Double integrals expressed in terms of Catalan’s and Aprey’s constants
Double integrals expressed in terms of Catalan’s and Aprey’s constants
Here we write down the double integral involving the reciprocal logarithmic function in terms of Catalan’s constant and use Mathematica to evaluate the expression
In[]:=
Integrate-1+Log[xy],{x,0,∞},{y,0,∞},Catalan
1/4
x
x
y
2
(1+x)
1/4
y
2
(1+y)
Out[]=
-1+Log[xy]yx,Catalan
∞
∫
0
∞
∫
0
1/4
x
x
y
2
(1+x)
1/4
y
2
(1+y)
Here we use Mathematica to numerical evaluate the double integral in terms of Catalan’s constant
In[]:=
NIntegrate-1+Log[xy],{x,0,∞},{y,0,∞},Catalan//N
1/4
x
x
y
2
(1+x)
1/4
y
2
(1+y)
Out[]=
{0.915966,0.915966}
Here we look at a 3D plot of the integrand
In[]:=
Plot3D-1+Log[xy],{x,0,1},{y,0,1},AxesLabelAutomatic
1/4
x
x
y
2
(1+x)
1/4
y
2
(1+y)
Out[]=
Here we write down the double integral involving the reciprocal logarithmic function in terms of Catalan’s constant and use Mathematica to evaluate the expression
In[]:=
NIntegrateLog[xy],{x,0,1,∞},{y,0,∞},Catalan-//N
-1+
1/4
x
1/4
y
3/4
x
2
(1+x)
1/4
y
2
(1+y)
7Zeta[3]
8π
Out[]=
{0.581167,0.581167}
Here we use Mathematica to numerical evaluate the double integral in terms of Catalan’s and Aprey’s constants
In[]:=
IntegrateLog[xy],{x,0,∞},{y,0,∞},Catalan-
-1+
1/4
x
1/4
y
3/4
x
2
(1+x)
1/4
y
2
(1+y)
7Zeta[3]
8π
Out[]=
Log[xy]yx,Catalan-
∞
∫
0
∞
∫
0
-1+
1/4
x
1/4
y
3/4
x
2
(1+x)
1/4
y
2
(1+y)
7Zeta[3]
8π
Here we look at a 3D plot of the integrand
In[]:=
Plot3DLog[xy],{x,0,1},{y,0,1},AxesLabelAutomatic
-1+
1/4
x
1/4
y
3/4
x
2
(1+x)
1/4
y
2
(1+y)
Out[]=
Here we take the difference of both equations to have an explicit integral for Aprey’s constant. Note the singularity at x=1
In[]:=
NIntegrateLog[xy],{x,0,1,∞},{y,0,∞},//N
-1+-1+x+
1/4
x
1/4
y
5/4
x
1/4
y
3/4
x
2
(1+x)
1/4
y
2
(1+y)
7Zeta[3]
8π
Out[]=
{0.334798,0.334798}
Here we use Mathematica to numerical evaluate the double integral in terms Aprey’s constants
In[]:=
IntegrateLog[xy],{x,0,∞},{y,0,∞},
-1+-1+x+
1/4
x
1/4
y
5/4
x
1/4
y
3/4
x
2
(1+x)
1/4
y
2
(1+y)
7Zeta[3]
8π
Out[]=
Log[xy]yx,
∞
∫
0
∞
∫
0
-1+x+-1+
5/4
x
1/4
y
1/4
(xy)
3/4
x
2
(1+x)
1/4
y
2
(1+y)
7Zeta[3]
8π
Here we look at a 3D plot of the integrand
In[]:=
Plot3DLog[xy],{x,0,1},{y,0,1},AxesLabelAutomatic
-1+-1+x+
1/4
x
1/4
y
5/4
x
1/4
y
3/4
x
2
(1+x)
1/4
y
2
(1+y)
Out[]=
References
[1] Robert Reynolds and Allan Stauffer
A Method for Evaluating Definite Integrals inTerms of Special Functions with Examples, International Mathematical Forum, Vol. 15, 2020, no. 5, 235- 244
References
[1] Robert Reynolds and Allan Stauffer
A Method for Evaluating Definite Integrals inTerms of Special Functions with Examples, International Mathematical Forum, Vol. 15, 2020, no. 5, 235- 244
[1] Robert Reynolds and Allan Stauffer
A Method for Evaluating Definite Integrals inTerms of Special Functions with Examples, International Mathematical Forum, Vol. 15, 2020, no. 5, 235- 244