Question 92522This question is from textbook The Road to Reality
: I find it difficult to grasp the idea of 'x raised to the power i (or multiple of i', or indeed 'x raised to the power of a general complex number'. I can understand 2^3, 3^0.5 etc., but what is the meaning of, say, 3^i, 3^(2+3i) etc. What are, say, '3^i', '3^(2+3i)', and why? How does this relate to Euler's formula?
The problem comes from reading 'The Road to Reality' by Roger Penrose, ISBN no 0-224-04447-8, p90 (section 5.2 entitled 'The Idea of the Complex Logarithm'). Real logarithms - OK: complex ones I just can't quite grasp.
This question is from textbook The Road to Reality
Answer by Nate(3500)   (Show Source):
You can put this solution on YOUR website! 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)
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