33.3. FOURIER SERIES 641

Definition 33.3.1 A function f : R→ R is called periodic of period T if for all x ∈ R,

f (x+T ) = f (x) .

An example of a periodic function having period 2L is x→ sin( kπ

L x)

and x→ cos( kπ

L x).

If you want to approximate a function with these periodic functions, then it is necessarythat it be periodic of period 2L. Otherwise it would not be reasonable to expect to be ableto approximate the function in any useful way with these periodic functions.

Before doing anything else, here are some important trig. identities.

sinacosb =12(sin(a+b)+ sin(a−b)) (33.2)

cosacosb =12(cos(a−b)+ cos(a+b)) (33.3)

sinasinb =12(cos(a−b)− cos(a+b)) (33.4)

These follow right away from the standard trig. identities for the sum of two angles. Hereis a lemma which gives an orthogonality condition.

Lemma 33.3.2 The following formulas hold. For m,n positive integers,

∫ L

−L

1√L

sin(mπ

Lx) 1√

Lsin(nπ

Lx)

dx =

{0 if m ̸= n1 if m = n

∫ L

−L

1√L

cos(mπ

Lx) 1√

Lcos(nπ

Lx)

dx =

{0 if m ̸= n1 if m = n∫ L

−Lsin(mπ

Lx)

cos(nπ

Lx)

dx = 0

Proof: Consider the first of these formulas. From one of the above trig. identities,∫ L

−Lsin(mπ

Lx)

sin(nπ

Lx)

dx =

12

∫ L

−Lcos((mπ

L− nπ

L

)x)− cos

((mπ

L+

nπ

L

)x)

dx

If m ̸= n, this clearly integrates to 0. If m = n, you have

12

∫ L

−L

(1− cos

(2nL

x))

dx = L

Thus ∫ L

−L

1√L

sin(nπ

Lx) 1√

Lsin(nπ

Lx)

dx = 1

The second formula works out the same way. Consider the third.∫ L

−Lsin(mπ

Lx)

cos(nπ

Lx)

dx =