1000 Solved Problems in Modern Physics

1.3 Solutions 41

which is consistent with Dirichlet’s theorem. Similar behavior is exhibited at

x=π,± 2 π...Figure 1.8 shows first four partial sums with equations

y=π/ 2

y=π/ 2 +2sinx

y=π/ 2 +2(sinx+(1/3) sin 3x)

y=π/ 2 +2(sinx+(1/3) sin 3x+(1/5) sin 5x)

1.19 By Problem 1.18,

y=

π

2

+ 2

(

sinx+

(

1

3

)

sin 3x+

(

1

5

)

sin 5x+

(

1

7

)

sin 7x+···

)

Putx=π/2 in the above series

y=π=

π

2

+ 2

(

1 −

1

3

+

1

5

−

1

7

+···

)

Henceπ 4 = 1 −^13 +^15 −^17 +···

1.20 The Fourier transform off(x)is

T(u)=

1

√

2 π

∫a

−a

eiuxf(x)dx

=

1

√

2 π

∫a

−a

1 .eiuxdx=

1

√

2 π

eiux

iu

∣

∣

∣

∣

a

−a

=

1

√

2 π

(

eiua−e−iua

iu

)

=

√

2

π

sinua

u

,u

= 0

Foru=0,T(u)=

√

2

πu.

The graphs of f(x) andT(u)foru=3 are shown in Fig. 1.9a, b, respec-

tively

Note that the above transform finds an application in the FraunHofer

diffraction.

̃f(ω)=Asinα/α

This is the basic equation which describes the Fraunhofer’s diffraction pat-

tern due to a single slit.

Fig. 1.9Slit function and its Fourier transform