Introduction to Numerical Methods for Time Dependent Differential Equations

APPENDIX A

AUXILIARY MATERIAL

A.1 Some useful Taylor series

We collect here some useful Taylor series used throughout the text, z denotes a complex variable, and the expression on the right denotes the convergence region.

1. Finite geometric series:

(A.1)

2. Geometric series:

(A.2)

3. Any branch of the logarithm has the expansion:

(A.3)

A.2 “” notation

Definition A.1 Let F(x) be a real function. We say that “F(x) is of order f(x) as x tends to a” and write F(x) = f(x)) as x → a, if and only if there exist two constants C > 0 and δ > 0 such that |F(x)| ≤ C |f(x)| when |x − a| < δ.

For example, if F(x) is n + 1 times differentiable around x = a, then by Taylor’s theorem,

where the remainder term satisfies R_n(x) = ((x − a)ⁿ⁺¹). Here, and many times when it is clear by context, we omit “as ...