A **Legendre function **is any solution of the Legendre equation.

## Where Does The Legendre Equation Appear?

**Legendre equations** (and their solutions) appear in electrostatic problems, wave functions for atoms, and many other applications. When separation of variables is used to solve a scalar wave equation in spherical coordinates, the following *associated Legendre equation* arises:

Substituting *cos*(*Θ*) = *x *gives:

If *m *= 0, the above equation reduces to the **ordinary Legendre equation**:

Legendre functions of the first and second kinds are solutions to this equation (Misra, 2007).

## Legendre Function: Types

There are two types: first kind *P _{v}*(

*z*), and second kind

*Q*(

_{v}*z*). Both types are linearly independent solutions to the

**ordinary Legendre equation**.

- The
**Legendre Function of the First Kind**is a solution to the Legendre equation. - The
**Legendre Function of the Second Kind**is also a solution to the equation, except that the solution is singular at the origin.

## Calculation

Calculating the Legendre function is usually performed with software. For example, this online calculator from Casio will calculate both *P _{v}*(

*z*) and

*Q*(

_{v}*z*) with just two inputs: a value for z and a specified degree.

## References

Abramowitz, M. and Stegun, I. A. (Eds.). “Legendre Functions.” Ch. 8 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th edition. New York: Dover, pp. 331-339, 1972.

Misra, D. Practical Electromagnetics: From Biomedical Sciences to Wireless Communication. Wiley, 2007.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 597-600, 1953.

Snow, C. Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. Washington, DC: U. S. Government Printing Office, 1952.