Geometry of Two-Variable Associated Legendre Polynomials

bound for y and x

4.

bound for z

60

cutting plane position y

2.5

polynomial degree n

5

integer order m

4

real order ν

1.5

(4,1.5)

P

5

(x, y) = -1.04482.39904

5

x

-3.32335xy

(4,1.5)

P

5

(x, 2.5) = 8.68058x-2.50652

5

x

On the left is a three-dimensional plot of a Legendre polynomial in two variables

x

and

y

; on the right is a two-dimensional plot of the surface cut by a plane perpendicular to the

y

axis. The exponential operator transform is defined in the Details.