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I've been dealing with a really mind buster, at least, for me.

Here it is

[tex]I_n(r,z)=\int_z^r\frac{p^n}{\sqrt{r^2-p^2}\sqrt{p^2-z^2}}\,dp[/tex]

where [tex]n[/tex] is an integer and [tex]0<z<r<1[/tex].

Mathematica tells me that the result is zero, but I'd like to know how to get there.

I've thought about elliptical integrals but Abramowitz doesn't help much beyond telling me that the Equivalent Inverse Jacobian Elliptic Function of

[tex]a\int_b^x\frac{dt}{\sqrt{(a^2-t^2)(t^2-b^2)}}[/tex]

is

[tex]nd^{-1}\left(\frac{x}{b}|\frac{a^2-b^2}{a^2}\right)[/tex]

and

[tex]a\int_x^a\frac{dt}{\sqrt{(a^2-t^2)(t^2-b^2)}}[/tex]

is

[tex]dn^{-1}\left(\frac{x}{a}|\frac{a^2-b^2}{a^2}\right)[/tex]

Whatever this means. I know nothing about these functions. nd and dn?

Any help?

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# One tough integral

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