Tagged Questions
0
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0answers
43 views
Which complex number cannot be written in polar form?
I'm really confused by this question. Is there such a number?
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2answers
87 views
What is the inverse $z^{-1}(z)$ of $z(\varphi)=e^{i\varphi}$ with $\varphi\in\Bbb N_0$?
Suppose I am given a complex number $z=x+iy\in\Bbb C$, with $\left|z\right|=1$, and I am told that $z=e^{i\varphi}$ for some integer $\varphi\in\Bbb N_0$ (the value of which is not given to me).
How ...
0
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3answers
86 views
Cartesian and Polar Coordinate
I should give the Cartesian Coordinates $(x,y)\in \mathbb{R\times R}$ and Polar Coordinates $(r,\varphi)\in R^+\times [0,2\pi)$ of the following Complex Numbers:
a) $z_{1}=-i$
b) $z_{2}=\sqrt{3}+i$
...
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votes
1answer
155 views
Converting to polar form
Write each of the given numbers in the polar form $re^{i\theta}$.
a.) $\frac{1-i}{3}$
b.) $-8\pi (1+\sqrt 3 i)$
For a, I got:
r = $\frac{\sqrt 2}{3}$ and $e^{i7\pi /2}$ since ...
1
vote
2answers
43 views
Polar coordinates - issue with direction denoted by angle
Convert $1-\sqrt{3}i$ to polar coordinates $(r,\varphi)$.
I started by computing $r=|1-\sqrt{3}i|=\sqrt{1^2+\sqrt{3}^2}=\sqrt{4}=2$. When I tried to compute the angle I did something like
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0
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0answers
93 views
How to find the formula of a spiral using least-squares(regression)?
Assume data from a plane which are roughly showing a spiral. I want employ the rationale of regression to find the parameters for the best fit by some spiral.
That means, I have to estimate the ...
1
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3answers
175 views
Expressing $e^z$ where $z=a+bi$ in polar form.
I am reading a passage of text that states:
"We can use the fact that $e^{a+bi}=e^a(\cos b+i\sin b)$ has polar form $\left<e^a,b \right>$ to verify that complex exponentials have various ...
1
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1answer
134 views
Complex Numbers and polar form
I am given the following information:
$$x[n]= s^n,\qquad n=0,\pm 1,\pm 2,\ldots$$
where $s=\sigma + j\omega = re^{i\theta}$ is a complex number in general.
I was wondering how the following is ...
0
votes
1answer
49 views
Find complex $z$ such that $z$ has the largest possible real part, and satisfies: $z^7 = -18-18i$
Find complex $z$ such that $z$ has the largest possible real part, and
satisfies the equation:
$z^7 = -18 -18i$
So, the 7th roots of $z = 18\sqrt{2}e^{i\frac{\frac{\pi}{4} + 2\pi k}{7}}$ ...
0
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3answers
146 views
Write $\cos(9x)$ in terms of powers of $\cos(x)$ [duplicate]
Possible Duplicate:
How to expand $\cos nx$ with $\cos x$?
Write $\cos(9x)$ in terms of powers of $\cos(x)$
I realize I could solve this by using De Moivre's and binomial expansion:
...
3
votes
2answers
73 views
Using polar form to prove $|z| = 1 \implies \text{Re}\left(\frac{1-z}{1+z}\right) = 0$
This was an answer provided to a question I asked previously. I followed the other approaches to the question; however, I couldn't seem to follow this one:
...
1
vote
3answers
367 views
A square root of i with negative imaginary part
In an ODE class, one assignment question says find the “rectangular” expression z = a + bi (with a and b real) and the “polar” expression |z|, Arg(z) where z is "a square root of i with negative ...
