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 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 1 Finite differences.
 2 Gauss-Hermite Integration.
 A. Gram-Schmidt orthogonalization.
 B. Definition and existence of orthogonal polynomials.
 C. Three-term recurrence relation for orthogonal polynomials.
 D. Orthogonal polynomials and quadrature rules.
 E. Extremal properties of orthogonal polynomials.
 F. Chebyshev polynomials.
 3 Asymptotic expansions.
 4 Monte-Carlo.
 5 Convex Analysis.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Gauss-Hermite Integration.

he following is an extremely efficient integration formula:

 (Gauss-Hermite Intergration)
Note that one can do the change of function to obtain more generic looking result.

The below values of are taken from [Abramowitz] , pages 890 and 924: What follows next is a fragment of theory of orthogonal polynomials that leads to the formula ( Gauss-Hermite Integration ). The proposition ( Gaussian quadrature rule ) provides the justification. There are several sections after ( Gaussian quadrature rule ) included for their importance for other applications within these Notes. The reference is [Gautschi] .

 A. Gram-Schmidt orthogonalization.
 B. Definition and existence of orthogonal polynomials.
 C. Three-term recurrence relation for orthogonal polynomials.
 D. Orthogonal polynomials and quadrature rules.
 E. Extremal properties of orthogonal polynomials.
 F. Chebyshev polynomials.
 Notation. Index. Contents.