  We investigate a new family of locally harmonic Maass forms which correspond
to periods of modular forms. They transform like negative weight modular forms
and are harmonic apart from jump singularities along infinite geodesics. Our
main result is an explicit splitting of the new locally harmonic Maass forms
into a harmonic part and a locally polynomial part that captures the jump
singularities. As an application, we obtain finite rational formulas for
suitable linear combinations of periods of meromorphic modular forms associated
to positive definite binary quadratic forms.
