7.5 Odd and Even Functions

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Remark 7.5.1. We shall also be limiting ourselves to functions defined over

intervals which are symmetric about the origin, i.e. PC[−a, a] for positive

a. This has the consequence that we can note some convenient properties of

even and odd functions.

Definition 7.5.1. f : [−a, a] −! R is even iff

8 x 2 [−a, a], f(−x) = f(x)

Definition 7.5.2. f : [−a, a] −! R is odd iff

8 x 2 [−a, a], f(−x) = −f(x)

Some more or less obvious remarks:

Proposition 7.5.1. The product of even functions is even, of odd functions

even, and the product of an even function with an odd function is odd.

Proof:

Go on. _

Proposition 7.5.2. The integral of an odd function over [−a, a] is zero. _

Proposition 7.5.3. The integral of an even function over [−a, a] is twice

the integral over [0, a]. _

Proposition 7.5.4. If f is even and differentiable then f0 is odd.

Proof:

Compose f with the map −x and apply the chain rule. _

7.6. FOURIER SERIES 145

Proposition 7.5.5. If f is odd and differentiable then f0 is even.

Proof:

Same argument. _

Proposition 7.5.6. The functions cos(nx) are even for all n 2 Z and the

functions sin(nx) are odd for all n 2 Z on the interval [−_, _] _

Remark 7.5.2. This may explain why I projected the sign function down

only on the sin(nx) terms. Projecting on the cos terms would have given me

zero.

Exercise 7.5.1. Go over all the proofs sketched above and fill in all the

gaps until you are satisfied that you believe the claims made or that your

scepticism is unappeasable.

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