Re: negative base raised to fractional exponent

On Tue, 16 Oct 2007 21:27:31 -0000, schaefer.mp@xxxxxxxxx wrote:

Does anyone know of an approximation to raising a negative base to a
fractional exponent? For example,

(-3)^-4.11111

since this cannot be computed without using imaginary numbers. The
approximation is needed in the context of a computer program. Any help
is appreciated.

So I guess your implying you don't have math library routines that
handle complex numbers?


Don't know if this is what you're looking for, but if you write -3 in
polar form it's

(-3 + i 0) = sqrt[ (-3)^2 + 0^2 ] e^[ i pi ] = 3 e^(i pi)

Raising that to the power -4.11111 gives

3^(-4.11111) e^(i -4.11111 pi)

= 3^(-4.11111)[ cos(-4.11111 pi) + i sin(-4.11111 pi) ]

= 0.0102680602... - i 0.00373722769...

The above is the so-called principal value. No complex number
arithmetic is required in the evaluation, but of course you do need
some standard math library routines.


If you know that all of your cases will be of the form

(-x)^y where x and y are real and x > 0,

then you have x^y * (-1)^y = x^y e^(i pi y)

which is

x^y * [ cos(pi y) + i sin(pi y) ]

.

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