2010-05-26, 12:41 AM
some power formulae:
a^i = (e^ln(a))^i = e^ln(a)i = cos(ln(a)) + isin(ln(a))
also, another thing - the mathematical rule that says (ab)^c = (a^c)(b^c) doesn't work with i. If it did, this is what would happen:
1^i = (-i^2)^i = (-1*i^2)^i = ((-1)^i)(i^2i) = (i^2i)(i^2i) = i^4i = (e^ipi/2)^4i = e^-2pi
this causes major problems with other mathematical rules, as well as the one above which states that:
1^i = cos(ln(1)) + isin(ln(1)) = cos(0) +isin(0) = 1
a^i = (e^ln(a))^i = e^ln(a)i = cos(ln(a)) + isin(ln(a))
also, another thing - the mathematical rule that says (ab)^c = (a^c)(b^c) doesn't work with i. If it did, this is what would happen:
1^i = (-i^2)^i = (-1*i^2)^i = ((-1)^i)(i^2i) = (i^2i)(i^2i) = i^4i = (e^ipi/2)^4i = e^-2pi
this causes major problems with other mathematical rules, as well as the one above which states that:
1^i = cos(ln(1)) + isin(ln(1)) = cos(0) +isin(0) = 1