g(x_)=
1/x
x
;CMRB=
∞
∑
k=1
k
(-1)
(g(k)-1)
In[]:=
g[x_]=x^(1/x);CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},​​WorkingPrecision100,Method"AlternatingSigns"];
g(x_)=
1/x
x
;CMRB=
∞
∑
k=1
k
(-1)
(g(k)-1)
g(x_)=
1/x
x
;w=
(t-b)
t-a
;​​CMRB=
b
∫
a
(b-a)Re(g(w+1)csc(πw))
2
(t-a)
t,a,b∈.
CMRB=
∞
∫
1
Re(g(t)csc(πt))
(t/(-)).​​CMRB=
∞
∫
0
Re(g[(1+tI)]csc(πt))
t​​CMRB
∞
∫
1
Re(g(t+1-)csc(π(t-1)))
t
g[x_]=x^(1/x);CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},​​WorkingPrecision100,Method"AlternatingSigns"];
In[]:=
g[x_]=x^(1/x);Tablew=
(t-b)
t-a
;​​CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},​​WorkingPrecision100],{a,0,5},{b,a+1,6}
Out[]=
{{-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
}}
In[]:=
g[x_]=x^(1/x);Tablew=
(t-b)
t-a
;​​CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},​​WorkingPrecision100],{a,4/10,5},{b,a+1,6}
Out[]=
{{-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
}}
In[]:=
a=E;b=Pi;
In[]:=
a=E;b=Pi;g[x_]=x^(1/x);w=
(t-b)
t-a
;​​Print[CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecision100]];Clear[a,b]
-9.3472×
-94
10
In[]:=
a=E;b=Pi;g[x_]=x^(1/x);w=
(t-b)
t-a
;​​Print[Integrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b}]];Clear[a,b]
π
∫

(-+π)Im
1
1+
(-π+t)
-+t
(1+((-π+t))/(-+t))
Csch[(π(-π+t))/(-+t)]
2
(-+t)
t
CMRB=
π
∫

(-+π)Im
1
1+
(-π+t)
-+t
(1+((-π+t))/(-+t))
Csch[(π(-π+t))/(-+t)]
2
(-+t)
t
In[]:=
g[x_]=x^(1/x);CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},​​WorkingPrecision100,Method"AlternatingSigns"];
In[]:=
a=E+PiI;b=Pi-EI;g[x_]=x^(1/x);(w=((t-b))/(t-a);​​Print[CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecision100]]);Clear[a,b]
-9.3472×
-94
10
+0.×
-107
10

In[]:=
g[x_]=x^(1/x);Tablew=
(t-b)
t-a
;​​CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},​​WorkingPrecision100],{a,0,5},{b,a+1,6}
Out[]=
{{-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
}}
In[]:=
g[x_]=x^(1/x);Tablew=
(t-b)
t-a
;​​CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},​​WorkingPrecision100],{a,4/10,5},{b,a+1,6}
Out[]=
{{-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
,-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
,-9.3472×
-94
10
},{-9.3472×
-94
10
}}
In[]:=
a=E;b=Pi;
In[]:=
a=E;b=Pi;g[x_]=x^(1/x);(w=((t-b))/(t-a);​​Print[CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecision100]]);Clear[a,b]
-9.3472×
-94
10
In[]:=
a=E+PiI;b=Pi-EI;g[x_]=x^(1/x);(w=((t-b))/(t-a);​​Print[CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecision100]]);Clear[a,b]
-9.3472×
-94
10
+0.×
-107
10

In[]:=
a=1;b=I;g[x_]=x^(1/x);w=
(t-b)
t-a
;​​Print[CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecision100]];Clear[a,b]
-9.3472×
-94
10
+0.×
-189
10

In[]:=
a=1;b=I;g[x_]=x^(1/x);w=
(t-b)
t-a
;​​Print[Integrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b}]];Clear[a,b]

∫
1
-
(1-)Im
1
1+
(-+t)
-1+t
1+
(-+t)
-1+t
Csch
π(-+t)
-1+t

2
(-1+t)
t
​
CMRB=

∫
1
-
(1-)Im
1
1+
(-+t)
-1+t
1+
(-+t)
-1+t

Csch
π(-+t)
-1+t

2
(-1+t)
t
​​​​​​
Somestrangeexamples:
CMRB=
π
∫

(-+π)Im
1
1+
(-π+t)
-+t
(1+((-π+t))/(-+t))
Csch[(π(-π+t))/(-+t)]
2
(-+t)
t.
CMRB=

∫
1
-
(1-)Im
1
1+
(-+t)
-1+t
1+
(-+t)
-1+t

Csch
π(-+t)
-1+t

2
(-1+t)
t.