I'm currently working with singular second order differential equations and I'm finding that the standard ODE textbooks available to me aren't very helpful. Most give rote definitions of ordinary/regular points of ODEs, and offer up only the Frobenius method for regular singular points(I'm working with singular, complex ODEs).

Some sources may mention complex analysis/branch cuts. Some may deal with singular ODEs. Some may discuss things geometrically. Nothing so far has mentioned Riemann Surfaces. It's all very disparate, and I find myself lacking a clear understanding of ODEs in the complex plain (and by extension, real valued ODEs as well).

What I'm looking for is an ODE textbook which gives a comprehensive treatment of ODEs in the complex plain, in particular one which discusses second order complex ODEs and their (3 (2?)) singular points. I would also like one which gives a more geometric treatment. Basically, I need a textbook which goes beyond series solutions and gives a general theory of complex (2nd order) ODEs.

Can anyone make any recommendations?


Maybe Einar Hille's Ordinary differential equations in the complex domain (Google books link)?

(Disclaimer: I haven't read it.)


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