What is the definition of a^i?

[3.14lwy]

(1)

what is the definition of a^i ? (where a is a real number , i = (-1)^(1/2) )

is this still a complex number ?




(2)

where can I find the prove of "a n degree polynomial has n roots" ??

 

[robert Ihnot]

1) a^i =e^(i(log(a)))=cos(log(a)) + isin(log(a))

2) A proof of the fundamental theorem of algebra, first formulated by Gauss is very difficult. A simpler proof can be found by the time you get into the study of Complex Functions, generally a Junior college study.

 

[3.14lwy]

thank you!

then , is i^i = e^[(i)*(pi/2)*(i)] = e^(-pi/2) ??

 

[Muzza]

Yes, it is.

 

[HallsofIvy]

With the proviso that aⁱ is, like most complex functions, not single valued.