Here's the secret though:
This formula can be interpreted as saying that the function
eix traces out the
unit circle in the
complex number plane as
x ranges through the real numbers. Here,
x is the
angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in
radians.
The original proof is based on the
Taylor series expansions of the
exponential function ez (where
z is a complex number) and of sin
x and cos
x for real numbers
x (see below). In fact, the same proof shows that Euler's formula is even valid for all
complex numbers
z.
A point in the
complex plane can be represented by a complex number written in
cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and
polar coordinates. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number
z =
x +
iy can be written as
http://upload.wikimedia.org/wikipedi...67a6bedb76.pnghttp://upload.wikimedia.org/wikipedi...b96a58c195.png where
http://upload.wikimedia.org/wikipedi...f135acbbf4.png the real part
http://upload.wikimedia.org/wikipedi...18a5f43fbb.png the imaginary part
http://upload.wikimedia.org/wikipedi...f6a2154e1d.png the
magnitude of
zhttp://upload.wikimedia.org/wikipedi...ae4008bf0f.png atan2(
y,
x) .
http://upload.wikimedia.org/wikipedi...8d7cc238fb.png is the
argument of
z—i.e., the angle between the
x axis and the vector
z measured counterclockwise and in
radians—which is defined
up to addition of 2π. Many texts write tan−1(
y/
x) instead of atan2(
y,
x) but this needs adjustment when
x ≤ 0.
Now, taking this derived formula, we can use Euler's formula to define the
logarithm of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation) that
http://upload.wikimedia.org/wikipedi...b27f237d89.png and that
http://upload.wikimedia.org/wikipedi...84b7c7aeb8.png both valid for any complex numbers
a and
b.
Therefore, one can write:
http://upload.wikimedia.org/wikipedi...f62e271d33.png for any
z ≠ 0. Taking the logarithm of both sides shows that:
http://upload.wikimedia.org/wikipedi...04019c4334.png and in fact this can be used as the definition for the
complex logarithm. The logarithm of a complex number is thus a
multi-valued function, because
http://upload.wikimedia.org/wikipedi...f356f63e04.png is multi-valued.
Finally, the other exponential law
http://upload.wikimedia.org/wikipedi...fd2cd505c9.png which can be seen to hold for all integers
k, together with Euler's formula, implies several
trigonometric identities as well as
de Moivre's formula.
That's the shortened version, just one play though.