Question 92522
Euler's formula is based upon the natural number, e. Also, the formula is really the only explanation of having a complex power.
While e^(xi) is given, we can manipulate it.
e^(xi)
(e^x)^i
Where as: e^x = 3 ~> x = ln(3)
Then:
e^(ln(3)*i) = cos(ln(3)) + sin(ln(3))i
~~~~
This relates to Euler's formula greatly as you may see....
This may help you:
Suggest that: p = cos(x) + sin(x)i
dp / dx = -sin(x) + cos(x)i
dp / dx = i^2*sin(x) + cos(x)i
dp / dx = [i*sin(x) + cos(x)]i
dp / dx = pi
(1 / p) dp / dx = i
ln(p) = xi + C
p = e^(xi + C)
cos(x) + sin(x)i = e^(xi + C)
cos(x) + sin(x)i = e^(xi) ~~~> using (0,1) as in (x,p)