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I'm currently studying complex analysis at a university (undergraduate) but my lecturer is doing a little bit of ordinary differential equation theory too, inclduing hypergeometric functions.

As I was studying this, I've felt that my lecturer's treatment of this theory is hardly rigrous. So I was looking for some very nice elegant textbooks that deal with this topic in a rigorous, clear and systematic manner, constructing the theory from the very basic notions, and especially books written in the definition-theorem-proof (kind of modern) format, or at least theorem-proof manner. But I've found very few.

My course gives Ince's book as a reference, which I think is qutie famous but personally too old, and as for this Ince's book, the theorems in the textbook are not fully self-containd, i.e., when you look up the theorems in the textbook, you need also to read the discussion around the theorems so as to know how the mathematical notations are used in the theorems within the context. And above all, I kind of felt that the treatment in Ince's is not that rigorous compared to other textbooks (including those in other areas) such as Rudin's. So I prefer if there is kind of a (modern) textbook that treats this theory in a crystal-clear manner.

Q1) Is there any textbook of this theory that fits the above description? Or is still the Ince's book the best in this area of study?

Q2) And in general, how should I study this topic? Should I study PDE? Or should I study holomorphis functions of several variables? So, could you give me some study-guide for this?

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  • $\begingroup$ This is perhaps duplicated in other questions such as here or here. $\endgroup$ – Epictetus Oct 13 '12 at 7:30
  • $\begingroup$ @Epictetus Oh.. but that one is about real ODE, isnt' it? $\endgroup$ – jachilles Oct 13 '12 at 7:32
  • $\begingroup$ Look at Lectures on Analytic Differential Equations. But you do need the theory of holomorphic multivariable functions. $\endgroup$ – Artem Jul 7 '17 at 23:53
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I think that Theory of Ordinary Differential Equations by Coddington and Levinson is a fairly standard/classical reference.

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I think Ince is the bomb, but perhaps the end-all be-all modern source is still to be written. I like how it gives a lot of theory and rigor but also a lot of computational tricks. It treats Hill's equation like it is just one more little fly to be swatted with the others.

I guess at a certain time, you have to start reading multiple books. Why not get what you can get out of Ince (have you really looked at it in detail?) and then also look at other books as well for the gaps in Ince.

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  • $\begingroup$ Actually this answer is way better than the other one since Ince is indeed is classic and the best way into the theory. $\endgroup$ – Artem Jul 8 '17 at 3:58

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