Common boundary conditions and operations

Standard Derivation Formulas at 0

You must know the following derivation formulas by heart in the neighbourhood of 0:

Function	Derivation formula of order n
e^x	1 + x + x²/2! + x³/3! + ... + x^n/n! + o(x^n)
sin(x)	x - x³/3! + x⁵/5! - ... + o(x^(2n+1))
cos(x)	1 - x²/2! + x⁴/4! - ... + o(x^(2n))
ln(1+x)	x – x²/2 + x³/3 – … + (-1)^(n+1) x^n/n + o(x^n)
1/(1-x)	1 + x + x² + ... + x^n + o(x^n)
(1+x)^a	1 + a*x + a(a-1)/2! * x² + ... + o(x^n)

Permitted operations on DLs

• Sum and product: DLs are added or multiplied by truncating to the common order n.
• Composition: if g(0) = 0, a DL of f can be composed with a DL of g (substitution).
• Division: two DLs can be divided if the denominator does not vanish at a (otherwise, factorise first).
• Integration: the DL of an antiderivative is obtained by integrating term by term (take care with the constant).
• Differentiation: generally not recommended, as it results in a loss of one order of precision and assumes that the DL of the function being differentiated exists.

Example: DL of e^x * cos(x) to order 2 at 0.

e^x = 1 + x + x^2/2 + o(x^2) and cos(x) = 1 - x^2/2 + o(x^2)

Product truncated to order 2: 1 + x + x²/2 - x²/2 + o(x²) = 1 + x + o(x²)

Common pitfall

When performing a product or a composition, each DL must be expanded to a sufficient order BEFORE truncation: omitting a term of order less than n in a factor may cause a useful term to disappear from the final result.