Identity

Makaveli

I`ve seen somewhere that e^(i*pi/2)=i
Is thes the only identity of this kind concerning these variables or there are others?(which may be useful when uing complex numbers in geometry)

**Posted:** Sat Mar 11, 2006 10:03 am

mathmanman

Well, there are :
$\bullet e^{i0} = 1$
$\bullet e^{2i\pi} = 1$
$\bullet e^{3i\frac{\pi}2} = -i$
$\bullet e^{2ik\pi} = 1$ ( $k \in \mathbb{N}$ )

And, the so-known... : $\boxed{e^{i\pi} = -1}$

Smile

**Posted:** Sat Mar 11, 2006 12:33 pm

boxedexe

I wouldn't consider these identities. Just use $e^{i\theta}=\cos\theta+i\sin\theta$ . Smile

Masoud Zargar

**Posted:** Sat Mar 11, 2006 1:58 pm

t0rajir0u

The expression $re^{i\theta}$ gives the vector in the complex plane with length $r$ and angle $\theta$ - hence, it has an $x$ (real) component of $r \cos \theta$ and a $y$ (imaginary) component of $r \sin \theta$ , thus:

$re^{i \theta} = r \left( \cos \theta + i \sin \theta \right)$

The fact that exponentials are being used in a clearly non-exponential way is difficult to justify without calculus.

Applications of this theorem:

$\left( re^{i \theta} \right)^n = r^n \left( \cos \theta + i \sin \theta \right)^n = r^n e^{i n \theta} = r^n \left( \cos n\th...$

This is a very useful property of complex numbers - it tells you that when you have a complex number $z$ in the complex plane, taking it to some power $n$ is equivalent to multiplying its angle by $n$ and taking its absolute value to the power of $n$ . Very powerful.

This theorem can also be used to extend $\cos, \sin$ to complex arguments, $\arccos, \arcsin$ to values larger than $1$ , and $\ln$ to negative and complex arguments. Smile

**Posted:** Sat Mar 11, 2006 5:07 pm

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