Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55)
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506 CONFLUENT HYPERGEOMETRIC FUNCTIONS 13.2.8 r(a)U(a, b, Z ) Similar integrals for Mc,,,(z) and Wc,,,(z) can be deduced with the help of.,13.1.32' and 13.1.33. Barnes-type Contour Integrals 13.2.9 for larg (-z)l<jr, a, b+O, -1, -2, . . . . The contour must separate the poles of I'(-s) from those of r(a+s); c is finite. 13.2.10 r(a)r(i+a-b)zau(a, b, Z ) 3a for larg z~<TJ a#O, -1, -2, . . ., b - u f l , 2, 3, . . . . The contour must separate the poles of r(--s) from those of r(u+s) and r(l+a-b+s). 13.3. Connections With Bessel Functions (see chapters 9 and 10) Bessel Functions as Limiting Cases If b and z are fixed, 13.3.1 lim{M(a, b, z/a)/r(b) ) =~*-+~1~-,(2,E) a+- 13.3.2 lim(M(a, b,-z/a)/r(b) ) =z++bJb-l(ZdZ) 13.3.3 a+ - lim { r(i+a-b) U(a, b, z/~)}=2z+-**K~-~(2l/z) a+ m 13.3.4 lim{ r(i+a-b)U(a, b, -z/u)) - - -7rier*bzf-fbHf21(2,E) (Yz>O) 13.3.5 - *-1(2rn (YZ<O) a+- -7rie-rfb&fbHt2) Expansions in Series 13.3.6 M(a, 6 , z)=e+T (b-aa-$)(+~)~-*+f (2b-2a-l),,(b-2fZ),, ( b - - a - 3 + n) 5 n!(b>,, * n-0 (-I),, Ib-a-f+n($') (b#0,-1J-2J * . a) 13.3.7 - 5 A,, (iz)+"( b -24 -)"Jb-,+,, (J(2zb -&a)) n=O where &=I, Ai=O, Az=$b, (la + 1 )&+I= (n + b- 1 ) A n -I + ( 2 ~ - a) A,, -2, (a real) 13.3.8 M(a,b,z) r(b) m n -0 =eh' C~Z"(-aZ)'"-b-n'~~-l+,,(2J(-UZ)) where co= 1, c, = - bh, c, = -+(2h-- 1)a +#b(b + l)h2, (n+ 1 ) c,,,, = [ (1 - 2h)n - bh]C,, +[(1-2h)u-h(h-l)(b+n-- 1)]C,,-1 -h(h- l)aC,,-2 (h real) 13.3.9 M(a, b, z)=2CR(a, b)l,,(z) where co = 1, c, (a, b) = 2a/b, n=O CJI+l(a, b)=2aC,,(a+l, b+l)/b-cC,-l(aJ 13.4. Recurrence Relations and Differential Properties 13.4.1 (b-a)M(a-l, b, z)+(2a-6+z)M(a, 6, 2) -aM(a+l, b, z)=O 13.4.2 b(b-l)M(a, b-1, z)+b(l-b-z)M(a, b, 2) +z(b--a)M(a, b+lJ z)=O 13.4.3 (l+a-b)M(a, b, z)-aM(a+l, b, 2) +(b-l)M(a, b-1, z)=O 13.4.4 bM(a, b, z)-bM(a-l, b, z)-zM(a, b+l, z)=O 13.4.5 b(a+z)M(a, 6, z)+z(a-b)Wa, b+1, z> -ubM(a+l, b, z)=O