Moivre-Laplace Formula
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Moivre-Laplace formula
exp(ix) = cos(x) + i sin(x),
where i = (-1)^(1/2) , and which is widely used in different items of mathematics is usually deduced from the Maclaurin expansions of the functions involved.
But the theory of Taylor (Maclaurin) expansions is a part of more general theory developed in the course of the functions of complex variable. As the Moivre-Laplace formula has numerous applications outside this theory, it seems reasonable to deduce it without references to Maclaurin series.
Problem.
To prove the Moivre-Laplace formula: exp(ix) = cos(x) + i sin(x) without use of the Maclaurin expansions.
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Solution
Moivre-Laplace formula
e ix = cos(x) + i sin(x),
where i = , and which is widely used in different items of mathematics is usually deduced from the Maclaurin expansions of the functions involved:
cos(x) = ; sin(x) = .
As -1 = i2 , we have:
cos(x) + i sin(x) = + and the sum of these two series can obviously be written as a single series
= ,
and the latter series is just the Maclaurin expansion of e ix .
But the theory of Taylor (Maclaurin) expansions is a part of more general ...
Solution Summary
The Moivre-Laplace Formula is prven without the use of Maclaurin expansions.