Integral identities and asymptotic calculations

For real $\mu ,\rho$ and $s>|\mu |+|\rho |$,

One derivation inserts the Schwinger representations for the two $K$-functions, integrates the Gaussian power of $x$, then sets the two positive Schwinger parameters equal to $tv,t(1-v)$. The $t$-integral is a gamma integral and the $v$-integral is a beta integral. The stated condition justifies absolute convergence. This also provides a direct proof of the identity used in the asymptotic sections; see Watson,GR,DLMF for the standard special-function framework.

In particular,

and at $\nu =0$ the continuous limit is $1/2$. Differentiation in $s$ at $s=2,\mu =\rho =0$ gives

For derivative moments, integration by parts gives, whenever the endpoint terms vanish,

The Bessel equation then reduces $K^{\prime 2}$ moments to the same Mellin family. These operations yield the coefficients of Theorem thm:large without an appeal to a symbolic run.

-data normalization. The identities

are the normalization checks behind (eq:Cfinite). They prevent a second angular normalization factor from being inserted into (eq:dG). That equation uses unnormalized exponentials in its displayed sum; (eq:resolvent) uses the normalized ${\psi }_j$'s.