33.6. INTEGRATING AND DIFFERENTIATING FOURIER SERIES 657
and that f ′ is piecewise continuous and 2L periodic. Let f denote the 2L periodic extensionof the above f . Then let the formal Fourier series for f ′ be
a0 +∞
∑n=1
an cos(nπx
L
)+
∞
∑n=1
bn sin(nπx
L
)Then by Theorem 33.6.2,∫ x
−Lf ′ (t)dt =
∫ x
−La0dt +
∞
∑n=1
anL
nπsin(nπx
L
)+
∞
∑n=1
bnL
nπ
((−1)n− cos
(nπxL
))Then a0 ≡ 1
2L∫ L−L f ′ (t)dt = 1
2L ( f (L)− f (−L)) = 0.
an ≡ 1L
∫ L
−Lf ′ (t)cos
(nπtL
)dt =
1L
f (t)cos(nπt
L
)|L−L
+1L
nπ
L
∫ L
−Lf (t)sin
(nπtL
)dt
=1L
nπ
L
∫ L
−Lf (t)sin
(nπtL
)dt = Bn
nπ
Lwhere Bn is the Fourier coefficient for f (t) . Similarly,
bn =1L
∫ L
−Lf ′ (t)sin
(nπtL
)dt =−1
Lnπ
L
∫ L
−Lf (t)cos
(nπtL
)dt =−nπ
LAn
where An is the nth cosine Fourier coefficient for f . Thus∫ x
−Lf ′ (t)dt =
∞
∑n=1
Bnnπ
LL
nπsin(nπx
L
)+
∞
∑n=1
(−nπ
LAn
) Lnπ
((−1)n− cos
(nπxL
))f (x)− f (−L) =
∞
∑n=1
Bn sin(nπx
L
)+
∞
∑n=1
An cos(nπx
L
)−
∞
∑n=1
An (−1)n
f (x) =∞
∑n=1
Bn sin(nπx
L
)+
∞
∑n=1
An cos(nπx
L
)+
(f (−L)−
∞
∑n=1
An (−1)n
)Thus that constant on the end is A0. It follows that
f (x) = A0 +∞
∑n=1
Bn sin(nπx
L
)+
∞
∑n=1
An cos(nπx
L
)and − nπ
L An = bn, Bnnπ
L = an and so
f ′ (x) =∞
∑n=1
an cos(nπx
L
)+
∞
∑n=1
bn sin(nπx
L
)=
∞
∑n=1
Bnnπ
Lcos(nπx
L
)+
∞
∑n=1
An
(−nπ
L
)sin(nπx
L
)=
∞
∑n=1
Bnddx
sin(nπx
L
)+
∞
∑n=1
Anddx
cos(nπx
L
)