denotes the solution of (31.2.1) that corresponds to the exponent 0 at and assumes the value 1 there. If the other exponent is not a positive integer, that is, if , then from §2.7(i) it follows that exists, is analytic in the disk , and has the Maclaurin expansion
Similarly, if , then the solution of (31.2.1) that corresponds to the exponent at is
When , linearly independent solutions can be constructed as in §2.7(i). In general, one of them has a logarithmic singularity at .
Solutions of (31.2.1) corresponding to the exponents 0 and at are respectively,
Solutions of (31.2.1) corresponding to the exponents and at are respectively,
Solutions (31.3.1) and (31.3.5)–(31.3.11) comprise a set of 8 local solutions of (31.2.1): 2 per singular point. Each is related to the solution (31.3.1) by one of the automorphisms of §31.2(v). There are 192 automorphisms in all, so there are equivalent expressions for each of the 8. For example, is equal to
which arises from the homography , and to