Friday, July 15, 2011

Verification

I posted two derivations for Euler's famous equation (here and here):

eix = cos x + i sin x

This can be verified in a particularly simple way.
The series representation of the exponential function:

ex =  1 +  x + x2/2! +  x3/3! + x4/4! + ..

is especially neat because each term in the series is the derivative of the term following, and the result of that is:

d/dx ex = ex

Which is, indeed, one definition of this function.
Substitution of ix leads to a simple shift in the pattern:

eix = 1 + ix - x2/2! - ix3/3! + x4/4! + ..

repeating with a period of 4. Powers of x like:

4n + 1 are multiplied by  i
4n + 2 -1
4n + 3 -i
4n 1

But remembering the series for sine and cosine, and multiplying the former by i:

  sin x =  x -  x3/3! + x5/5! - x7/7! + ..
i sin x = ix - ix3/3! + ..
cos x = 1 - x2/2! + x4/4! - x6/6! + ..

Adding:

cos x + i sin x = 1 + ix - x2/2! - ix3/3!+ x4/4! + ..
= eix

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