Steven G
Elite Member
- Joined
- Dec 30, 2014
- Messages
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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??
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??