2
votes
1answer
31 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.
3
votes
4answers
165 views

Show that $|z+1|\le|z+1|^2 +|z|$ for all $z \in \mathbb{C}$

Question: Show that $|z+1|\le|z+1|^2 +|z|$ for all $z \in \mathbb{C}$ So far I have, Suppose $1\le|z+1|$ $|z+1|\le|z+1|^2$ $|z+1|\le|z+1|^2+|z|$ Now I must show $|z+1|<1$ but this is where ...
1
vote
0answers
32 views

Zeta function universality: How to compute the shift parameter for simple functions?

I've come across Zeta function universality. For a nice function $f$ in a nice subset $U$ of the complex strip between real $0$ and $1$, one can find a real $t$, such the zeta function $\zeta$ shifted ...
0
votes
2answers
46 views

What is the Laurent series of the complex absolute value?

What is the Laurent series of the function $f(z) = |z|$? It seems to be ill defined at $z=0$. Are there any other expansion techniques applicable for this function at $z=0$?
1
vote
1answer
48 views

When is $|f(x)|$ equivalent to $f(|x|)$

Specifically for functions of a complex variable. Are there any rules of thumb?
0
votes
0answers
26 views

Questions about $|f(1+a+bi)|<|f(1+a)|$

Let $a,b >0$ and $|*|$ denote the absolute value. Let $f(z)$ be a realvalued analytic function defined for $Re(z)>1.$ For any $a,b$ we have $|f(1+a+bi)|<|f(1+a)|$. Some questions : $1)$ If ...
1
vote
1answer
270 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 ...
-1
votes
1answer
535 views

Describe the set of points on the complex plane…

Describe the set of points on the complex plane for which $|z-2| + |z+2|=4$... So, I know you can solve this instantly, just by using definition, but I want to do it the long way.. So, $$|x- i*y ...
4
votes
2answers
158 views

How do we know that $|i!| = \sqrt{\pi \operatorname{csch} \pi}$?

(Source: Wolfram Alpha) Or, to write it out in full, $$|i!| = \sqrt{\frac{2\pi e^\pi}{e^{2\pi} - 1}}$$ How is this identity derived? Also, knowing this, could we find the exact values for the real ...
1
vote
1answer
412 views

how to find absolute value for complex fraction

I have a Fourier transfer equation $H(jw) = \frac{jwL}{(jw)^2LC+jw\frac{L}{R}+1}$, and I need to find frequency to make $|H(jw)|$ is max. I know I should take the derivative of $|H(jw)|$ then find ...
4
votes
3answers
776 views

Proving two integral inequalities

Can anyone help me to prove that these integral inequalities hold? Here $x$ is a real value: $$ \left| \int_a^b\ f(x) dx \right| \leq \int_a^b\ |f(x)| dx $$ Here $z$ is a complex value: $$ \left| ...
6
votes
2answers
207 views

Sum of two absolute values in complex plane

I'm trying to find out all $z \in C$ that satisfy the following condition: $|z+1|+|z-i|=3$ I understand that $|z|=r$ represents a circle with a radius of $r$. I also understand that $|z+1|=r$ can ...
0
votes
1answer
791 views

derivative of absolute value of a complex function

If $f:U\subset\mathbb{C}\mapsto\mathbb{C}$, where $f(x+iy)=u(x,y)+iv(x,y)$ is a meromorphic function and if $f$, $f'$, and $f''$ are not zero in the strip $a<x<b$, can we get ...