Case: April 2009

e (mathematical constant) - Wikipedia, the free encyclopedia:

The exponential function e^x may be written as a Taylor series

$e^{x} = 1 + {x \over 1!} + {x^{2} \over 2!} + {x^{3} \over 3!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

Because this series keeps many important properties for e^x even when x is complex, it is commonly used to extend the definition of e^x to the complex numbers. This, with the Taylor series for sin and cos x, allows one to derive Euler's formula:

$e^{ix} = \cos x + i\sin x,\,\!$

which holds for all x. The special case with x = π is known as Euler's identity:

$e^{i\pi}+1 =0 .\,\!$

Consequently,

$e^{i\pi}=$

from which it follows that, in the principal branch of the logarithm,

$\log_e (-1) = i\pi.\,\!$

Furthermore, using the laws for exponentiation,

$(\cos x + i\sin x)^n = \left(e^{ix}\right)^n = e^{inx} = \cos (nx) + i \sin (nx),$

which is de Moivre's formula.

The case,

$\cos (x) + i \sin (x)\,\!$

is commonly referred to as Cis(x).

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