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The question is when the following integral converges or diverges:

$\int_{-1}^{0} x^{-p}$ for $p > 0$.

I am stuck at the case where $0 < p < 1$, where I end up with

$$\frac{1}{1-p} - \frac{(-1)^{1-p}}{1-p}$$

I am not sure how to split this funciton. should it be from 0 < p < 1 or p <= 1. I cant tell what to do at the value '1'.

AlphaDJog
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1 Answers1

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The problem with non-integer powers of negative numbers is that you'll need to use complex numbers.

Also, in the complex numbers, $(-1)^z$ for a given value of $z$ is a multi-valued function.

You could always choose one particular value and, for example, represent $(-1)^p$ as $e^{i \pi p}$.

Robby the Belgian
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