logarithm of a complex number? I have a task to study a function like this one: F ( z

Kiana Dodson

Kiana Dodson

Answered question

2022-06-23

logarithm of a complex number?
I have a task to study a function like this one:
F ( z ) = ln ( e i z 4 ) z 3
I'm trying to simplify this: since the exponential is the inverse function of ln ( ) can we simplify it to be:
i z
log ( z ) = l n ( r ) + i θ
But my problem is with :
ln ( e i z ) = ln ( e i x y ) can we simplify it to: i x y?

Answer & Explanation

Paxton James

Paxton James

Beginner2022-06-24Added 25 answers

The function z i z 4 is one branch of ln e i z 4 . All branches of ln e i z 4 are therefore
f k ( z ) = i z 4 + 2 π i k , k Z .
The possible branches of F are hence
F k ( z ) = i z + 2 π i k z 3 , k Z .
F 0 ( z ) = i z is the only entire branch, all others have a pole (of order 3) in 0.
"But my problem is with : ln ( e i z ) = ln ( e i x y ) can we simplify it to: i x y?"
If you are free to choose the branch of the logarithm, you can choose that branch. Otherwise, either list all possible branches, i x y + 2 π i k , k Z , or if a specific branch is prescribed, use that.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?