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 sizzling plasmas, was once based in 1946 whilst 1. D. Landau released his research of linear (small amplitude) waves in such plasmas. It used to be no longer until eventually a few ten to 20 years later, in spite of the fact that, with impetus from the then swiftly constructing managed fusion box, that enough awareness was once dedicated, in either theoretical and experimental study, to clarify the significance and ramifications of Landau's unique paintings. in view that then, with advances in laboratory, fusion, area, and astrophysical plasma examine, we now have witnessed vital devel opments towards the knowledge of various 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 concepts and together with quite a lot of appli cations. As such, it really is now usually brought in collage physics and engineering curricula on the senior and first-year-graduate degrees. an important prerequisite for all of contemporary plasma experiences is the below status oflinear waves in a temporally and spatially dispersive medium equivalent to a plasma, together with the kinetic (Landau) concept description of such waves. instructing adventure has often proven that scholars (seniors and first-year graduates), while first uncovered to the kinetic conception of plasma waves, have problems in facing the necessary sophistication in multidimensional advanced variable (singular) integrals and transforms.

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If the sum over the Bm terms is infinite, then the point a is said to be an essential singularity. 7. Meromorphic Functions A functionJ(z) which has poles only in a region D of the complex plane is said to be meromorphic in D. Rational fractions are examples of functions that are meromorphic in the full complex plane. B. Residues The coefficient B 1 of the first term of the singular part of the Laurent expansion [see Eqs. 179) One notes that Residue[J(;::),a] = B1. 180) Residues Theorem: IfJ(;::) is a meromorphic function in the region D, then for any closed path L in D J.

7B) Similarly, for K2 we obtain -zwox) Z K2 = exp ( - - c - "2 sgn(x). 79) Substituting from Eqs. 79) into Eg. 74) gives ia M(x,t) = Be sgn(x) {exp[ -i(wot - kox)] + exp[i(wot + kox)]}, (LBO) 23 THE COOKBOOK where ko = wolc. 81 ) Observation of this equation shows that we have a standing-wave pattern for both x < 0 and x > O. That is, in both half-spaces we have waves propagating in both directions. This nonphysical result is the consequence of the loss of causality of Eq. (l. 73), and as we will show in the next section is due to the loss of the transient motion in Eq.

Then Eq. (1. 92) becomes R(ro) R(ro') + i/(OJ) = -I . P f+oo -, - 2m - 00 ro - ro 1 dOJ' + _I 2n P f /(ro') +00 - ,dro' - 00 ro - ro i +"2 R(ro) +"2 /(ro). 93) Splitting Eq. 93) into real and imaginary parts, we obtain 1 R(ro) =-P n f+oo -/(ro') ,-dro; -ooro-ro 1 f+ 00 R(ro') - ,dro. 94) /(ro) = - - P n - 00 OJ - OJ The relations given by Eq. 94) are known as the Kramers-Kronig relations. They show that for the Laplace transform of a stable system the real and imaginary parts are related in a well-defined manner.