DeMoivre's Theorem

[Steven G]

Euler's formula states that e^(ix) = cos(x) + i*sin(x) for all x

Now I let x=nt.

On one hand e^(i*nt) = cos(nt) + i*sin(nt), while on the other hand I get

e^(i*nt) = (e^(i*t))^n = (cos(t) + i*sin(t))^n.

Now these two last equations are valid for all x (including x=nt)

Equating the two results we get (cos(t) + i*sin(t))^n = cos(nt) + i*sin(nt) for all nt.

But this last equation is only valid for n being an integer. Where did this extra constraint come from??

 

[Dr.Peterson]

See https://en.wikipedia.org/wiki/De_Moivre's_formula#Failure_for_non-integer_powers,_and_generalization

The problem is that e^(i*nt) = (e^(i*t))^n is not true for non-integer n, simply because a non-integer power (the RHS) is multi-valued.