By S. Yu. Slavyanov

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13). 3. The Whittaker functions and their asymptotics. The confluent hypergeometric equation contains the term with the first derivative. The description of asymptotic solutions of second order equations becomes more simple if this term vanishes. 1), and introduce the parametersµ and x: µ = (c -1)/2, x= c/2-a. 19) 1 1/4 -µ 2 ) y(z)=O, y"(z)+ ( --+~+ 4 z z2 which does not contain the first derivative. 21) Y1{z) = Mx,µ(z) = e-z 2 l 2zµ+lq>(µ- X + 1/2,2µ+ 1,z), Y2{z) = Wx,µ(z) = e-z 2 / 2zµ+i/ 2w(µ - x + 1/2, 2µ + 1, z).

19) 1 1/4 -µ 2 ) y(z)=O, y"(z)+ ( --+~+ 4 z z2 which does not contain the first derivative. 21) Y1{z) = Mx,µ(z) = e-z 2 l 2zµ+lq>(µ- X + 1/2,2µ+ 1,z), Y2{z) = Wx,µ(z) = e-z 2 / 2zµ+i/ 2w(µ - x + 1/2, 2µ + 1, z). 19). The Stokes lines are the positive and negative real axes. ,,. r(µ- x + 1/2) . r(-µ- x + 1/2) z , k=l 92(µ,x,z) =91(µ,-x,z). I. µ (z ) -_ e-z/2 z X Y1 ( x, µ, z ) , Iargzl :::; 11' - e. 18): r(2µ + 1) e-z/2zxei1r(µ-x+1/2)g (µ x z) 1 r(µ + x + 1/2) ' ' + r(2µ + 1) ezl2z-(µ+x+1/2)g (µ x z) 2 ' ' ' r(µ - x + 1/2) e:::; Iargzl:::; 11' - e, r(2µ + 1) e-z/2zXe-i1r(µ-x+1/2)g (µ X z) 1 r(µ + x + 1/2) ' ' + r(2µ + 1) ez/2z-(µ+x+1/2)g (µ x z) 2 r(µ - x + 1/2) ' ' ' -11' + e:::; Iargzl:::; -€, r(2µ + 1) -x/2 x [ Mx,µ (x ) -r(µ + x + l/ 2) e x cos 11'(µ- x + 1/2)]g1(µ,x,x) + Mx,µ(-x) r(2µ + 1) exl2x-(µ+x+1/2)g (µ x x) 2 ' ' r(µ - x + 1/2) ' r(2µ + 1) x/2 x r(µ + x + 1/2) e x g1(µ, x.

3. PARABOLIC CYLINDER FUNCTIONS AND THEIR ASYMPTOTICS 29 e 3. The following notations are used: ( = z 2 /4, = x 2 /4, ao = 1, bo = 1, bk(11) = ak(-11-l), ak+i(11) = (k2" + v(~k 1 >)ak(11) r(2k-v) or ak (11 ) = k! 2a,. r(-v). /2ir 2-v+l(-(v+l)/2e< r(-11) x ( 1+ D,,(z) 2"ei11'v/2(v/2ef. f2ir 2-v-le-(v+l)/2e-F. 2r(-11) x (1 + 2" COS11"11e-F. ( 1 + z=-x f ~(-1)kbk(11)e-k) (-l)kak(11)e-k) k=l + rY::) 2-v-lef. ( 1 + ~bk(11)e-k) We see that on the positive real axis the asymptotics of the function V,,(x) contains only the dominant part.