Indefinite Integral of $\dfrac {x^{p-1}}{x^{2m} - a^{2m}}$

Question

I am looking at a bunch of related integrals found in Spiegel's "Mathematical Handbook of Formulas and Tables", (Schaum, 1968), item $14.336$.

It's a complicated and messy old beast:

For $m \in \mathbb Z$ such that $m \ge 1$:

$\displaystyle \int \dfrac {x^{p-1} \, \mathrm d x}{x^{2m} - a^{2m}} = \frac 1 {2 m a^{2 m - p}} \sum_{k=1}^m \cos \frac {k p \pi} m \ln \left({x^2 - 2 a x \cos \frac {k \pi} m + a^2}\right)$

$\displaystyle - \frac 1 {m a^{2 m - p}} \sum_{k=1}^m \sin \frac {k p \pi} m \arctan \left({\frac {x - a \cos \frac {k \pi} m}{a \sin \frac {k \pi} m}}\right)$

$\displaystyle + \frac 1 {2 m a^{2 m - p}} (\ln (x - a) + (-1)^p \ln (x + a)) + C$

for all $0 < p \le 2 m$.

(Note that he glosses over the sign of $x$, presenting the expression for positive $x$ only -- I will explore the negative $x$ case later.)

I thought of $2$ different approaches:

1. Factorise the denominator using the standard result:

$\displaystyle x^{2 n} - y^{2 n} = (x - y) (x + y) \prod_{k \mathop = 1}^{n - 1} \left({x^2 - 2 x y \cos \dfrac {k \pi} n + y^2}\right)$

and then explore the possibility of doing a partial fraction expansion. However, this did not work out so well, as I was then unable to see how to simplify the expression on the top of the resulting terms in all but the $x - a$ and $x + a$ denominators, so I didn't proceed further with that.

2. Prove it by induction on $m$, which first concerns establishing the base case:

$\displaystyle \int \frac {\mathrm d x} {x^2 - a^2} = \frac 1 {2 a} \ln \left({\dfrac {x - a} {x + a}}\right) + C$ (that is: $m = 1$, $p = 1$)

$\displaystyle \int \frac {x \, \mathrm d x} {x^2 - a^2} = \ln \left({x^2 - a^2}\right) + C$ (that is: $m = 1$, $p = 2$)

But when I put $m = 1$ into the above expression, I get:

$\dfrac 1 {2 a^{2 - p} } ( (-1)^p \ln (x^2 + 2 a x + a^2) ) + \dfrac 1 {2 a^{2 - p} } (\ln (x - a) + (-1)^p \ln (x + a) )$

But that pesky term on the left, in $(-1)^p \ln (x^2 + 2 a x + a^2)$ is clearly incorrect.

This arises from the $\cos \dfrac {k p \pi} m \ln \left({x^2 - 2 a x \cos \dfrac {k \pi} m + a^2}\right)$ expression in the given solution.

But for integer $p$ and $m = 1$, that $\cos \dfrac {k p \pi} m$ term, which I would hope to vanish, defiantly remains in place.

It doesn't help much changing it to $\cos \dfrac {k p \pi} {2 m}$, because this time it does not vanish for $p = 2$.

So either Spiegel has reported this result wrong, or I'm misunderstanding something.

Should that cosine in fact be a sine? Then it would vanish away like I want it to.

For $m = 2$ and $p = 1$ you get this:

$\dfrac 1 {4 a^3} \ln \left( {\dfrac {x - a} {x + a} }\right) - \dfrac 1 {2 a^3} \arctan \dfrac x a$

which is consistent with the result quoted. I have gone a little further, trying $p = 2, 3, 4$, but now it is looking as though the cosine term is completely erroneous and perhaps should not be there at all, because any contribution it makes is consistently absent.

I have gone no further with higher $m$ values.

Answer 1

The sane way to do this is using complex functions. The integrand is a rational function with simple poles at $x = \omega_k a$ where $\omega_k =e^{\pi i k /m} $, $k = 0,1,\ldots, 2m-1$ are the $2m$'th roots of unity:

$$ f(x) = \frac{x^{p-1}}{x^{2m} - a^{2m}} = \frac{x^{p-1}}{\prod_{k=0}^{2m-1} (x - \omega_k a)}$$

Its residue at $x=\omega_k$, is then $$\text{res}(f, \omega_k a) = \frac{(\omega_k a)^{p - 2m}}{2m}$$ so that $$f(x) = \sum_{k=0}^{2m-1} \frac{(\omega_k a)^{p-2m}}{2m (x - \omega_k a)} $$ which has an indefinite integral $$ F(x) = \sum_{k=0}^{2m-1} \frac{(\omega_k a)^{p-2m}}{2m} \log (x - \omega_k a) $$ For positive real $x$ and $a$ we can write $$\log(x - \omega_k a) = \frac{1}{2} \log(x^2 - 2a \cos(2\pi k /m) x + a^2) - i \arctan \left(\frac{a \sin(2 \pi k/m)}{x - a \cos(2\pi k /m)}\right)$$ $$ (\omega_k a)^{p-2m} = a^{p-2m} \left(\cos(p k \pi/m) + i \sin(p k \pi/m)\right)$$ and the formula follows.

Answer 2

There is a typo in the formula

\begin{align} I(m,p) =\int \dfrac {x^{p-1} \, \mathrm d x}{x^{2m} - a^{2m}} & = \frac 1 {2 m a^{2 m - p}} \sum_{k=1}^{m-1} \cos \frac {k p \pi} m \ln \left({x^2 - 2 a x \cos \frac {k \pi} m + a^2}\right)\\ & \>\>\>\>\>- \frac 1 {m a^{2 m - p}} \sum_{k=1}^{m-1}\sin \frac {k p \pi} m \arctan \left({\frac {x - a \cos \frac {k \pi} m}{a \sin \frac {k \pi} m}}\right)\\ & \>\>\>\>\>+ \frac 1 {2 m a^{2 m - p}} (\ln (x - a) + (-1)^p \ln (x + a)) \end{align} where the summation should run from $1$ to $m-1$, instead of $m$. Then, the followings check out $$I(1,1)=\int \frac {\mathrm d x} {x^2 - a^2} = \frac 1 {2 a} \ln \left({\dfrac {x - a} {x + a}}\right)$$

$$I(1,2)=\int \frac {x \, \mathrm d x} {x^2 - a^2} = \ln \left({x^2 - a^2}\right)$$

$$I(2,1)=\int \frac {\, \mathrm d x} {x^4 - a^4} = \dfrac 1 {4 a^3} \ln \left( {\dfrac {x - a} {x + a} }\right) - \dfrac 1 {2 a^3} \arctan \dfrac x a$$