An Introduction to the Linear Theories and Methods of Electrostatic Waves in Plasmas pp 1-47 | Cite as

# The Cookbook Everything You Always Wanted to Know About Fourier, Laplace, and Hilbert Transforms, but Were Afraid to Ask ...

## Abstract

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.

## Keywords

Analytic Continuation Meromorphic Function Inverse Fourier Transform Hilbert Transform Causal Function## Preview

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