On the Determination of the Free Electron Distribution of An Ionized Gas

Irvin W Kay

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-16- The numbers x. Are assumed to be distinct, i. E. , it is assumed that there are 1 no multiple roots of the equation (U. L6). If this is not so, r(k) can be modified slightly so that the multiple roots of (I-1. I6) split up into simple roots, and one can consider the original case as a limiting form in which two or more roots of (U. 16) coalesce. It is obvious that all conditions on t(k) are satisfied by (U. L^) in any case, even if one ignores its derivation.
The phase law of (2. 8) provid...es the means for obtaining an expression for the function p(k) once t(k) is assumed to be given by (Li, l5); . M n (U. I7) p(k) ' ^^^:-- ^ .
TT (k-\) TT (k-x. ) i=l ^ j=l "^ Notice that p(k), unlike r(k), has poles in the upper half plane: the x.. Since no proper eigenfunctions can exist it is clear from Section 3 that the function V(x) does not vanish identically for all values of x larger than some finite number; i. E. , V(x) must be different from zero in the positive x direction at arbitrarily large values of x.

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