Prove that
\[\huge{(e^w)^z=e^{[(w+2ni\pi)(z)]}}\]
but \[\huge (e^w)^z\ne e^{(wz)}\] where w and z are complex.

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- 2bornot2b

- schrodinger

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- 2bornot2b

- myininaya

\[\text{ Do you recall } e^{ i \theta}= \cos(\theta) +i \sin(\theta)\]

- myininaya

\[e^{i \theta}=\cos(\theta)+i \sin(\theta)=\cos(\theta+2 n \pi)+ i \sin( \theta + 2 n \pi))=e^{i(\theta+2 n \pi)}\]

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## More answers

- myininaya

And of course n is an integer

- myininaya

Try to use that in this proof
I would write w as a+bi by the way until the end when you write a+bi back as w

- 2bornot2b

So you want me to start with \(\large (e^{a+ib})^z\)
Sorry for the late reply @myininaya I was away.

- 2bornot2b

What do I do next @myininaya ?

- KingGeorge

The first thing you need to show is not too bad. Since \(w\) is complex, \(w=w+2ni\pi \). Hence\[\Large (e^w)^z=e^{(w+2ni\pi)~z}\]by using a simple substitution.

- 2bornot2b

@KingGeorge, is it just substitution? Perhaps you didn't notice I have replaced the ( )

- KingGeorge

Herp derp.

- KingGeorge

My ideas aren't panning out :(

- anonymous

What do you mean by complex? \(w=x+iy\)?

- 2bornot2b

@Ishaan94, Yes.

- anonymous

Does the fact that w and z are complex numbers restrict the use of euler's identity?

- experimentX

why \( (e^w)^z \not = e^{wz} \) ???

- 2bornot2b

I don't think so @Ishaan94
@experimentX see bullet three after visiting the following link http://en.wikipedia.org/wiki/Exponentiation#Failure_of_power_and_logarithm_identities

- 2bornot2b

Certainly you can use euler's identity

- anonymous

I know that \(e^{x} =e^{y}\) doesn't imply that x equals y, but I still don't understand it. Hmm okay maybe I know why. \(e^{i\theta} = \cos \theta +i\sin \theta\).
\[\sin 30 = \sin150,\,\text{but}\, 150\neq30.\]\[\sin \theta=\sin (2n\pi + \theta)\]

- anonymous

You can only have one unique value of theta in a period by adding 2npi you change the period. Or, you complete the circle and reach the point where you started from.

- 2bornot2b

But what does it have to do with this complex identity ?

- anonymous

hmm, maybe you have to go back to the definition in terms of complex log

- 2bornot2b

@estudier Do you mean the definition of \(a^b\) where both a and b are complex?

- anonymous

Problem seems to be you cannot equate a multi value f with a single value f....

- anonymous

A lot. This is how you represent a complex number in complex plane. |dw:1336309768854:dw|You can only get the same theta value if you complete circle.

- anonymous

z^az^b = e^(alnz)e^(blnz) = e^a(lnz +2pi i n) etc

- 2bornot2b

@estudier Let me tell you the problem I am facing. I was going through the definition of \(a^b\) which is defined as \(e^{b Loga}\). Now I already know that \(a^b=(e^{Loga})^b\) Does that mean \(e^{Loga~b}=(e^{Loga})^b\)

- anonymous

Then z^(a+b) = e^(a+b) ln z = e^(a+b)( ln z + 2 pi i k) = e^(a+b)ln ze^2pi i k(a+b) ->
z^(a+b) is in z^az^b (might be same might not)

- anonymous

I really don't like thi stuff.........:-(

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