  The three-loop four-point function of stress-tensor multiplets in N=4 super
Yang-Mills theory contains two so far unknown, off-shell, conformal integrals,
in addition to the known, ladder-type integrals. In this paper we evaluate the
unknown integrals, thus obtaining the three-loop correlation function
analytically. The integrals have the generic structure of rational functions
multiplied by (multiple) polylogarithms. We use the idea of leading
singularities to obtain the rational coefficients, the symbol - with an
appropriate ansatz for its structure - as a means of characterising multiple
polylogarithms, and the technique of asymptotic expansion of Feynman integrals
to obtain the integrals in certain limits. The limiting behaviour uniquely
fixes the symbols of the integrals, which we then lift to find the
corresponding polylogarithmic functions. The final formulae are numerically
confirmed. The techniques we develop can be applied more generally, and we
illustrate this by analytically evaluating one of the integrals contributing to
the same four-point function at four loops. This example shows a connection
between the leading singularities and the entries of the symbol.
