The Hermite Polynomials

Definition: \(H_n(x) = (-1)^n \; e^{x^2} \frac{\textrm{d}^n}{\textrm{d}x^n} e^{-x^2} \)

• Choose a Hermite Polynomial from the menu.
• Use button "Plot Hermite Polynomial" to plot the choosen polynomial.
• The program optimizes the display range for the last plotted polynomial to give a good representation of it.
• One can replot a polynomial to rescale the display for it.
• "Clear Plot Area" will delete all plotted polynomials.
• Below the plot area there are all used polynomials shown.

The first nine Hermite Polynomials used here:

• \( H_0(x) = 1 \)
• \( H_1(x) = 2 x \)
• \( H_2(x) = 4 x^2 - 2 \)
• \( H_3(x) = 8 x^3 - 12 x \)
• \( H_4(x) = 16 x^4 - 48 x^2 + 12 \)
• \( H_5(x) = 32 x^5 - 160 x^3 + 120 x \)
• \( H_6(x) = 64 x^6 - 480 x^4 + 720 x^2 - 120 \)
• \( H_7(x) = 128 x^7 - 1344 x^5 + 3360 x^3 - 1680 x \)
• \( H_8(x) = 256 x^8 - 3584 x^6 + 13440 x^4 - 13440 x^2 + 1680 \)