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Question on e raised to a complex power

Prove that, if z is a complex number that e^z x e^z* (complex conjugate) is real. Furthermore what can we say about e^z / e^z*?

I have done the other questions in the exercise, but these I'm struggling with.

Any help would be much appreciated, thanks!:biggrin:
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Avatar for Notnek
Notnek
21
Original post by Student10011
Prove that, if z is a complex number that e^z x e^z* (complex conjugate) is real. Furthermore what can we say about e^z / e^z*?
What have you tried? Can you post all your working?
Reply 2
Avatar for rmclare95
rmclare95
Original post by Student10011
Prove that, if z is a complex number that e^z x e^z* (complex conjugate) is real. Furthermore what can we say about e^z / e^z*?

I have done the other questions in the exercise, but these I'm struggling with.

Any help would be much appreciated, thanks!:biggrin:


I'm not the best at complex numbers but this is what I got.

I made z= a+bi and z*=a-bi

For the first proof e^z x e^z*. I wrote e^(a+bi) × e^(a-bi). This would then come to e^a x e^bi × e^a x e^-bi. Leading to the outcome of e^2a. Proving the first part.

Did you get something similar?
Reply 3
Avatar for Student10011
Student10011
OP
Original post by rmclare95
I'm not the best at complex numbers but this is what I got.

I made z= a+bi and z*=a-bi

For the first proof e^z x e^z*. I wrote e^(a+bi) × e^(a-bi). This would then come to e^a x e^bi × e^a x e^-bi. Leading to the outcome of e^2a. Proving the first part.

Did you get something similar?


Ahh, yeah that works. Yea, I've done the other part of the question now as well - thanks for the help!

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