1000 Solved Problems in Modern Physics

1.2 Problems 23

Fig. 1.2Saw-tooth wave

1.19 Use the result of Problem 1.18 for the Fourier series for the square wave to
prove that:

1 −

1

3

+

1

5

−

1

7

+··· =

π

4

1.20 Find the Fourier transform off(x)=

{

1 ,|x|<a

0 ,|x|>a

1.21 Use the Fourier integral to prove that:
∫∞

0

cosaxdx

1 +a^2

=

π

2

e−x

1.22 Show that the Fourier transform of the normalized Gaussian distribution

f(t)=

1

τ

√

2 π

e

−t^2

2 τ^2 , −∞<t<∞

is another Gaussian distribution.

1.2.3 Gamma and Beta Functions

1.23 The gamma function is defined by:

Γ(z)=

∫∞

0

e−xxz−^1 dx,(Re z>0)

(a) Show thatΓ(z+1)=zΓ(z)

(b) And ifzis a positive integern, thenΓ(n+1)=n!

1.24 The Beta functionB(m,n) is defined by the definite integral:

B(m,n)=

∫ 1

0

xm−^1 (1−x)n−^1 dx

and this defines a function ofmandnprovidedmandnare positive. Show

that:

B(m,n)=

T(m)T(n)

T(m+n)