Basic Calculus and Analysis - Computer Graphics and Geometric Modeling - page 894

() = 0 lim

k

Rx

Æ•

for all x ΠI .

Analytic functions are properly discussed in complex analysis. See Appendix E.

Let f : (a,b) Æ R be of class C • and let c Œ (a,b). The power series Definition.

•

1
0 n

n

Â

() n

() - ( ) f cxc

!

n =

is called the Taylor series for f about c .

D.3 Differential Equations

For the sake of completeness we shall state the two theorems about solutions to dif-
ferential equations that are needed in this topic. This section uses a few concepts from
Sections 4.2 and 4.3. Note that the order of an (ordinary) differential equation is the
order of the highest derivative appearing in the equation.

Problem 1: Let D be a connected open subset of R n+1 and assume that f 1 , f 2 ,...,
f n : D Æ R are continuous functions. We want to find an open interval (a,b) and func-
tions j i : (a,b) Æ R , so that

(1) (t,j 1 (t),j 2 (t),...,j n (t)) ΠD , and
(2) j i ¢(t) = f i (t,j 1 (t),j 2 (t),...,j n (t)).

for t Π(a,b).

The equations in Problem 1 are called a system of n ordinary differential equations
of the first order and the differentiable functions j i (t), if they exist, are called solutions
to the system.

D.3.1. Theorem. Let (t 0 ,x 1 ,x 2 ,...,x n ) ΠD . Then there exists an e>0 and unique
continuously differentiable functions j i :(t 0 -e,t 0 +e) Æ R that are solutions to
Problem 1 and satisfy j i (t 0 ) = x i .

Proof. See [CodL55].

Problem 2: Let D be a connected open subset of R n+1 and assume that f : D Æ R
is a continuous function. We want to find an open interval (a,b) and a function
j : (a,b) Æ R , so that

(1) (t,j(t),j¢(t),...,j (n-1) (t)) Œ D , and
(2) j (n) (t) = f(t,j(t),j¢(t),...,j (n-1) (t)).

for t Π(a,b).