By W. D. Jones, H. J. Doucet, J. M. Buzzi (auth.)

Modern plasma physics, encompassing wave-particle interactions and collec tive phenomena attribute of the collision-free nature of scorching plasmas, used to be based in 1946 while 1. D. Landau released his research of linear (small amplitude) waves in such plasmas. It was once no longer until eventually a few ten to 20 years later, although, with impetus from the then speedily constructing managed fusion box, that adequate awareness was once dedicated, in either theoretical and experimental examine, to clarify the significance and ramifications of Landau's unique paintings. on account that then, with advances in laboratory, fusion, house, and astrophysical plasma examine, we've got witnessed vital devel opments towards the certainty of numerous linear in addition to nonlinear plasma phenomena, together with plasma turbulence. this day, plasma physics stands as a well-developed self-discipline containing a unified physique of robust theoretical and experimental recommendations and together with a variety of appli cations. As such, it truly is now often brought in college physics and engineering curricula on the senior and first-year-graduate degrees. an important prerequisite for all of recent plasma reviews is the lower than status oflinear waves in a temporally and spatially dispersive medium similar to a plasma, together with the kinetic (Landau) idea description of such waves. instructing event has frequently proven that scholars (seniors and first-year graduates), whilst first uncovered to the kinetic idea of plasma waves, have problems in facing the necessary sophistication in multidimensional advanced variable (singular) integrals and transforms.

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**Example text**

Let us define a region to be a nonempty, connected subset of the complex plane. Then, iff(z) is analytic and a single-valuedfunction of Z for any Z belonging to a region D of C,J( z) is said to be holomorphic or analytic in D. We will denote the class of all holomorphic functions in D as A(D). If D = C, thenf(z) is said to be an entirefunction. For n being any positive integer,f(z) = zn is singlevalued and analytic in the whole complex plane and is thus said to be an 41 THE COOKBOOK entire function.

63) in the form If. M(k t> 0) = - e _. 64 ) where we have used the mathematical identity, I 1(1 w6 - 0 2 = - 20 0 - Wo 1) . 65 ) + 0 + Wo We leave it as an exercise for the reader to show, using the theory ofresidues and the recipe given by Eq. 32) for inverse Fourier transforms, that 1-exp[-i(0-wo)t] 0- Wo ------'---=---------'------"-'--'- - . ::. - Fz. -+ -i exp (-iwox) - - - u(x 0+ Wo e c + et), where u(x - xo) = 1 for x between 0 and Xo and = 0 elsewhere. Thus, using Eq. 66) and noting that the exponential function exp( -iwot) is an invariant with respect to the spatial inverse Fourier transformation, we can write M in real-space and time coordinates as M(x, t > 0) ilf.

Moreover, we may assume that (1 is a tensor, so that the vectorsj and E are not generally colinear. The relation given by Eq. 7), however, is not the most general one for a linear homogeneous medium. As a matter of fact, Eq. 7) states that the current at a given position T and at a given time t, is a function of the electric field at this position and time. One may expect, however, because of the charged-particle motion, that the current is not a function of only the local WAVES IN A CONDUCTIVITY-TENSOR-DEFINED MEDIUM 51 electric field.

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