g(x_)=;CMRB=(g(k)-1)
1/x
x
∞
∑
k=1
k
(-1)
In[]:=
g[x_]=x^(1/x);CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},WorkingPrecision100,Method"AlternatingSigns"];
g(x_)=;CMRB=(g(k)-1)
1/x
x
∞
∑
k=1
k
(-1)
g(x_)=;w=;CMRB=t,a,b∈.
1/x
x
(t-b)
t-a
b
∫
a
(b-a)Re(g(w+1)csc(πw))
2
(t-a)
CMRB=
∞
∫
1
Re(g(t)csc(πt))
(t/(-)).CMRB=∞
∫
0
Re(g[(1+tI)]csc(πt))
tCMRB∞
∫
1
Re(g(t+1-)csc(π(t-1)))
tg[x_]=x^(1/x);CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},WorkingPrecision100,Method"AlternatingSigns"];
In[]:=
g[x_]=x^(1/x);Tablew=;CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecision100],{a,0,5},{b,a+1,6}
(t-b)
t-a
Out[]=
{{-9.3472×,-9.3472×,-9.3472×,-9.3472×,-9.3472×,-9.3472×},{-9.3472×,-9.3472×,-9.3472×,-9.3472×,-9.3472×},{-9.3472×,-9.3472×,-9.3472×,-9.3472×},{-9.3472×,-9.3472×,-9.3472×},{-9.3472×,-9.3472×},{-9.3472×}}
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In[]:=
g[x_]=x^(1/x);Tablew=;CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecision100],{a,4/10,5},{b,a+1,6}
(t-b)
t-a
Out[]=
{{-9.3472×,-9.3472×,-9.3472×,-9.3472×,-9.3472×},{-9.3472×,-9.3472×,-9.3472×,-9.3472×},{-9.3472×,-9.3472×,-9.3472×},{-9.3472×,-9.3472×},{-9.3472×}}
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In[]:=
a=E;b=Pi;
In[]:=
a=E;b=Pi;g[x_]=x^(1/x);w=;Print[CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecision100]];Clear[a,b]
(t-b)
t-a
-9.3472×
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In[]:=
a=E;b=Pi;g[x_]=x^(1/x);w=;Print[Integrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b}]];Clear[a,b]
(t-b)
t-a
π
∫
1
1+
(-π+t)
-+t
(1+((-π+t))/(-+t))
2
(-+t)
CMRB=(-+π)ImCsch[(π(-π+t))/(-+t)]t
π
∫
1
1+
(-π+t)
-+t
(1+((-π+t))/(-+t))
2
(-+t)
In[]:=
g[x_]=x^(1/x);CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},WorkingPrecision100,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},WorkingPrecision100]]);Clear[a,b]
-9.3472×+0.×
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In[]:=
g[x_]=x^(1/x);Tablew=;CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecision100],{a,0,5},{b,a+1,6}
(t-b)
t-a
Out[]=
{{-9.3472×,-9.3472×,-9.3472×,-9.3472×,-9.3472×,-9.3472×},{-9.3472×,-9.3472×,-9.3472×,-9.3472×,-9.3472×},{-9.3472×,-9.3472×,-9.3472×,-9.3472×},{-9.3472×,-9.3472×,-9.3472×},{-9.3472×,-9.3472×},{-9.3472×}}
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In[]:=
g[x_]=x^(1/x);Tablew=;CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecision100],{a,4/10,5},{b,a+1,6}
(t-b)
t-a
Out[]=
{{-9.3472×,-9.3472×,-9.3472×,-9.3472×,-9.3472×},{-9.3472×,-9.3472×,-9.3472×,-9.3472×},{-9.3472×,-9.3472×,-9.3472×},{-9.3472×,-9.3472×},{-9.3472×}}
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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},WorkingPrecision100]]);Clear[a,b]
-9.3472×
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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},WorkingPrecision100]]);Clear[a,b]
-9.3472×+0.×
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10
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In[]:=
a=1;b=I;g[x_]=x^(1/x);w=;Print[CMRB-NIntegrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecision100]];Clear[a,b]
(t-b)
t-a
-9.3472×+0.×
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10
In[]:=
a=1;b=I;g[x_]=x^(1/x);w=;Print[Integrate[Re[g[(1+w)]Csc[πw]](t-a)^-2*(b-a),{t,a,b}]];Clear[a,b]
(t-b)
t-a
∫
1
(1-)ImCsch
1
1+
(-+t)
-1+t
1+
(-+t)
-1+t
π(-+t)
-1+t
2
(-1+t)
CMRB=-t
∫
1
(1-)ImCsch
1
1+
(-+t)
-1+t
1+
(-+t)
-1+t
π(-+t)
-1+t
2
(-1+t)
Somestrangeexamples:
CMRB=(-+π)ImCsch[(π(-π+t))/(-+t)]t.
π
∫
1
1+
(-π+t)
-+t
(1+((-π+t))/(-+t))
2
(-+t)
CMRB=-t.
∫
1
(1-)ImCsch
1
1+
(-+t)
-1+t
1+
(-+t)
-1+t
π(-+t)
-1+t
2
(-1+t)