Yuri M. Galperin's An Introduction to Modern Solid State Physics PDF
By Yuri M. Galperin
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11) for process (5) is stable: = L2 (r) no more slowly than h, and W = h + S-l is the Q = 1. , if (6) then lim sup Ilgn - hi I = 0(£). (7) n+ex> Remark 1. §4. Actually process (5) converges in r is smooth. 4) is = fr 11(t)dt 47T£e r st r l. (1) can be the surface of a metallic body or the surface of a metallic screen (an infinitely thin body). body. if An 1. 4, the following theorem follows. Theorem 1. ~n+l = Let 11n = (I - Al)~n + 1, a(t)~n' ~O where aCt) is defined in (2), =1 (3) Al~ = fr(47T£erst)-la(t)~(t)dt.
Let (11) Then a 3 1 nZ =--=ala Z kj' From (3) it follows that k . - < 1.. < n-n- 1i1/k. j. jn- l :5.. nZ(kj)-l. Thus (13) 46 Therefore ;a; < a < _1_ al - 2 - (14) For fixed a l and This means that a 2 , the parameter a 3 is uniquely determined by (12). can be tabulated as a function of a l and a 2 . In Table 1 the results are given for n = 10. In the horizontal line the e values of a l are given. In the vertical line the values of a 2 are At the intersections the values of C(a l ,a 2) are given.
A convergent geometric series. 32). K. E f (22) an' Process (22) converges no more slowly than If K. J = 1 then equation (20) for some E H such that (23) f ds = O. This solution satisfies the condition r Jr a ds = O. (24) There is only one solution of equation (20) with in the class of functions a E H satisfying (24). , a O = f. f satisfying a O satisfies condition (24), The process converges at least as fast as a convergent geometric series. (23) This solution can be 22 A proof of Theorem 2 is similar to the proof of Theorem 1 and can be left to the reader as an exercise.
An Introduction to Modern Solid State Physics by Yuri M. Galperin