0
votes
1answer
19 views

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$, where $aj+b$ is a complex number, and $|f(x)|$ is the modulus function. In the past I've been calculating $|(aj+b)^{-1}|$ by multiplying the numerator and ...
2
votes
1answer
30 views

If $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ are complex numbers, then $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$

Let $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ be two complex numbers. Ahlfors says that $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$. I don't understand why that is. Any help would be greatly appreciated.
-1
votes
2answers
59 views

Sketching a set of complex numbers and deducing the value of $|z +1 - i|$ for such numbers

The point $P$ represents the complex number $z$. a) Given that $\arg(\frac{z-2i}{z+2}) = \frac{\pi}{2}$ , sketch the locus of $P$. Ok so I've sketched this and this is what it looks like : b) ...
3
votes
4answers
87 views

Solve $z^2 - iz = |z - i|$

I have the equation: $z^2 - iz = |z - i|$ The solutions are $i$, $-\sqrt3/2 + i/2$, $\sqrt3/2 + i/2$ Can someone please walk me through or give me a hint...
0
votes
1answer
46 views

Complex number in polar coordinates

I have to get $\Im$, $\Re$, the absolut value as well as the argument $\phi$ of the complex number $$z = \left(-\frac{1}{\sqrt2}+\sqrt\frac{3}{2}i\right)^8$$ I do this by transforming $z' = ...
0
votes
4answers
162 views

Find $z$ such that $|z+1|+ |z-1|=4$

I have this problem: Find all points of the complex plane wich satisfy: $$|z+1| + |z-1| = 4 $$ I know this is an ellipse with foci 1 and -1, and i know the answer is : $$3 x^2+4 y^2 \leq 12$$ but ...
0
votes
0answers
20 views

Derivative of squared Fourier transform

I haven't found any relative to this, so I would like to get some help. I have a function $h(x) = |\mathcal{F} [P(x) e^{ic+iZ(x)a}]|^2 $ and I would like to find the derivative with respect to the ...
1
vote
1answer
261 views

Maximum of an absolute value complex function

I'm working my way through Marsden's Basic Complex Analysis book and I can't solve this problem. It's problem 23 of section 1.2 if that helps. Let $a$ be a complex number, find the maximum of ...
3
votes
2answers
203 views

Complex number with z to the power of 4

I have to find all $z\in C$ for which BOTH of the following is true: 1) $|z|=1$ 2) $|z^4+1| = 1$ I understand that the 1) is a unit circle, but I can't find out what would be the 2). Calculating ...
1
vote
1answer
53 views

What is the minimum of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$ over all complex numbers?

Find the Least value of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$ My try:: Let $A(2,2)$ and $B(-1,5)$ and $C(6,-2)$ and $P(x,y)$ be a point Here $A,B$ and $C$ are the point ...
1
vote
1answer
85 views

What is the modulus of a number?

What is the exact definition of the modulus of a number? As far as I know, it is the distance between the origin and the point associated with this number. So if $z=a+bi \in \Bbb ...
1
vote
2answers
175 views

Calculating the absolute value of a complex number - am I right?

To calculate the absolute value of a complex number u must use the following formular $(a^2+b^2)^½$=|a+bi| So for instance with -4-5i would have the absolute value ...
4
votes
3answers
165 views

Is there an alternate definition for $\{ z \in \mathbb{C} \colon \vert z \vert \leq 1 \} $.

Is there a method of constructing a subset of a reasonably arbitrary ring so that when the construction is applied the $\mathbb{C}$ the result is $B = \{ z \in \mathbb{C} \colon |z| \leq 1 \} $? My ...
2
votes
1answer
111 views

Simplifying $|a+b|^2 + |a-b|^2$

I want to simplify $|a+b|^2 + |a-b|^2$ where $a, b \in \mathbb{C}$. I've used Wolfram Alpha to get $$ |a+b|^2 + |a-b|^2 = 2\left(|a|^2 + |b|^2\right) $$ I'm trying to understand the steps involved in ...
2
votes
1answer
323 views

When should I put an absolute value sign around a function?

In Griffiths' introduction to QM book, I see $\int f^* f\ dx$ often written as $\int | f |^2 dx $, whereas $\int f^* g\ dx$ is understandably just that. Should I always write the absolute value sign ...