By MAURICE ROSEAU (Eds.)
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5 function F(s) which satisfies F ( s ) s e L ( 0 , -f oo), Vce]c 1 ? c 2 [. c_} 2 We set Cc +ioo I = ζιπ ^ c _ i o o and pt e ^(p)dp, c x < c < c 2 , F(i)=/(-logt), t > 0 . 6, it can be ci pointed out that f ( t ) is independent of c and / ( i ) e " is square integrable over ( — oo, + oo) for each ce]c c 2 [, which ends the proof. 45) and consider the set ^ of functions + F(s) which have a compact support in R , vanish in some neighborhood of 0 and are indefinitely differentiable. 47) p + kl - ds from which it follows, on the assumption that the support of F(s) is included in the compact interval [a, b \ (0 < a < b) that k l \p ^(p)\ Rci +10 + i 00 2πι J C l _ i o o On the other hand the change r->r Γ(νν) for Re W > C 1 1 >0, r>0. 28) leads to ( 2 2 8 ) 38 SPECIAL FUNCTIONS [Ch. 30) ο with Re w > Re u > 0. )-^ + Y H ' or with the change ι-»τ, (1 + 1 ) ~ τ 1 = also known as the beta Euler integral. 5. 11) if q is odd. 17) we get The order of summation and integration can be reversed as is readily 3 Ch. 34) l-Erf(y(a/r))-=-M «>^. z ZKlJ c z Since we can obviously replace the contour C by the straight line which joins c —ioo to c + ioo, with c>0, we get l-Erf(y(fl/i)) = e-^^> and - {l-Erf(y(a/r))}di, pi -*<"> = p J f ο in Re ρ > 0. Some elementary properties of Jv(t). 41) let us assume z— t to be real positive and make the change of variable C = iw/2; hence L . 48) for any integer rc. 47) with respect to t Ch. 49) = I(Jv-l(t)-Jv+1(t)). 47) provides and finally νΛ(ί) = ^ ( Λ - . ( 0 + Λ . 52) ( t v _ v( t ) ) ' = - r j _ ( v _ u ( r ) . All of these relations can be extended to complex values of t by analytic continuation. 2 to t ^ J ^ t ) - * — , Ρ Rev^O. 53) 2 2 = 0, z = a(t -b )K t>b, a, b real 44 [Ch. 2 SPECIAL FUNCTIONS 1 Tc + ioo 2 C A \ T 2 - B c>0.