# Understanding a formula

• Dec 27th 2008, 01:17 PM
Truthbetold
Understanding a formula
i^-i= $\displaystyle \sqrt{e^\pi}$

I read here, The number e, that this formula is not understood.

"In 1864 Benjamin Peirce had his picture taken standing in front of a blackboard on which he had written the formula above. In his lectures he would say to his students:-
Gentlemen, we have not the slightest idea what this equation means, but we may be sure that it means something very important.:
Is there a meaning to this equation?

I tried to understand it with my calculator, but failed horribly.
i^-i= $\displaystyle \sqrt{e^\pi} =about 4.787$
With my limited understanding of i, that makes absolutely sense to me whatsoever.

Thanks!
• Dec 27th 2008, 01:41 PM
Jhevon
Quote:

Originally Posted by Truthbetold
i^-i= $\displaystyle \sqrt{e^\pi}$

I read here, The number e, that this formula is not understood.

"In 1864 Benjamin Peirce had his picture taken standing in front of a blackboard on which he had written the formula above. In his lectures he would say to his students:-
Gentlemen, we have not the slightest idea what this equation means, but we may be sure that it means something very important.:
Is there a meaning to this equation?

I tried to understand it with my calculator, but failed horribly.
i^-i= $\displaystyle \sqrt{e^\pi} =about 4.787$
With my limited understanding of i, that makes absolutely sense to me whatsoever.

are you familiar with Euler's formula? $\displaystyle e^{i \theta} = \cos \theta + i \sin \theta$.
with that we can see that $\displaystyle i = e^{i(\pi / 2 + 2 n \pi)}$ for any integer $\displaystyle n$. since all will give the same value, we can take one, $\displaystyle e^{i \pi / 2}$ is the nicest.
thus we have, $\displaystyle i^{-i} = (e^{i \pi / 2})^{-i} = e^{\pi / 2} = (e^\pi)^{1/2} = \sqrt{e^\pi}$