  I study the Feynman integrals needed to compute two-loop self-energy
functions for general masses and external momenta. A convenient basis for these
functions consists of the four integrals obtained at the end of Tarasov's
recurrence relation algorithm. The basis functions are modified here to include
one-loop and two-loop counterterms to render them finite; this simplifies the
presentation of results in practical applications. I find the derivatives of
these basis functions with respect to all squared-mass arguments, the
renormalization scale, and the external momentum invariant, and express the
results algebraically in terms of the basis. This allows all necessary two-loop
self-energy integrals to be efficiently computed numerically using the
differential equation in the external momentum invariant. I also use the
differential equations method to derive analytic forms for various special
cases, including a four-propagator integral with three distinct non-zero
masses.
