  Fractional calculus generalizes the derivative and antiderivative operations
of differential and integral calculus from integer orders to the entire complex
plane. Methods are presented for using this generalized calculus with Laplace
transforms of complex-order derivatives to solve analytically many differential
equations in physics, facilitate numerical computations, and generate new
infinite-series representations of functions. As examples, new exact analytic
solutions of differential equations, including new generalized Bessel equations
with complex-power-law variable coefficients, are derived.
