(formula relating exponentials to trigonometric functions)
Euler's formula is an equation worked out by mathematician Leonhard
Euler regarding complex numbers that is instrumental to
the applied mathematics of physics, engineering, and related disciplines.
It relates exponentials to trigonometric functions, yielding a means
to convert an equation of trigonometric functions into an equation of
exponential functions, and vice versa, thus giving the mathematician
the choice of the methods of working with an equation, such as
solving for some variable of interest or calculating with the
function. The formula:
eix = cos x + i sin x
e - Euler's number, a particular irrational number that begins 2.7182.... (It is specifically the unique real number such that the function f(x) = ex yields its own slope.)
i - the square root of minus 1.
sin, cos - trigonometric functions.
x - any real number.
The formula was worked out using series expansions, a method of
comparing such functions and finding out how they are related, which
showed that given the notion of i, this formula produced
consistent and useful results. Among the formula's implications is
this equation:
eiπ = -1
or
eiπ + 1 = 0
This latter form of this equation is known as Euler's identity.