What is the definition of a^i?

1. Dec 11, 2004

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" ??

2. Dec 11, 2004

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. Dec 12, 2004

3.14lwy

thank you!

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

4. Dec 12, 2004

Muzza

Yes, it is.

5. Dec 12, 2004

HallsofIvy

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