Formal Power Series - Operations On Formal Power Series - Formal Differentiation of Series

Formal Differentiation of Series

Given a formal power series

in R], we define its formal derivative, denoted Df or, by

Df = \sum_{n \geq 1} a_n n X^{n-1}.

The symbol D is called the formal differentiation operator. The motivation behind this definition is that it simply mimics term-by-term differentiation of a polynomial.

This operation is R-linear:

D(af + bg) = a \cdot Df + b \cdot Dg

for any a, b in R and any f, g in R]. Additionally, the formal derivative has many of the properties of the usual derivative of calculus. For example, the product rule is valid:

D(fg) = f \cdot (Dg) + (Df) \cdot g,

and the chain rule works as well:

D(f\circ g ) = \left( Df\circ g\right) \cdot Dg,

whenever the appropriate compositions of series are defined (see above under composition of series).

Thus, in these respects formal power series behave like Taylor series. Indeed, for the f defined above, we find that

(D^k f)(0) = k! a_k,

where Dk denotes the kth formal derivative (that is, the result of formally differentiating k times).

Read more about this topic:  Formal Power Series, Operations On Formal Power Series

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