In 1861 Von Dr. Anton Winkler published the derivation of the first two integrals in [2]. Reynolds and Stauffer recently [1] derived these famous integrals. In this Notebook we will look at using Mathematica to plot these integrals.​1)
∞
∫
0
log(
-2παy

+1)
2
y
+1
y=-πlog
-α
α
Γα+
1
2

2π
+α
​​2)
∞
∫
0
log(1-
-2παy

)
2
y
+1
y=-
1
2
πlog
-2α-1
α
2
Γ(α+1)
2π
+2α
​​3)
∞
∫
0
logtanh
αy
2

2
a
+
2
y
y=
π
2a
log
aα
2
Γ
aα+π
2π

2π
2
Γ
aα
2π
+1
​​where Re(α)>0 and Re(a)>0.​References​1) Reynolds, R.; Stauffer, A. Derivation of Logarithmic and Logarithmic Hyperbolic Tangent Integrals Expressed in Terms of Special Functions. Mathematics 2020, 8, 687​2) Anton, W. Eigenschaften Einiger Bestimmten Integrale; Hof, K.K., Ed.; Bavarian State Library: Bavaria, Germany, 1861
In[]:=
winckler1[α_]:=NIntegrate
Log[1+
-2πyα

]
1+
2
y
,{y,0,∞},N-πα+Log
Gamma
1
2
+α
α
α
2π

In[]:=
winckler1closed[α_]:=Integrate
Log[1+
-2πyα

]
1+
2
y
,{y,0,∞},-πα+Log
Gamma
1
2
+α
α
α
2π

In[]:=
winckler2[α_]:=NIntegrate
Log[1-
-2πyα

]
1+
2
y
,{y,0,∞},N-
1
2
π2α+Log
2
Gamma[1+α]
2π
1+2α
α

In[]:=
winckler2closed[α_]:=Integrate
Log[1-
-2πyα

]
1+
2
y
,{y,0,∞},-
1
2
π2α+Log
2
Gamma[1+α]
2π
1+2α
α

In[]:=
winckler1[1+]
Out[]=
{0.0670191-0.0633635,0.0670191-0.0633635}
In[]:=
winckler2[2+]
Out[]=
{-0.104555+0.0516243,-0.104555+0.0516243}
In[]:=
Plot3DRe
Log[1+
-2πyα

]
1+
2
y
,{α,0,3},{y,0,4}
Out[]=
In[]:=
Plot3DIm
Log[1+
-2πyα

]
1+
2
y
,{α,0,3},{y,0,4}
Out[]=
In[]:=
Plot3DRe
Log[1-
-2πyα

]
1+
2
y
,{α,0,3},{y,0,4}
Out[]=
In[]:=
Plot3DIm
Log[1-
-2πyα

]
1+
2
y
,{α,0,3},{y,0,4}
Out[]=
In[]:=
AnimatePlot3DRe
Log[1+
-2πy(a+b)

]
1+
2
y
,{a,0,1},{b,-4,4},AxesLabelAutomatic,{y,0,5}
Out[]=
y
In[]:=
AnimatePlot3DIm
Log[1+
-2πy(a+b)

]
1+
2
y
,{a,0,1},{b,-4,4},AxesLabelAutomatic,{y,0,5}
Out[]=
y