# Gauss-Hermite integration Calculator

## Calculates the integral of the given function f(x) over the interval (-∞,∞) using Gauss-Hermite integration.

 $\normal{\large\int_{\small -\infty}^{\hspace{25}\small \infty}}e^{-x^2}f(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}f(x_i)\\{\large\int_{\small -\infty}^{\hspace{25}\small \infty}}g(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}e^{x_i^2}g(x_i),\hspace{10}g(x)=e^{-x^2}f(x)\\$
 g(x)=e-x2f(x)
 g(x) f(x)
 partition n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 Generally, this integral can be rapidly and accurately calculated.The integrand f(x) is assumed to be analytic and non-periodic.$\normal Gauss-Hermite\ integration\\[10](1)\ {\large\int_{\small -\infty}^{\hspace{25}\small \infty}}f(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}e^{x_i^2}f(x_i)\\[10]\hspace{30} nodes\ x_i:\hspace{20}H_n(x_i)=0\\\hspace{30} weight\hspace{20} w_i={\large\frac{2^{n+1}{\large n!\sqrt{\pi}}}{[H_{n+1}(x_i)]^2}}\\$

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