Begin2.DVI

Now interchange the roles of summation and integration on the right hand side of

equation (12.119) to obtain

ζ(z) =

∑∞

r=1

1

rz

=^1

Γ(z)

∫∞

0

xz−^1

∑∞

r=1

e−rx dx (12.120)

where now the summation on the right hand side of equation (12.120) is the geo-

metric series ∞

∑

r=1

e−rx =e−x+e−^2 x+e−^3 x+··· = e

−x

1 −e−x

Consequently, the equation (12.120) simplifies to the form

ζ(z)Γ(z) =

∫∞

0

xz−^1

e−x

1 −e−xdx (12.121)

Observe that the Gamma function and Zeta function properties dictate that z be

restricted such that z
= 1, 0 ,− 1 ,− 2 ,− 3 ,... in using equation (12.121).

Product property of the Gamma function

The Gamma function satisfies the product property that

Γ

(

1

n

)

Γ

(

2

n

)

Γ

(

3

n

)

Γ

(

4

n

)

···Γ

(

n− 1

n

)

=

(2π)n−^21

n^12

(12.122)

In order to derive this result we develop the following background material.

Example 12-7.

Show that

sin nθ = 2n−^1 sin θsin(θ+π

n

) sin(θ+^2 π

n

) sin(θ+^3 π

n

)···sin( θ+ (n−1)π

n

) (12.123)

and

θlim→ 0

sin nθ

sin θ

=n= 2n−^1 sin(

π

n

) sin(

2 π

n

) sin(

3 π

n

)···sin(

(n−1)π

n

) (12.124)

Solution

The following proof follows that presented in the reference Hobson^3

First show that the function

x^2 n− 2 xncos nθ + 1 (12.125)

(^3) Ernest William Hobson, A treatise on plane trigonometry , 5th Edition, Cambridge University
Press, 1921, Pages 117-119.