  For the Laplacian $\Delta$ defined on a p.c.f. self-similar fractal, we give
an explicit formula for the resolvent kernel of the Laplacian with Dirichlet
boundary conditions, and also with Neumann boundary conditions. That is, we
construct a symmetric function $G^{(\lambda)}$ which solves $(\lambda
\mathbb{I} - \Delta)^{-1} f(x) = \int G^{(\lambda)}(x,y) f(y) d\mu(y)$. The
method is similar to Kigami's construction of the Green kernel in
\cite[\S3.5]{Kig01} and is expressed as a sum of scaled and "translated" copies
of a certain function $\psi^{(\lambda)}$ which may be considered as a
fundamental solution of the resolvent equation. Examples of the explicit
resolvent kernel formula are given for the unit interval, standard Sierpinski
gasket, and the level-3 Sierpinski gasket $SG_3$.
