A2 Mathematical Definitions And Notation

A.2.1. Taylor expansion

We assume that the function r(x) can be expressed as an expansion at the point xA in terms of an infinite series thus:

r(x) = Co + ci (x - xa) + C2 (x - xa)2 + C3 (x - xa )3 + . . ., where the cs are constants. To find c0 we set x = xA to yield c0 = r(xA). To find c1 we differentiate once with respect to x and then set x = xA to yield c1 = (dr/dx)A, where the subscript A indicates that dr/dx is evaluated at x = xA. Carrying on we see that we can write (dmr/dxm)A = m!cm and hence:

2! Vdx2A

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