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Additional info for The Lorentz group and harmonic analysis (The Mathematical physics monograph series)
Example text
12) 0 + (1+--1 x3i2 8761 + . . ) s i n ( < - : ) ] . 4. 4 Primitive of Scorer functions Gordon (1970) also gives some primitives implying the inhomogeneous function Gi(x). The primitive Gi [ a ( x+ P)] dx seems unable t o be expressed simply in terms of Airy functions. Nevertheless, we can calculate 0 1 . J zGi [a(. + p)]dx = + 2Gz' [a(z+ P)] an- -P 0 1 + P)]dx x x = - (- - P) + -Gal on- 2 / + Gi [a(. P)] d z x2Gi [a(. 14) [a(x a2 1 --Gi [a(. 13) + P)] + P2 J + P)] Gi [a(. 5 + P)1 J Gi [Q(X + P)] dz.
We obtain 1 . 1 Gi(-z) = -Bz(z) - -h(z). 141) The ascending series of Gi'(z) is deduced (like for Gi(z)) thanks t o the ascending series of Si'(-z)and Hi'(z). 142) 8! 629 1 2! 5! = -T (Xl + - +x3 - + - 3x6 Hi(-x) M 1 T X c co n=O + (-l)n(3n)! 1 3"n! x3n . 1 2! 8! 5! 143) + ... 129), we can obtain the expansion of Gi(-x) and H i ( x ) . 50). Olver (1954) gives the asymptotic series under an equivalent form 1 [ c 1 =1+ 2 73-X TX O0 (3s+2)! s! 145) , x -+ -m. 4 Zeros of the Scorer functions In an interesting paper on the zeros of the Scorer functions, Gil, Segura and Temme (2003) gave several important results on the subject.
9! 5 g(z) = z (1 +,-&x3 --z 7! + h ( - z ) = - z 2 ( -1+ - T 3 2 5! 8 +zg + . . 9 8! 2. 129). We obtain 1 . 1 Gi(-z) = -Bz(z) - -h(z). 141) The ascending series of Gi'(z) is deduced (like for Gi(z)) thanks t o the ascending series of Si'(-z)and Hi'(z). 142) 8! 629 1 2! 5! = -T (Xl + - +x3 - + - 3x6 Hi(-x) M 1 T X c co n=O + (-l)n(3n)! 1 3"n! x3n . 1 2! 8! 5! 143) + ... 129), we can obtain the expansion of Gi(-x) and H i ( x ) . 50). Olver (1954) gives the asymptotic series under an equivalent form 1 [ c 1 =1+ 2 73-X TX O0 (3s+2)!