Expansion Problems Arising From the Watson Transformation
Expansion Problems Arising From the Watson Transformation
Eugene Pflumm
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V)yq. (y)dy ^1^^'^) + 2 sin nv / G(y. Z)i^(z, v)zg(z)dz f(y)'iy J(y;>v)y'' = Y (x, v) r 4 ^ (. 1 ^l(y, v)f(y) X '^(yjv)y' 2 2 sin rtv 2 ^o(^^^) f(y)dy r '^(y^v)y'' X / G(y, z)i^(z, v)zq(z)dz 2 sin jtv G(x, y)^„(y, v)yq(y)dy 'i-^(y, v)fiy)dy X wl (y;. V)y'' + - h sin^jtv f G(x, y)i (y, v)yq(y)dy . R ^^^^^ T G(y, z)? ( z, v)zq(z)dz a '^(y'^)y a = |^+$2+$5+li, • 27 We note that r ^. (y, v)f(y) J^=^^(x, v) / -i- -5- dy "^(y. V)y . A. ^l(y, v)f(y) u(y^v)y dy which was considered in proof of Theorem ...1. Hence lim — r / ^ ^ vdv = m — > 00 f(x) 2 m We will now show that the contributions to the contour integral due to Jp, ^ and ^, tend toward zero. Consider X y I2 =F-iI^ V-'^) / rr^ / G(y, . Z)?^(z, v)z 00 m Similarly ve find fel 2 2 h sin jtv G(x, y)5'Jy, v)yq(y)dy r f(y)dy " ^a^y. V)y ^ 2 / G(y, z)Y^(z, v)zq(z)dz const for V along r^ const, J{ev M^^ for V along r^ and 15; 2 sin 3tv G(x, y)T, (y, v)yq(y)dy / \(y, v)f(y)dy '^(y^-vjy const for V along P, const . J^ev 5 M«^ for V along r 2* 30 - We conclude therefore lim / ( ^, + ^)vdv = 0.
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