CMRB=0.18785964246206712024...
n=1or25.65665403510586285599...
g(x_)=
n/x
x
;
-1
2
Im
∞
∫
(-∞)
g(1-t)
sin(πt)
t=CMRB.
g(x_)=
n/x
x
;
1
2
Im
∞
∫
(-∞)
g(1+t)
sin(πt)
t=CMRB.
g(x_)=
n/x
x
;​​
1
4
Im
∞
∫
(-∞)
g(t+1)-g(1-t)
sin(πt)
t=CMRB.
In[]:=
n={1,25.6566540351058628559907};
In[]:=
g[x_]=x^(n/x);​​-1/2Im[N[NIntegrate[(g[(1-t)])/(Sin[πt]),{t,-InfinityI,InfinityI},WorkingPrecision60],20]]
Out[]=
{0.18785964246206712025,0.18785964246206712025}
In[]:=
g[x_]=x^(n/x);​​1/2Im[N[NIntegrate[(g[(1+t)])/(Sin[πt]),{t,-InfinityI,InfinityI},WorkingPrecision60],20]]
Out[]=
{0.18785964246206712025,0.18785964246206712025}
In[]:=
g[x_]=x^(n/x);​​1/4Im[N[NIntegrate[(g[(1+t)]-(g[(1-t)]))/(Sin[πt]),{t,-InfinityI,InfinityI},WorkingPrecision60],20]]
Out[]=
{0.18785964246206712025,0.18785964246206712025}
In[]:=
g[x_]=x^(1/x);CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},WorkingPrecision->100,Method->"AlternatingSigns"];a=0;b=I;g[x_]=x^(1/x);v=
t
1+t+t
;​​PrintCMRB--NIntegrate
Re[
-v
v
Csc[Pi/v]]
2
t
,{t,a,b},WorkingPrecision->100;Clear[a,b]
-9.3472×
-94
10
v=
t
t+t+1
;​​CMRB=
∞
∑
k=1
k
(-1)
(
1/k
k
-1)=-

∫
0
Re
-v
v
csc
π
v

2
t
t=
1
2
(-)
∞
∫
(-∞)
Re
-v
v
csc
π
v

2
t
t.
v=
t
t+t+1
;​​CMRB=
∞
∑
k=1
k
(-1)
(
1/k
k
-1)=
1
2
(-)
∞
∫
(-∞)
Re
-v
v
csc
π
v

2
t
t.
v=
t
t+t+1
;​​CMRB=
∞
∑
k=1
k
(-1)
(
1/k
k
-1)=-

∫
0
Re
-v
v
csc
π
v

2
t
t
g(x_)=
1/x
x
;​​
1
4
Im
∞
∫
(-∞)
g(t+1)-g(1-t)
sin(πt)
t=CMRB.
In[]:=
g[x_]=x^(1/x);CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},WorkingPrecision->100,Method->"AlternatingSigns"];a=-InfinityI;b=InfinityI;​​g[x_]=x^(1/x);(v=t/(1+t+tI);​​Print[CMRB-(-I/2NIntegrate[Re[v^-vCsc[Pi/v]]/(t^2),{t,a,b},WorkingPrecision->100])]);Clear[a,b]
-9.3472×
-94
10
In[]:=
CMRB-(-INIntegrate[Re[v^-vCsc[Pi/v]]/(t^2),{t,0,InfinityI},WorkingPrecision->100]+INIntegrate[Re[v^-vCsc[Pi/v]]/(t^2),{t,I,InfinityI},WorkingPrecision->100])
Out[]=
-9.3472×
-94
10
CMRB=-
∞
∫

Re
-v
v
csc
π
v

2
t
t-
∞
∫
0
Re
-v
v
csc
π
v

2
t
t
In[]:=
CMRB-(-INIntegrate[Re[v^-vCsc[Pi/v]]/(t^2),{t,0,I},WorkingPrecision->100])
Out[]=
-9.3472×
-94
10
CMRB=-

∫
0
Re
-v
v
csc
π
v

2
t
t
CMRB=-
∞
∫

Re
-v
v
csc
π
v

2
t
t-
∞
∫
0
Re
-v
v
csc
π
v

2
t
t=-

∫
0
Re
-v
v
csc
π
v

2
t
t
In[]:=
1/-I
Out[]=

g(x_)=
1/x
x
;v=
t
t+t+1
;​​CMRB=
∞
∫

Re
-v
v
csc
π
v

2
t
t-
∞
∫
0
Re
-v
v
csc
π
v

2
t
t=

∫
0
Re
-v
v
csc
π
v

2
t
t.
CMRB=
∞
∑
k=1
k
(-1)
(
1/k
k
-1)
In[]:=
g[x_]=x^(1/x);CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},WorkingPrecision->100,Method->"AlternatingSigns"];a=-InfinityI;b=InfinityI;g[x_]=x^(1/x);v=
t
1+t+t
;​​PrintCMRB--2NIntegrate
Re[
-v
v
Csc[Pi/v]]
2
t
,{t,a,b},WorkingPrecision->100;Clear[a,b]
-9.3472×
-94
10
1
2
(-)
∞
∫
(-∞)
Re
-v
v
csc
π
v

2
t
t
In[]:=
g[x_]=x^(1/x);CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},WorkingPrecision->100,Method->"AlternatingSigns"];a=0;b=I;g[x_]=x^(1/x);v=
t
1+t+t
;​​PrintCMRB--NIntegrate
Re[
-v
v
Csc[Pi/v]]
2
t
,{t,a,b},WorkingPrecision->1000;Clear[a,b]
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in t near {t} = {0.937500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}. NIntegrate obtained 1007 and 1013 for the integral and error estimates.
-9.3472×
-94
10
In[]:=
CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},WorkingPrecision->1000,Method->"AlternatingSigns"];
In[]:=
CMRB-N-NIntegrate
Re[
-v
v
Csc[Pi/v]]
2
t
,{t,0,},WorkingPrecision->2000,1000
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in t near {t} = {0.×
-2051
10
+2060}. NIntegrate obtained 2058 and 2063 for the integral and error estimates.
Out[]=
1.136904579174515915810652944398114538901009851043650092031804195061599764129×
-923
10