 
Summary: 18.014ESG Notes 2
Pramod N. Achar
Fall 1999
1 The Trigonometric Functions
Consider the following properties which might be satisfied by a given pair of functions u, v : R R:
Du = v Dv = u (1)
u(0) = 0 v(0) = 1 (2)
Theorem 1.1. There exists a pair of functions u, v satisfying (1) and (2).
Proof. Deferred. We will do this after we develop some theory of power series.
Theorem 1.2. If there is a pair of functions satisfying (1) and (2), it is unique.
Before we prove this, we need to establish the following:
Lemma 1.3. Suppose that f and g are two functions such that Df = g and Dg = f. Then f2
+ g2
is a
constant.
Proof. Let us compute the derivative of f2
+ g2
:
D(f2
+ g2
