Problem Statement: What is the correct way of computing the argument of the following equation?
Relevant Equations: I am trying to compute the argument ##\Phi## of the equation
$$\frac{r-\tau\exp\left(i\varphi\right)}{1-\tau r\exp\left(i\varphi\right)} \tag{1}$$
which using Euler's equation...
Homework Statement
Find all $$n \in Z$$, for which $$ (\sqrt 3+i)^n = 2^{n-1} (-1+\sqrt 3 i)$$
Homework Equations
$$ (a+b i)^n = |a+b i|^n e^{i n (\theta + 2 \pi k)} $$
The Attempt at a Solution
First I convert everything to it`s complex exponential form: $$ 2^n e^{i n (\frac {\pi}{3}+ 2\pi...
Homework Statement
Write ##5-3i## in the polar form ##re^\left(i\theta\right)##.
Homework Equations
$$
|z|=\sqrt {a^2+b^2}
$$
The Attempt at a Solution
First I've found the absolute value of ##z##:
$$ |z|=\sqrt {5^2+3^2}=\sqrt {34} $$.
Next, I've found $$ \sin(\theta) = \frac {-3} {\sqrt...
Homework Statement
Show that
$$\int_C e^zdz = 0$$
Let C be the perimeter of the square with vertices at the points z = 0, z = 1, z = 1 +i and z = i.
Homework Equations
$$z = x + iy$$
The Attempt at a Solution
I know that if a function is analytic/holomorphic on a domain and the contour lies...
Homework Statement
A particle of energy E moves in one dimension in a constant imaginary potential -iV where V << E.
a) Find the particle's wavefunction \Psi(x,t) approximating to leading non-vanishing order in the small quantity \frac{V}{E} << 1.
b) Calculate the probability current density...
The problem
I would like to solve:
$$ \bar{z} = z^n $$ where ##n## is a positive integer.
The attempt
## z = r e^{i \theta} \\ \\ \overline{ r e^{i \theta} } = r^n e^{i \theta n} \\ r e^{-i \theta} = r^n e^{i \theta n} ##
## r = r^n \Leftrightarrow true \ \ if \ \ n=1 \ \ or \ \ r=1##
##...
I am trying to find out the interference condition between tool and a part. The below attached snapshot is the equation between interference and machine feed. At dy/dx = 0, I will have max. interference, which I intend to find. Except x and y every alphanumeric character in the following...
Mod note: Fixed all of the radicals. The expressions inside the radical need to be surrounded with braces -- { }
(This question is probably asked a lot but I could not find it so I'll just ask it myself.)
Does the square root of negative numbers exist in the complex field? In other words is...
Homework Statement
I just cant seem to get the right answer. z^4+80i=0
looking at the complex plane u see the radius=r=80 (obviously)
using De Moivre extension: z^n=(r^(1/n))(cos((x/n)+k2pi/n)-isin((x/n)+k2pi/n)
z1=((80)^(1/4))(cos(3pi/8)+isin(3pi/8)
shouldnt this be a root?
z2=...
ok having major problems. i can easily solve z^2 + pz +a+bi=0 solutions but that extra qiz is really annoying me.
z^2 + 3z+4iz-1+5i=0
(z+2i)^2+3z-5+5i=0
z+2i = w, z=w-2i
w=-3(w-2i)+5-5i
then im not getting anything sensible for solving x and yi. what am i doing wrong?