g(x_)=;CMRB=(g(k)-1)
1/x
x
β
β
k=1
k
(-1)
In[ο]:=
g[x_]=x^(1/x);CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},ββWorkingPrecisionο’100,Methodο’"AlternatingSigns"];
g(x_)=;CMRB=(g(k)-1)
1/x
x
β
β
k=1
k
(-1)
g(x_)=;w=;ββCMRB=οt,a,bβο¦.
1/x
x
ο(t-b)
t-a
b
β«
a
(b-a)Re(g(w+1)csc(Οw))
2
(t-a)
CMRB=
οβ
β«
1
Re(g(t)csc(Οt))
(οt/(-ο)).ββCMRB=β
β«
0
Re(g[(1+tI)]csc(οΟt))
οtββCMRBοβ
β«
1
Re(g(οt+1-ο)csc(οΟ(t-1)))
οtg[x_]=x^(1/x);CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},ββWorkingPrecisionο’100,Methodο’"AlternatingSigns"];
In[ο]:=
g[x_]=x^(1/x);Tableοw=;ββCMRB-NIntegrate[Re[g[(1+w)]Csc[Οw]](t-a)^-2*(b-a),{t,a,b},ββWorkingPrecisionο’100],{a,0,5},{b,a+1,6}ο
ο(t-b)
t-a
Out[ο]=
{{-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ},{-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ},{-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ},{-9.3472Γ,-9.3472Γ,-9.3472Γ},{-9.3472Γ,-9.3472Γ},{-9.3472Γ}}
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In[ο]:=
g[x_]=x^(1/x);Tableοw=;ββCMRB-NIntegrate[Re[g[(1+w)]Csc[Οw]](t-a)^-2*(b-a),{t,a,b},ββWorkingPrecisionο’100],{a,4/10,5},{b,a+1,6}ο
ο(t-b)
t-a
Out[ο]=
{{-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ},{-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ},{-9.3472Γ,-9.3472Γ,-9.3472Γ},{-9.3472Γ,-9.3472Γ},{-9.3472Γ}}
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In[ο]:=
a=E;b=Pi;
In[ο]:=
a=E;b=Pi;g[x_]=x^(1/x);w=;ββPrint[CMRB-NIntegrate[Re[g[(1+w)]Csc[Οw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecisionο’100]];Clear[a,b]
ο(t-b)
t-a
-9.3472Γ
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In[ο]:=
a=E;b=Pi;g[x_]=x^(1/x);w=;ββPrint[Integrate[Re[g[(1+w)]Csc[Οw]](t-a)^-2*(b-a),{t,a,b}]];Clear[a,b]
ο(t-b)
t-a
Ο
β«
ο
1
1+
ο(-Ο+t)
-ο+t
(1+(ο(-Ο+t))/(-ο+t))
2
(-ο+t)
CMRB=(-ο+Ο)ImοCsch[(Ο(-Ο+t))/(-ο+t)]οοοt
Ο
β«
ο
1
1+
ο(-Ο+t)
-ο+t
(1+(ο(-Ο+t))/(-ο+t))
2
(-ο+t)
In[ο]:=
g[x_]=x^(1/x);CMRB=NSum[(-1)^k(g[k]-1),{k,1,Infinity},ββWorkingPrecisionο’100,Methodο’"AlternatingSigns"];
In[ο]:=
a=E+PiI;b=Pi-EI;g[x_]=x^(1/x);(w=(ο(t-b))/(t-a);ββPrint[CMRB-NIntegrate[Re[g[(1+w)]Csc[Οw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecisionο’100]]);Clear[a,b]
-9.3472Γ+0.Γο
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10
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In[ο]:=
g[x_]=x^(1/x);Tableοw=;ββCMRB-NIntegrate[Re[g[(1+w)]Csc[Οw]](t-a)^-2*(b-a),{t,a,b},ββWorkingPrecisionο’100],{a,0,5},{b,a+1,6}ο
ο(t-b)
t-a
Out[ο]=
{{-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ},{-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ},{-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ},{-9.3472Γ,-9.3472Γ,-9.3472Γ},{-9.3472Γ,-9.3472Γ},{-9.3472Γ}}
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In[ο]:=
g[x_]=x^(1/x);Tableοw=;ββCMRB-NIntegrate[Re[g[(1+w)]Csc[Οw]](t-a)^-2*(b-a),{t,a,b},ββWorkingPrecisionο’100],{a,4/10,5},{b,a+1,6}ο
ο(t-b)
t-a
Out[ο]=
{{-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ},{-9.3472Γ,-9.3472Γ,-9.3472Γ,-9.3472Γ},{-9.3472Γ,-9.3472Γ,-9.3472Γ},{-9.3472Γ,-9.3472Γ},{-9.3472Γ}}
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In[ο]:=
a=E;b=Pi;
In[ο]:=
a=E;b=Pi;g[x_]=x^(1/x);(w=(ο(t-b))/(t-a);ββPrint[CMRB-NIntegrate[Re[g[(1+w)]Csc[Οw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecisionο’100]]);Clear[a,b]
-9.3472Γ
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10
In[ο]:=
a=E+PiI;b=Pi-EI;g[x_]=x^(1/x);(w=(ο(t-b))/(t-a);ββPrint[CMRB-NIntegrate[Re[g[(1+w)]Csc[Οw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecisionο’100]]);Clear[a,b]
-9.3472Γ+0.Γο
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In[ο]:=
a=1;b=I;g[x_]=x^(1/x);w=;ββPrint[CMRB-NIntegrate[Re[g[(1+w)]Csc[Οw]](t-a)^-2*(b-a),{t,a,b},WorkingPrecisionο’100]];Clear[a,b]
ο(t-b)
t-a
-9.3472Γ+0.Γο
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In[ο]:=
a=1;b=I;g[x_]=x^(1/x);w=;ββPrint[Integrate[Re[g[(1+w)]Csc[Οw]](t-a)^-2*(b-a),{t,a,b}]];Clear[a,b]
ο(t-b)
t-a
ο
β«
1
(1-ο)ImοCschοοο
1
1+
ο(-ο+t)
-1+t
1+
ο(-ο+t)
-1+t
Ο(-ο+t)
-1+t
2
(-1+t)
β
CMRB=-οt
ο
β«
1
(1-ο)ImοCschοοο
1
1+
ο(-ο+t)
-1+t
ο1+ο
ο(-ο+t)
-1+t
Ο(-ο+t)
-1+t
2
(-1+t)
ββββββ
Somestrangeexamples:
CMRB=(-ο+Ο)ImοCsch[(Ο(-Ο+t))/(-ο+t)]οοοt.
Ο
β«
ο
1
1+
ο(-Ο+t)
-ο+t
(1+(ο(-Ο+t))/(-ο+t))
2
(-ο+t)
CMRB=-οt.
ο
β«
1
(1-ο)ImοCschοοο
1
1+
ο(-ο+t)
-1+t
ο1+ο
ο(-ο+t)
-1+t
Ο(-ο+t)
-1+t
2
(-1+t)