Introduction​​​​In this work we will look at an interesting definite integral where the general case was derive in closed form in terms of the Lerch function by Reynolds and Stauffer [1]. ​​​​
0
∫
∞
m
y
log(log(a y))
1+α
2 m+2
y
y=
π
2
α
(m+1)
(log(
2
(-2  (m+1) log(a)+ log(α)-π)
2
Γ(-
2  (m+1) log(a)- log(α)+π
4 π
)
2
(-2  (m+1) log(a)+ log(α)-3 π)
2
Γ(
-2  (m+1) log(a)+ log(α)+π
4 π
-1)
)+log(
2  π
m+1
))
​​​​where a, m, and α are general complex numbers. We will look at a particular case of the original integral and focus on the log-gamma function as a closed form solution. ​​​​References​​​​Reynolds, R.; Stauffer, A. A Definite Integral Involving the Logarithmic Function in Terms of the Lerch Function. Mathematics 2019, 7, 1148.​​​​https://www.mdpi.com/2227-7390/7/12/1148​​
In[]:=
D
m
y
k
Log[ay]
1+
2+2m
y
α
,
k

1+m
1+k
π
-
k
2
HurwitzZeta-k,
1
2
1+
π-(2+2m)Log[a]+Log[α]
2π
+
k
2
HurwitzZeta-k,
π-(2+2m)Log[a]+Log[α]
4π

(1+m)
α
,k/.{k0}//FullSimplify
In[]:=
reyn[a_,α_,m_]:=NIntegrate
m
y
Log[Log[ay]]
1+
2+2m
y
α
,{y,0,1/a,∞},N
πLog
2π
1+m
+Log
2
Gamma-
π+2(1+m)Log[a]-Log[α]
4π

-π-2(1+m)Log[a]+Log[α]
4π
Gamma-1+
π-2(1+m)Log[a]+Log[α]
4π

-3π-2(1+m)Log[a]+Log[α]
4π

2(1+m)
α

In[]:=
reynclosed[a_,α_,m_]:=Integrate
m
y
Log[Log[ay]]
1+
2+2m
y
α
,{y,0,∞},
πLog
2π
1+m
+Log
2
Gamma-
π+2(1+m)Log[a]-Log[α]
4π

-π-2(1+m)Log[a]+Log[α]
4π
Gamma-1+
π-2(1+m)Log[a]+Log[α]
4π

-3π-2(1+m)Log[a]+Log[α]
4π

2(1+m)
α

Here we will look at some interesting evaluations using Mathematica with plots. We will also plot the singularities and use Mathematica to integrate around these singularities to achieve the correct numerical result.
In[]:=
reyn[2+,1+,2+]
Out[]=
{0.112662+0.235263,0.112662+0.235263}
Example 1
In[]:=
reyn[2,16,1]
Out[]=
{-0.20121+0.308425,-0.20121+0.308425}
In[]:=
reynclosed[2,16,1]//Simplify
Out[]=

∞
∫
0
yLog[Log[2y]]
1+16
4
y
y,
1
16
πLog[π]+Log
2
Gamma-
1
4

9
2
Gamma-
3
4


In[]:=
PlotRe
yLog[Log[2y]]
1+16
4
y
,{y,0,1},GridLines{{1/2},None},GridLinesStyleDirective[Red,Dashed],ExclusionsNone
Out[]=
In[]:=
PlotIm
yLog[Log[2y]]
1+16
4
y
,{y,0,1},GridLines{{1/2},None},GridLinesStyleDirective[Red,Dashed],ExclusionsNone
Out[]=
Example 2
In[]:=
reyn[3,3^4,1]
Out[]=
{-0.0894266+0.137078,-0.0894266+0.137078}
In[]:=
reynclosed[3,3^4,1]//Simplify
Out[]=

∞
∫
0
yLog[Log[3y]]
1+81
4
y
y,
1
36
πLog[π]+Log
2
Gamma-
1
4

9
2
Gamma-
3
4


In[]:=
PlotRe
yLog[Log[3y]]
1+81
4
y
,{y,0,1},GridLines{{1/3},None},GridLinesStyleDirective[Red,Dashed],ExclusionsNone
Out[]=
In[]:=
PlotIm
yLog[Log[3y]]
1+81
4
y
,{y,0,1},GridLines{{1/3},None},GridLinesStyleDirective[Red,Dashed],ExclusionsNone
Out[]=
Example 3
In[]:=
reyn[2,2^6,2]
Out[]=
{-0.0936076+0.102808,-0.0936076+0.102808}
In[]:=
reynclosed[2,2^6,2]//Simplify
Out[]=

∞
∫
0
2
y
Log[Log[2y]]
1+64
6
y
y,
1
48
πLog
2π
3
+Log
2
Gamma-
1
4

9
2
Gamma-
3
4


In[]:=
PlotRe
2
y
Log[Log[2y]]
1+64
6
y
,{y,0,1},GridLines{{1/2},None},GridLinesStyleDirective[Red,Dashed],ExclusionsNone
Example 4
Example 5