Regular singular point

Abstract: First--Order of Differential Equations. Second--Order Linear Equations. Linear Equations with Constant Coefficients. Power Series Solutions. Plane Autonomous Systems. Existence and Uniqueness Theorems. Approximate Solutions. Efficient Numerical Integration. Regular Singular Points. Sturm--Liouville Systems. Expansions in Eigenfunctions. Appendices. Bibliography. Index.

Abstract: This paper is a brief historical review of linear singular systems, followed by a survey of results on their solution and properties. The frequency domain and time domain approaches are discussed together to sketch an overall picture of the current status of the theory.

Abstract: Let the function f be defined on I=[a,b] and, possibly, be singular at an interior point c∈(a,b). Recall that the improper integral was defined by $$\int\limits_{a}^{b} {f\left( x \right)} dx: = \mathop{{\lim }}\limits_{{\mathop{{{{\varepsilon }_{1}} \to 0}}\limits_{{{{\varepsilon }_{1}} > 0}} }} \int\limits_{a}^{{c - {{\varepsilon }_{1}}}} {f\left( x \right)} dx + \mathop{{\lim }}\limits_{{\mathop{{{{\varepsilon }_{2}} \to 0}}\limits_{{{{\varepsilon }_{2}} > 0}} }} \int\limits_{{c + {{\varepsilon }_{2}}}}^{b} {f\left( x \right)dx,} $$ if both limits exist (cf. §6.1.3). By Remark 6.1.2a, the improper integral exists for f (x): = |x-c|s with s>-1. For \( f(x): = \frac{1}{{x - c}}({\text{i}}{\text{.e}}{\text{.,s = - 1}}) \) (i.e, s=-1) one obtains $$\int\limits_{a}^{{c - {{\varepsilon }_{1}}}} {\frac{1}{{x - c}}dx} + \int\limits_{{c + {{\varepsilon }_{2}}}}^{b} {\frac{1}{{x - c}}dx = \log \frac{{b - c}}{{c - a}} + \log \frac{{{{\varepsilon }_{1}}}}{{{{\varepsilon }_{2}}}}.} $$ (7.1.1)