The Cookbook Everything You Always Wanted to Know About Fourier, Laplace, and Hilbert Transforms, but Were Afraid to Ask ...
Electromagnetic waves in plasmas, like electromagnetic waves in space and in all other media, must obey Maxwell’s equations. Even for propagation in free space the resulting wave equation is a linear second-order partial-differential equation. Thus, the equations describing wave propagation in a complex medium such as a plasma can be complicated and difficult to solve directly. A technique that is widely used for solving such linear differential equations for the case of infinite homogeneous plasmas involves the so-called Fourier and Laplace transforms. Basically, this technique maps the real-space and time variables (r,t) of the wave—which are differentially related in the wave equation—to a complex space (ω,k), where the variables ω and k are algebraically related. In this way, one arrives relatively easily at an expression for E(ω,k), for example, where E is the electric field of the wave as expressed in (ω,k) space. Application of Fourier-Laplace transforms to other equations that must be satisfied by the wave-plasma system allows other quantities, such as f 1(ω,k) and n 1(ω,k), the first-order perturbations of the particle distribution and particle density, respectively, to be similarly computed in (ω,k) space.
KeywordsAnalytic Continuation Meromorphic Function Inverse Fourier Transform Hilbert Transform Causal Function
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