In this notebook we look at a pair of definite integrals similar to ones in Bierens de Haan, Table 147.​1.
1
∫
0
loglog
1
x

2
(x+1)
log
1
x

x=
π
2
2
ζ-
1
2
log(2)-(2
2
-1)
′
ζ
-
1
2
+ζ-
1
2
(0)
ψ
1
2
​​2.
1
∫
0
1
2
(x+1)
log
1
x

x=(1-2
2
)
π
ζ-
1
2
​​​ReferenceBierens de Haan, D., Nouvelles Tables d’int ́egrales d ́efinies, Amsterdam, 1867
In[]:=
NIntegrate
Log[Log[1/x]]
2
(x+1)
Log[1/x]
,{x,0,1},
π
2
2
Zeta[-1/2]Log[2]-2
2
-1(Zeta'[-1/2]+Zeta[-1/2]PolyGamma[0,1/2])//N
Out[]=
{-0.875784,-0.875784+2.5047×
-16
10
}
In[]:=
Plot
Log[Log[1/x]]
2
(x+1)
Log[1/x]
,{x,0,1},AxesLabelAutomatic
Out[]=
In[]:=
NIntegrate
1
2
(x+1)
Log[1/x]
,{x,0,1},
π
1-2
2
Zeta[-1/2]//N
Out[]=
{0.673718,0.673718-8.25067×
-17
10
}
In[]:=
Plot
1
2
(x+1)
Log[1/x]
,{x,0,1},AxesLabelAutomatic
Out[]=