Euler defined the exponential function for
 complex numbers
and discovered it's relationship to the 
trigonometric functions
for any real number
Euler's formula states that the complex 
exponential function
It can be set using the following formula
...as a special case of formula, which is 
known as Euler's Identity
Euler's formula was published in 1748 by
in his book Introductio in analtsin 
Infinitorum, section 138
about how the imaginary exponential
are expressed in terms of sine and cosine
this formula was described by Richard Feynman
as the most remarkable formula in mathematics
it lists the main algebraic operations
with the constants...
zero
one
the number and it is also known 
as Euler's number
the imaginary unit and use 
the greek letter pi
to refer to the ratio between the length of the circunference and the length of it's diameter
in 1988 readers of the journal 
Mathematicall Intelligencer threw the formula 
as the most beautiful formula in history
we can this identity by a method 
of development of functions in power series
the sine function of x
function of the cosine of x
and the exponential
we see that the exponential function
 is elevated to a variable named x
if we equate tha variable x to the 
complex number i by pi
then we have an exponential 
series high i for pi
the terms of the exponential 
series we are in manner
complex units that have an exponent 
greater than can be reemplaced
since i squared equals minus one
I can change the square by a minus sign
the i cubed as i we have 
the i to the fourth gate 
and i to the fifth remains i
as the series is a sum of the terms we use 
the commutativity of addition to 
rearrange the terms
and also use the associative 
property to group terms
you note that we have factored the 
complex unit i
describes the first association cosine function
the second describes the function of
 association within
simplifying we obtain the following equality
know that the sine of pi is equal to zero
equality is now reduced to the 
following expression
where the cosine of pi is equal to minus 
one and the sine of pi is equal to zero 
then we have e raised i by pi equals minus one
we have finally obtained the Euler identity
this expression expresses a few mathematical 
symbols infinite beauty worthy of a 
genius like Euler
after his death in 1783 was an ambitious 
proyect to publish all of his scientific work
composed of more than 800 treaties, 
which makes it the most prolific 
mathematician in history
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