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