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

((−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

L

∫ L

−Lf (t)sin

(nπtL

)dt

=1L

L

∫ L

−Lf (t)sin

(nπtL

)dt = Bn

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

)

33.6. INTEGRATING AND DIFFERENTIATING FOURIER SERIES 657and 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’ beao-+ Yauco (7 ")+¥o sin (“ *)Then by Theorem 33.6.2,[fae = [ate + nin ("P)Ende (-ar-oo("E)Then ap = +f", f (t)dt = 4 (f (L)—f (-L)) =0.an = r | £Ooos ven+i "£0 sin ( mm at=i fr F(t) sin ( mm at = Bywhere B,, is the Fourier coefficient for f (t) . Similarly,1bn =i [#0 sin ( mY a= pt cos ( me) at = <a Anwhere A, is the n’” cosine Fourier coefficient for f. Thus[fae = YB = sin (“™)+E (OT An) ge ("08 (72)f(x) —f(-L) = Y Basin (“T +h cos (“ *\- Lavnix — h=D sin (“ L ")+ Yan cos (“ L sie BanThus that constant on the end is Ao. It follows thatf(x)= Aut Basin (™ r *)+ Yan cos ( a)and —"FAn = bn, Bn = dn and soancos (“™ "+L sin (“ *)sco) Fy (8) snas (“)+ Yas 7005 ( T)csfi) ==llin|Ms~TMsw&3|2=)Byll