Computation of some transcendental integrals from path signatures

Abstract

It is shown that if $\gamma$ is a path of finite $p$ variation ($1\leq p< 2$) in a euclidean vector space and $f,g,h$ are Lipschitz functions on the trace of $\gamma$ then $s\mapsto F(s)=\int_\gamma f^sg dh$ defines an entire holomorphic function provided the convex hull of the image of $f$ does not contain zero. If in addition $|\log z|\leq \log 2$ on the convex hull of the image of $f$ then for any $s\in \mathbf{C}$, $F(s)$ can be computed from the nonnegative integer values $\{F(k)\}_{k\in \mathbf{N}}$. If in addition to these hypotheses each of $f,g,h$ is a polynomial, then the values $F(k)$ are computable directly from the signature of $\gamma$ thus all values of $F(s)$ are computable from the signature. As a special case the winding number of a closed path $\gamma$ around an affine submanifold of codimension two is computed from finitely many terms of the signature provided certain estimates are satisfied.