HERMITE_PRODUCT_DISPLAY
Display 2D Cartesian Products of Hermite Polynomials

HERMITE_PRODUCT_DISPLAY is a MATLAB program which displays an image of a function created by the Cartesian product of two Hermite polynomials, such as f(x,y) = h(3,x) * h(1,y).

There are five types of Hermite polynomial available. Perhaps the best behaved are "Hen(n,x)" and "Hf(n,x)", which don't blow up within the plotting interval as fast as the other functions do.

The physicist's Hermite polynomial H(n,x) can be defined by:

        H(n,x) = (-1)^n exp(x^2/2) * d^n/dx^n ( exp(-x^2/2) )
      

The normalized physicist's Hermite polynomial Hn(n,x) is scaled so that

        Integral ( -oo < X < +oo ) exp ( - X^2 ) * Hn(M,X) Hn(N,X) dX = delta ( N, M )
      

The probabilist's Hermite polynomial He(n,x) is related to H(n,x) by:

        He(n,x) = H(n,x/sqrt(2)) / sqrt ( 2^n )
      

The normalized probabilist's Hermite polynomial Hen(n,x) is scaled so that

        Integral ( -oo < X < +oo ) exp ( - 0.5*X^2 ) * Hen(M,X) Hen(N,X) dX = delta ( N, M )
      

The Hermite function Hf(n,x) is related to H(n,x) by:

        Hf(n,x) = H(n,x) * exp(-x^2/2) / sqrt ( 2^n * n! * sqrt ( pi ) )
      

        Integral ( -oo < X < +oo ) Hf(M,X) Hf(N,X) dX = delta ( N, M )
      

Usage:

hermite_product_display ( 'name', i, j )

• 'name' is 'h', 'hn', 'he', 'hen', or 'hf';
• i is the index of the polynomial in the X direction;
• j is the index of the polynomial in the Y direction.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Related Data and Programs:

FEM_BASIS_Q4_DISPLAY, a MATLAB program which displays a basis function associated with a linear quadrilateral ("Q4") mesh.

FEM_BASIS_T3_DISPLAY, a MATLAB program which displays a basis function associated with a 3-node triangle "T3" mesh.

FEM_BASIS_T4_DISPLAY, a MATLAB program which displays a basis function associated with a 4-node triangle "T4" mesh.

FEM_BASIS_T6_DISPLAY, a MATLAB program which displays a basis function associated with a 6-node triangle "T6" mesh.

HERMITE_POLYNOMIAL, a MATLAB library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

POLYGONAL_SURFACE_DISPLAY, a MATLAB program which displays a surface in 3D described as a set of polygons;

Source Code:

• hermite_product_display.m, displays a 2D Hermite product such as H(i,x)*H(j,y).

Examples and Tests:

Here we plot all the possible products of orders 0 through 5, using the "hen" polynomial:

• hen_00.png, Hen(0,x)*Hen(0,y);
• hen_01.png, Hen(0,x)*Hen(1,y);
• hen_02.png, Hen(0,x)*Hen(2,y);
• hen_03.png, Hen(0,x)*Hen(3,y);
• hen_04.png, Hen(0,x)*Hen(4,y);
• hen_05.png, Hen(0,x)*Hen(5,y);
• hen_11.png, Hen(1,x)*Hen(1,y);
• hen_12.png, Hen(1,x)*Hen(2,y);
• hen_13.png, Hen(1,x)*Hen(3,y);
• hen_14.png, Hen(1,x)*Hen(4,y);
• hen_15.png, Hen(1,x)*Hen(5,y);
• hen_22.png, Hen(2,x)*Hen(2,y);
• hen_23.png, Hen(2,x)*Hen(3,y);
• hen_24.png, Hen(2,x)*Hen(4,y);
• hen_25.png, Hen(2,x)*Hen(5,y);
• hen_33.png, Hen(3,x)*Hen(3,y);
• hen_34.png, Hen(3,x)*Hen(4,y);
• hen_35.png, Hen(3,x)*Hen(5,y);
• hen_44.png, Hen(4,x)*Hen(4,y);
• hen_45.png, Hen(4,x)*Hen(5,y);
• hen_55.png, Hen(5,x)*Hen(5,y);

You can go up one level to the MATLAB source codes.