m-107 



Let f (^, u) be expressed as a power series in a in the form 



00 



$a,u) = y^. a" $„(\,u) (III-133) 



,U) = ^ a"l^ 



Since (III- 132) must be true for all o, it then follows that 



m n " " 



(-ioC(g,r,))" '_-i(a?+Bri) V^ [iC(§,Tl)] ( ( -i(X?+3Ti)n^ n Nn^ n 



; ^ e + / V ; lie v $ (X,u ) dX du = 



m! -*— ' n! | ) m-n^ ' 



(III- 134) 



Solving this for $ (X,u) for m=0, 1, 2, we have 



$q(X,u) = - 6(X - a) 6(n-3) (III-135a) 



f.(X.M)=2iY |'c(5.ri)e-^<^^+^^n" (III-135b) 



^.(X,M) = 2v[c(?.ri)]-*^v[c(?.ri)e-^^^^^^^>]") (III-135c) 



where f ' denotes the inverse Fourier transform of f and f*g denotes the 

 convolution of f and g. 



To see how these are obtained, we consider the solution for $2(X,|j), 

 assuming ?n(X, u) and ^^ (X, \j.) have been obtained as given above. Then writing 

 out (III- 134) for m = 2, we have 



Iq +Ii +I2 +I3 =0 (III- 136a) 



where 



T [-1 yC(?,ti)] ^ -i(a?+ 0ri) (III-136b) 



^^ 2! ^ 



Arthur a.IlittlcJnir. 



S-7001-0307 



