By V. P. Havin, S. V. Hruscev, N. K. Nikol'skii, V. I. Vasyunin
Read or Download Linear and Complex Analysis Problem Book: 199 Research Problems (English, German and French Edition) PDF
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Additional resources for Linear and Complex Analysis Problem Book: 199 Research Problems (English, German and French Edition)
Example text
J1.. - n + 2)t HIl - n+1 r(~ + 1)r().. + J1.. + 2) 1 = r(~ + 1)r(J1.. + 1) = t 10 (t - r().. or g(t) r().. + 1) = r(fJ + l)r().. _ A-,8+1 fJ) t . 25) A more symmetric form is obtained by introducing the functions x P- 1 o/p(x) = r(~)' ffie fJ > O. 28) provided ffie ().. - ~) > o. 50 2. 29) are usually referred to as operators of fractional integration. 19) we have \112(g) = 10r 10(I g(t) dt dtl , and more generally \IIn(g) = 1 111 ... 110-1g(t) dt dtn-l ... dtl . 30) Consequently, if f3 is a fractional number, \11,8 (g) can be thought of as the result of "integrating g f3 times".
Z-a) g(Z) A- z SlDrrV + (Z - a)l-v k df -dz f(~) d~ (Z - ~)I-v . f (~ - f f (~) d~ (z - ~)l-v C az a)2- Y -V(b ~ - z zV sin2 yrr - krr 2(z - a)2-Y-V(b - z)Y {!!...
Example 7. The integral we obtain 101 dx / x a is divergent if ex 2: 1. p. If ex t dx = _1_ , Jo xa 1- ex ex > 1. p. 1 1 dx - =0. 104) 28 1. Reference Material Example 8. l b a = __1_ + _1_, dx b- c (x - c)2 a- c a < c < b. v. l b (b-C) . 106) This is not casual. The exact rules for the differentiation of finite part integrals can be obtained within the distributional context. p. f d7: (7: -~) C k = 0, k = 2, 3, 4, .... 88) remains valid when poles of any order, ai, ... p. 108) where bl , . , bn are the poles of F inside C.