A question about complex number and e

SUPERMAN2

I don't know how we have this equality:
$e^{\frac{i\pi}{2}}=i$
Please help.May be you can give me the proof or hint.Thank you very much.

**Posted:** Thu May 21, 2009 3:20 am

boxedexe

It directly follows by the well-known Euler formula $e^{ix} = \cos x + i\sin x$ , which can be proved using power series (or the definition of sine and cosine) or integration. See this for a brief discussion of this formula.

**Posted:** Thu May 21, 2009 4:25 am

Chgk

Here is a nice, short proof that uses differential equations:

Observe that $f(z) = i sin(iz) + cos(iz)$ satisfies
$f(z) + f'(z) = 0$ as well as the initial value condition $f(0) = 1$
It can be proven that any function defined over C and that satifies both of these conditions must be equal to $e^{ - z}$

Hence $e^{ - z} = f(z) = i sin(iz) + cos(iz)$ Replace $x = iz$ and obtain Euler's equality.

**Posted:** Fri May 22, 2009 12:33 pm

t0rajir0u

See my comments about rotation matrices here.

**Posted:** Fri May 22, 2009 12:43 pm

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