# §14.2 Differential Equations

## §14.2(i) Legendre’s Equation

Standard solutions: , , , , , . and are real when and , and and are real when and .

## §14.2(ii) Associated Legendre Equation

(14.2.2) reduces to (14.2.1) when . Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations , , , , .

, , and are real when , , and , and ; and are real when and , and .

Unless stated otherwise in §§14.214.20 it is assumed that the arguments of the functions and lie in the interval , and the arguments of the functions , , and lie in the interval . For extensions to complex arguments see §§14.2114.28.

## §14.2(iii) Numerically Satisfactory Solutions

Equation (14.2.2) has regular singularities at , −1, and , with exponent pairs , , and , respectively; compare §2.7(i).

When , and , and are linearly independent, and when they are recessive at and , respectively. Hence they comprise a numerically satisfactory pair of solutions (§2.7(iv)) of (14.2.2) in the interval . When , or , and are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair.

When and , and are linearly independent, and recessive at and , respectively. Hence they comprise a numerically satisfactory pair of solutions of (14.2.2) in the interval . With the same conditions, and comprise a numerically satisfactory pair of solutions in the interval .