In this Note book we look at the below integral and a few plots. We also look at some values for a, real, complex and purely imaginary.
In[]:=
Integrate
xLog[Log[ax]]
1+
4
x
,{x,0,∞},
1
8
ππ+2Log[π]+4Log
Gamma
3
4
-
Log[a]
π

Gamma
1
4
-
Log[a]
π

//TraditionalForm
Out[]//TraditionalForm=

∞
∫
0
xlog(log(ax))
4
x
+1
x,
1
8
π4log
Γ
3
4
-
log(a)
π
Γ
1
4
-
log(a)
π
+π+2log(π)
Note the singularity at x=1/a when a is real. So we have to integrate around the singularity.
In[]:=
genr[a_]:=NIntegrate
xLog[Log[ax]]
1+
4
x
,{x,0,1/a,∞},N
1
8
ππ+2Log[π]+4Log
Gamma
3
4
-
Log[a]
π

Gamma
1
4
-
Log[a]
π


In[]:=
geni[a_]:=NIntegrate
xLog[Log[ax]]
1+
4
x
,{x,0,∞},N
1
8
ππ+2Log[π]+4Log
Gamma
3
4
-
Log[a]
π

Gamma
1
4
-
Log[a]
π


In[]:=
genclosed[a_]:=Integrate
xLog[Log[ax]]
1+
4
x
,{x,0,∞},
1
8
ππ+2Log[π]+4Log
Gamma
3
4
-
Log[a]
π

Gamma
1
4
-
Log[a]
π


In[]:=
Plot3DRe
xLog[Log[ax]]
1+
4
x
,{a,-1,1},{x,0,2},AxesLabel{"a","x"}
Out[]=
In[]:=
Plot3DIm
xLog[Log[ax]]
1+
4
x
,{a,-1,1},{x,0,2},AxesLabel{"a","x"}
Out[]=
In[]:=
genr[-1]
Out[]=
{0.920857+1.2337,0.920857+1.2337}
In[]:=
genclosed[-1]
Out[]=

∞
∫
0
xLog[Log[-x]]
1+
4
x
x,
1
8
ππ+2Log[π]+4Log
Gamma
7
4

Gamma
5
4


In[]:=
genr[2]
Out[]=
{-0.401387+0.384812,-0.401387+0.384812}
In[]:=
genclosed[2]//FullSimplify
Out[]=

∞
∫
0
xLog[Log[2x]]
1+
4
x
x,
1
8
ππ+2Log[π]+4Log
Gamma
3
4
-
Log[2]
π

Gamma
1
4
-
Log[2]
π


In[]:=
genr[3]
Out[]=
{-0.0681565+0.17382,-0.0681565+0.17382}
In[]:=
genclosed[3]//FullSimplify
Out[]=

∞
∫
0
xLog[Log[3x]]
1+
4
x
x,
1
8
ππ+2Log[π]+4Log
Gamma
3
4
-
Log[3]
π

Gamma
1
4
-
Log[3]
π


In[]:=
genr[-1+]
Out[]=
{0.716083+1.1279,0.716083+1.1279}
In[]:=
genclosed[-1+]//FullSimplify
Out[]=

∞
∫
0
xLog[Log[(-1+)x]]
1+
4
x
x,
1
8
ππ+2Log[π]+Log
4
Gamma
3
2
-
Log[2]
2π

4
Gamma1-
Log[2]
2π


In[]:=
genr[2]
Out[]=
{0.474316+0.946769,0.474316+0.946769}
In[]:=
genclosed[2]//FullSimplify
Out[]=

∞
∫
0
xLog[Log[2x]]
1+
4
x
x,
1
8
ππ+2Log[π]+4Log
Gamma
5
4
-
Log[2]
π

Gamma
3
4
-
Log[2]
π


Here we use the the formula without the singularity to yield the evaluation
In[]:=
geni[]
Out[]=
{0.425391+1.2337,0.425391+1.2337}
In[]:=
genclosed[]//FullSimplify
Out[]=

∞
∫
0
xLog[Log[x]]
1+
4
x
x,
1
8
ππ+Log
2
π
4
Gamma
5
4

4
Gamma
3
4

