  In this paper, we construct Shintani lifts from integral weight weakly
holomorphic modular forms to half-integral weight weakly holomorphic modular
forms. Although defined by different methods, these coincide with the classical
Shintani lifts when restricted to the space of cusp forms. As a side effect,
this gives the coefficients of the classical Shintani lifts as new cycle
integrals. This yields new formulas for the $L$-values of Hecke eigenforms.
When restricted to the space of weakly holomorphic modular forms orthogonal to
cusp forms, the Shintani lifts introduce a definition of weakly holomorphic
Hecke eigenforms. Along the way, auxiliary lifts are constructed from the space
of harmonic weak Maass forms which yield a "fractional derivative" from the
space of half-integral weight harmonic weak Maass forms to half-integral weight
weakly holomorphic modular forms. This fractional derivative complements the
usual $\xi$-operator introduced by Bruinier and Funke.
