Mathematical Methods in Science and Engineering, 2nd Edition

Chapter 1Legendre Equation and Polynomials

Legendre polynomials, are the solutions of the Legendre equation:

1.1

They are named after the French mathematician Adrien-Marie Legendre (1752–1833). They are frequently encountered in physics and engineering applications. In particular, they appear in the solutions of the Laplace equation in spherical polar coordinates.

1.1 Second-Order Differential Equations of Physics

Many of the second-order partial differential equations of physics and engineering can be written as

1.2

where some of the frequently encountered cases are:

1. 1. When and are zero, we have the Laplace equation:
   1.3
2. which is encountered in many different areas of science like electrostatics, magnetostatics, laminar (irrotational) flow, surface waves, heat transfer and gravitation.
3. 2. When the right-hand side of the Laplace equation is different from zero, we have the Poisson ...