Calculus How To

Legendre Function: Definition, Examples

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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:
legendre equation
Substituting cos(Θ) = x gives:
legendre 2

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

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

Legendre Function: Types

There are two types: first kind Pv(z), and second kind Qv(z). Both types are linearly independent solutions to the ordinary Legendre equation.

;egendre function second kind

Graph of the first five Legendre functions of the second kind. Image: Geek 3| Wikimedia Commons.

  • 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.


Calculating the Legendre function is usually performed with software. For example, this online calculator from Casio will calculate both Pv(z) and Qv(z) with just two inputs: a value for z and a specified degree.


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.

Stephanie Glen. "Legendre Function: Definition, Examples" From Calculus for the rest of us!

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