Physical Foundations of Cosmology

84 The hot universe

of Integrals, Series, and Products(San Diego: Academic Press, 1994). The result
for purely imaginaryβcan be analytically continued to the realβand we obtain

J∓(−^1 )=

{

π

(

α^2 −β^2

)− 1 / 2

+(ln(α/ 4 π)+C)+O

(

α^2 ,β^2

)

,

−(ln(α/π)+C)+O

(

α^2 ,β^2

)

,

(3.40)

for bosons and fermions respectively. HereC≈ 0 .577 is Euler’s constant and
O

(

α^2 ,β^2

)

denotes terms which are quadratic and higher order inαandβ.

Problem 3.9Verify that the next subleading correction to (3.40) is

∓

7 ζ( 3 )

8 π^2

(α^2 + 2 β^2 ),

whereζis the Riemann zeta function.

To determineJ∓(^1 )andJ±(^3 )from (3.39) and (3.40), we need the “initial conditions”
J∓(ν)(α= 0 ,β).Settingα=0 and changing the integration variables, we can rewrite
the expression in (3.34) as

J∓(ν)(0,β)=

∫∞

0

(y+β)ν+(y−β)ν

ey∓ 1

dy+

∫^0

−β

(y+β)ν

ey∓ 1

dy−

∫β

0

(y−β)ν

ey∓ 1

dy.

(3.41)
Replacingyby−yin the last integral and noting that

1

ey∓ 1

+

1

e−y∓ 1

=∓ 1 ,

we obtain for oddν

J∓(ν)(0,β)=

∫∞

0

(y+β)ν+(y−β)ν

ey∓ 1

dy∓

βν+^1

ν+ 1

. (3.42)

It follows that

J∓(^1 )(0,β)=

{ 1

3 π

(^2) −^1
2 β
(^2) ,
1
6 π
(^2) + 1
2 β
(^2). (3.43)
Substituting (3.40) into (3.39) and taking into account (3.43), one finds

J∓(^1 )=

⎧

⎪⎪⎪

⎨

⎪⎪⎪

⎩

1

3

π^2 −

1

2

β^2 −π

√

α^2 −β^2 −

1

2

α^2

(

ln

(α

4 π

)

+C−

1

2

)

+α^2 O,

1

6

π^2 +

1

2

β^2 +

1

2

α^2

(

ln

(α

π

)

+C−

1

2

)

+α^2 O,

(3.44)