By F. Oberhettinger
This ebook includes tables of integrals of the Mellin rework variety z-l J (a) 1> (z) q,(x)x dx o t because the substitution x = e- transforms (a) into (b) 1> (z) the Mellin rework is usually often called the 2 sided Laplace rework. using the Mellin rework in numerous difficulties in mathematical research is definitely proven. Parti cularly common and potent is its software to difficulties bobbing up in analytic quantity thought. this is often in part on account that if ¢(z) similar to a given q,(x) by means of (a) is understood, then ¢(z) belonging to xaq,(x) or extra common to P xaq,(x ) (p actual) is also recognized. (See relatively the principles in sections 1. 1 and a pair of. 1 of this booklet. ) a listing of significant contributions conce~ning Mellin trans varieties is further on the finish of the advent. Latin letters (unless another way acknowledged) denote actual optimistic numbers whereas Greek letters denote complicated parameters in the given diversity of validity. the writer is indebted to Mrs. Jolan Eross for her tireless attempt and endurance whereas typing this manuscript. Oregon country collage Corvallis, Oregon may possibly 1974 Fritz Oberhettinger Contents half I. Mellin Transforms advent. . . • . • • • . • . . . . . . . . . . . . • • • • . . . • . • . . • • • . • . 1 a few functions of the Mellin rework research. ••. •••. . . •. •. . . . •• . • . . . . . . ••. . . . . •• 6 1. 1 normal formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eleven 1. 2 Algebraic features and Powers of Arbitrary Order . . . thirteen 1. three Exponential features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Z) ]-1 (lb 2 -a 2 1) -~Zq y '" (2ab) -1(a 2+b 2+x2) - ~ z- 1 (b 2+a 2 ) Ib2-a21 0 < Re z < 1 I. 43 v < 1 x > 1 x < 1 x > 1 B(v+l,z) [ljJ(z)-ljJ(v+1+z)] Re z > 0 > -2 (l-x)V(logx)2 Re x B (v+l, z) {[ljJ (z) -ljJ (V+1+z»)2 +ljJ' (z) -ljJ' (v+1+z)} > -1 e -at (logt) 2 Re z > 0 r (z) a -z { [ljJ (z) -loga) 2 +ljJ I (z) } Re z > 0 40 I. 51 2 log [c + (x+a) 2] c 2+(x+b) O
4 -ax -bx (b+x)-le- ax Exponential Functions b b -z -z -z r (z) Be z > 0 y(z,ab) Be z > 0 r(z ,ab) x > a eabbz-lr(z)r(l_z,ab) Be z > 0 I. 18 l-exp(-ax-p)p > 0 -p -1 a zip r(-z/p) O
14 "[\)I' -b log (l+x) x < 1 0 x > 1 Re z > -1 z -1 -1 z-l csc('ITz) [a 10ga 10gb-'ITcot('ITz) (a [10g2-~\)I (l+~z) + z-l ~\)I(~+~z)] z-l -b z-l )] 36 I. 21 (l+x) -11og(1+x 2 ) (logx) -2