# hermite_product_display

hermite_product_display, a MATLAB code 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 ) )
```
and is scaled so that:
```        Integral ( -oo < X < +oo ) Hf(M,X) Hf(N,X) dX = delta ( N, M )
```

### Usage:

hermite_product_display ( 'name', i, j )
where
• '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.

### Languages:

hermite_product_display is available in a MATLAB version.

### Related Data and Programs:

fem_basis_q4_display, a MATLAB code which displays a basis function associated with a linear quadrilateral ("Q4") mesh.

fem_basis_t3_display, a MATLAB code which displays a basis function associated with a 3-node triangle "T3" mesh.

fem_basis_t4_display, a MATLAB code which displays a basis function associated with a 4-node triangle "T4" mesh.

fem_basis_t6_display, a MATLAB code which displays a basis function associated with a 6-node triangle "T6" mesh.

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

hermite_product_polynomial, a MATLAB code which defines Hermite product polynomials, creating a multivariate polynomial as the product of univariate Hermite polynomials.

polygonal_surface_display, a MATLAB code which displays a surface in 3D described as a set of polygons;