Certain Dual Integral Equations And Sonines Integrals
Certain Dual Integral Equations And Sonines Integrals
Arthur S Peters
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1?) satisfies the first equation (3. 1). For the second equation of the pair we have to evaluate . 0 /^2 .. 2v-v-l, . -^— — p^-L^ ^ C 757 C /(1-t) (z -tC ) dt " J^=l C=l ■>oo iv 2-7v+v, ^''^■^ '0 00 ITT 2' = ^ Xd o-2A-ld ^A+ir^v-A ITT iTTV d? ^A+l/ ^v-Aj^^^^^g )jA+l(^^^)d^ -e ^ z^Rv+l) ^A d ;, -2A-ld . 2A+ir-'-, -, ^^A, 2 .^2. -v-l, . The Integrals on the lei't hand sides of (3-29) and (5-50) converge if Iz > and if v+2 > |v|, a condition which is satisfied if V > -1. [The representation of the...se integrals in terms of integrals over (Ojl), as shown, is valid if A > -1. However, it is clear that by expressing for example / ^^~ K (^ze~ ' )J (|Od| as J Q V A higher derivatives, v;ith respect to ^, of a similar integral; the term (l-t)'"" in the representations can be made to appear e''> 20 as (1-t) . ] Under the interpretation (3-25) the results (3. 29) and (3. 30) lead to ^ " ^ 2^'-\n:>. -, xH-i d^2A+Y(\_^)K(^, ^2_^^2p-^ M C=i ^P-u:if'\^ ^-2A-ld p2?^irn ^-^^ 2.. 2v^J-, + /F(C)^7C ^C /(l-t)(x -tC )dtd but if X > 1 the right he.
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