  We introduce a method for calculating individual elements of matrix
functions. Our technique makes use of a novel series expansion for the action
of matrix functions on basis vectors that is memory efficient even for very
large matrices. We showcase our approach by calculating the matrix elements of
the exponential of a transverse-field Ising model and evaluating quantum
transition amplitudes for large many-body Hamiltonians of sizes up to $2^{64}
\times 2^{64}$ on a single workstation. We also discuss the application of the
method to matrix inverses. We relate and compare our method to the
state-of-the-art and demonstrate its advantages. We also discuss practical
applications of our method.
