Question

according to eulers thorem the complex number eix can be written as
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Answer 1

Answer:

Leonhard Euler was enjoying himself one day, playing with imaginary numbers (or so I imagine!), and he took this well known Taylor Series (read about those, they are fascinating):

ex = 1 + x + x22! + x33! + x44! + x55! + ...

And he put i into it:

eix = 1 + ix + (ix)22! + (ix)33! + (ix)44! + (ix)55! + ...

And because i2 = −1, it simplifies to:

eix = 1 + ix − x22! − ix33! + x44! + ix55! − ...

Now group all the i terms at the end:

eix = ( 1 − x22! + x44! − ... ) + i( x − x33! + x55! − ... )

And here is the miracle ... the two groups are actually the Taylor Series for cos and sin:

cos x = 1 − x22! + x44! − ...

sin x = x − x33! + x55! − ...

And so it simplifies to:

eix = cos x + i sin x

He must have been so happy when he discovered this!

And it is now called Euler's Formula

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