proof

**Mhmh96** · March 1st 2012, 01:03 PM

**Plato** · March 1st 2012, 01:19 PM

To prove this you need to know the principal value of a complex exponential.
$z^w=\exp(w\cdot\text{Log}(z))$

Recall that $\text{Log}(i)=\ln(1)+\frac{i\pi}{2}.$

**Mhmh96** · March 1st 2012, 01:47 PM

Ok,but how can i start?

**Plato** · March 1st 2012, 02:11 PM

Originally Posted by Mhmh96

Ok,but how can i start?

What do you mean by that?

What does $i\text{Log}(i)=~?$ .

Then what is $\exp(i\text{Log}(i))~?$

**Mhmh96** · March 1st 2012, 02:19 PM

Yes, i have no background about how to prove it so i want to know the proof in details ,if that ok.

**Plato** · March 1st 2012, 02:28 PM

Originally Posted by Mhmh96

i have no background about how to prove it so i want to know the proof in details ,if that ok.

Well that is not going to happen here.

Because if you have have no background then you would not begin to understand the proof.

**Mhmh96** · March 1st 2012, 02:34 PM

I guess that is right,thanks anyway.

**Mhmh96** · March 2nd 2012, 07:34 AM

I think i found the proof ,but i still have one more problem

when we raise i to the power i ,the result should be real number,right?

**Soroban** · March 4th 2012, 07:01 AM

Hello, Mhmh96!

$\text{Prove: }\:i^i \:=\:\dfrac{1}{\sqrt{e^{\pi}}}$

$\text{Let }\,z \:=\:i^i$

$\text{Take logs: }\:\ln(z) \:=\:\ln(i^i) \quad\Rightarrow\quad \ln(z)\:=\:i \ln(i) \;\;{\bf[1]}$

$\text{Given: }\:e^{i\theta} \:=\:\cos\theta + i\sin\theta$

$\text{Let }\theta = \tfrac{\pi}{2}\!:\;\;e^{i\frac{\pi}{2}} \;=\;\cos\tfrac{\pi}{2} + i\sin\tfrac{\pi}{2} \quad\Rightarrow\quad e^{i\frac{\pi}{2}} \:=\:i$

$\text{Take logs: }\:\ln\left(e^{i\frac{\pi}{2}}\right) \:=\:\ln(i) \quad\Rightarrow\quad i\tfrac{\pi}{2}\ln(e) \:=\:\ln(i)$

. . $\text{Hence: }\:\ln(i) \:=\:i\tfrac{\pi}{2}$

$\text{Substitute into }{\bf[1]}\!:\;\;\ln(z) \:=\:i\left(i\tfrac{\pi}{2}\right) \quad\Rightarrow\quad \ln(z)\:=\: -\tfrac{\pi}{2}$

$\text{Therefore: }\:z \;=\;e^{-\frac{\pi}{2}} \;=\;\dfrac{1}{e^{\frac{\pi}{2}}} \;=\; \dfrac{1}{(e^{\pi})^{\frac{1}{2}}} \;=\;\dfrac{1}{\sqrt{e^{\pi}}}$

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