  For integral representations of associated Legendre functions in terms of
modified Bessel functions, we establish justification for differentiation under
the integral sign with respect to parameters. With this justification,
derivatives for associated Legendre functions of the first and second kind with
respect to the degree are evaluated at odd-half-integer degrees, for general
complex-orders, and derivatives with respect to the order are evaluated at
integer-orders, for general complex-degrees. We also discuss the properties of
the complex function $f:{\mathbb C}\setminus\{-1,1\}\to{\mathbb C}$ given by
$f(z)=z/(\sqrt{z+1}\sqrt{z-1})$.
