_0})}e^{s}$
when $s=i\omega$?
I know that if the $e^s$ wasn't there, I'd be able to ...
