C.1 Factorials and Series Expansions

The factorial function is

so, for example, 5!=1 2 3 4 5 = 120 . The factorial 0! = 1. The number of ways of choosing k distinct objects from a given set of n objects is given by the binomial coefficient

Multifactorials are the product of integers in steps of two ( n!!), three ( n!!!), and so on. The double factorial n!! is defined recursively as

so, for example, 5!! = 1 3 5 = 15 and 6!! = 2 4 6 = 48.

The Taylor series expansion for a smooth function f( x) about x ₀ is

where f ^{( n )} is the n-th derivative of the function f( x) at the position x ₀. A Maclaurin expansion is a special case of the Taylor expansion in which x ₀ = 0.

Well known expansions are

The Binomial series expansion requires x < 1 and is

or, in slightly more compact notation,

Stirling s formula is log( n!) n log( n) ? n for n ? 1

C.2 Differentiation

The chain rule for the product of two differentiable functions f( x) and g( x) may be expressed as:

or using the notation where ( indicates a derivative we may write

If the ratio of two...