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18.014ESG Notes 2 Pramod N. Achar
 

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

  

Source: Achar, Pramod - Department of Mathematics, Louisiana State University

 

Collections: Mathematics