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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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