exponential function related to trig??? (1 Viewer)

[freaking_out]

i heard that the exponential/logarithmic functions are related to trig. i know that this is extension stuff, but i am curious as to how they are related?

 

[Affinity]

some relations:

you might need to read up something about power series.

1. Maclaurin Series:

Exp[x] = 1/0! + x/1! + x^2/2! + ... + x^n/n! + ...
Cos[x] = 1/0! - x^2/2! + x^4/4! - x^6/6! + ...
Sin[x] = x - x^3/3! + x^5/5! - x^7/7! + ...

and from these you can see that

Exp[ix] = Cos[x] + i*Sin[x] where i = sqrt(-1)

and

Cos[x] = (Exp[ix] + Exp[-ix])/2
Sin[x] = (Exp[ix] - Exp[-ix])/2i

I vaguely remember some more relations are mentioned in an analysis textbook.

 

[McLake]

And a little extension:

e^(i*pi) = -1

 

[Affinity]

Suggest reading:
Any calculus textbook that covers power series / any analysis textbook.
go through "mean-value theorem" , then "taylor series"