ABSTRACT
Since [(z), L(z), and llf(z) are all holomorphic in D(u), the functions +([) and $,([) are holomorphic in the corresponding [ domain A(u), say. 11.2 Can the differential equation (1 1.02) be satisfied formally by a series of the previous form (7.05)? By direct trial we find that the answer is negative, unless +(C) and the 1(1,(0 of odd suffix all vanish. Nor can (1 1.02) be satisfied by the more general expansion
unless 4 (0 vanishes. To discover a suitable series solution we first use the method of $3.1 to arrive at a
in which $(u,[) is any function which is bounded in A(u) when lul+ a. Provided that #(O) = &and temporarily we assume this to be the case-we may choose
$ (4 C) = 42 (1)/(4C), thereby making the right-hand side of (1 1.05) a perfect square. On taking the square root and integrating, we derive
giving
where 1 @(O = jp p$ do.
