document.write( "Question 92522This question is from textbook The Road to Reality
\n" ); document.write( ": 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? \r
\n" ); document.write( "\n" ); document.write( "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. \n" ); document.write( "
Algebra.Com's Answer #67358 by Nate(3500)\"\" \"About 
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.
\n" ); document.write( "While e^(xi) is given, we can manipulate it.
\n" ); document.write( "e^(xi)
\n" ); document.write( "(e^x)^i
\n" ); document.write( "Where as: e^x = 3 ~> x = ln(3)
\n" ); document.write( "Then:
\n" ); document.write( "e^(ln(3)*i) = cos(ln(3)) + sin(ln(3))i
\n" ); document.write( "~~~~
\n" ); document.write( "This relates to Euler's formula greatly as you may see....
\n" ); document.write( "This may help you:
\n" ); document.write( "Suggest that: p = cos(x) + sin(x)i
\n" ); document.write( "dp / dx = -sin(x) + cos(x)i
\n" ); document.write( "dp / dx = i^2*sin(x) + cos(x)i
\n" ); document.write( "dp / dx = [i*sin(x) + cos(x)]i
\n" ); document.write( "dp / dx = pi
\n" ); document.write( "(1 / p) dp / dx = i
\n" ); document.write( "ln(p) = xi + C
\n" ); document.write( "p = e^(xi + C)
\n" ); document.write( "cos(x) + sin(x)i = e^(xi + C)
\n" ); document.write( "cos(x) + sin(x)i = e^(xi) ~~~> using (0,1) as in (x,p) \n" ); document.write( "
\n" );