The number is the base of the power and is its exponent.
Splitting the exponent in its real and imaginary parts one obtains
This shows that both the modulus and the argument (http://planetmath.org/Complex) of the general power are in general multivalued. The modulus is unique only if , i.e. if the exponent is real; in this case we have
Let . If one lets the point go round the origin anticlockwise, gets an addition and hence the has been multiplied by a having the modulus , and we may say that has come to a new branch.
, where is a positive integer, coincides with the root (http://planetmath.org/CalculatingTheNthRootsOfAComplexNumber) of .
(with ); all these values are positive real numbers, the simplest of them is .
(with ) also are situated on the positive real axis.
(with ); all these are (meaning here that their imaginary parts are distinct from 0), situated on the circumference of the unit circle such that all points of the circumference are accumulation points of the sequence of the (see this entry (http://planetmath.org/SequenceAccumulatingEverywhereIn11)).
(with ), are situated on the half line beginning from the origin with the argument radians.
|Date of creation||2013-03-22 14:43:17|
|Last modified on||2013-03-22 14:43:17|
|Last modified by||pahio (2872)|
|Defines||base of the power|