| There should be no great disagreement to say that linear differential equations are important both within mathematics or in their applications to other disciplines. Historically, differential equations were first written down by Newton ever since he invented the differential calculus in his search for the understanding of physical world. Thus, it is natural to treat both the independent and dependent variables of the differential equation and its solutions to be real. However, experience shows us that many important DEs and their solutions have their natural domain on the complex plane C and that they assume complex values in general. For example, the Bessel equation, Hermite equation, Laguerre equation, etc, are all equations that have their natural domains on or part of C. Apart from these familiar examples, there are also less familiar but by no means less important differential equations that have not been well-understood. Prominent examples are the Mathieu equation, Lame equation, and Hill's equation, see [3]. Each of them is at least one and a half century old. These equations sit at a higher level than the previously mentioned equations and so they are much harder to treat and with less tools available. Bank and Laine discovered in the 1980s that the Nevanlinna value distribution theory could be applied to treat special cases of the Hill's equation which have periodic potentials. They discovered that the zero-distribution of solutions have close relation with the quantization of the corresponding DEs. Chiang and Ismail were able to show that this Hill's equation can be solved by special functions of the confluent hypergeometric type in 2006. They also unified the Nevanlinna approach and the classical special function approach.;The main focus of this thesis is to consider special cases of certain non-homogeneous Hill's equation and to show that one can use Nevanlinna theory viewpoint (subnormality and oscillatory) to discover new "function-theoretic quantization" criteria of the equations. New facts about the corresponding special function, namely the Lommel functions, are also established. *Please refer to dissertation for footnotes. |
