Regular Singular Points

Regular Singular Points. If cither of the functions a(a ), fi x) is not analytic at the point x = a, wc say that this point is a singular point of the differential equation. When the functions a ir), f)(x) are of such a nature that the differential equation may be written in the form... 

If x = a is a regular singular point of the equation (3.1) it can he shown that there exists at least one solution of the form ... 

Tiius the regular singular points of the hypcrgcometrie equation are -... 

We shall now consider the form of the solutions in the neighbourhood of tile regular singular points. 

Since Legendre s equation has a regular singular point at ft = 1 we nmy on the basis of equation (3.8) take the... 

Here, the familiar eigenvalues of the orbital angular momentum operators have already been incorporated, reminding the reader that their form and numerical values are determined by the periodicity condition + 2n) = (0) for the first one, and by guaranteeing good behavior for the second at its regular singularity points, 9 = 0 and n. 

For regular singular points, a series solution of the differential equation can be obtained by the method of Frobenius. This is based on the following generalization of the power series expansion ... 

Series Solutions Near a Regular Singular Point... 

In Section 3.5.1, the Frobenius series method was discussed with regard to differential equations with regular singular points. In this section, a method is given that effectively deals with differential equations with ordinary points. To illustrate the importance of this method consider the relatively harmless-looking first-order initial value problem... 

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