Raising a negative real base to an irrational number. Example: y = ( &#x2212;<!-- − -

herbariak1

herbariak1

Answered question

2022-05-19

Raising a negative real base to an irrational number.
Example:
y = ( 2 ) π

Answer & Explanation

bideanbarrenaf5

bideanbarrenaf5

Beginner2022-05-20Added 8 answers

The definition used in complex analysis is a b = exp ( b log ( a ) ) where log ( a ) is any branch of the complex logarithm of a. Thus this is a multivalued function.

Thus in your example, log ( 2 ) = ln ( 2 ) + i ( 2 n + 1 ) π where n is an arbitrary integer, and so
( 2 ) π = exp ( π ( ln ( 2 ) + i ( 2 n + 1 ) π ) ) = 2 π ( cos ( ( 2 n + 1 ) π 2 ) + i sin ( ( 2 n + 1 ) π 2 ) )

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