Template Notebook
Generate
f[x_]=(-1)^x(x^(1/x)-1);
g[x_]=x^(1/x)
Out[]=
1
x
x
CMRB=N[NSum[f[n],{n,1,Infinity},Method"AlternatingSigns",WorkingPrecision57],50]
Out[]=
0.18785964246206712024851793405427323005590309490012
CMRB-INIntegrate[(g[(-tI+1)]-g[(tI+1)])/(Exp[Pit]-Exp[-Pit]),{t,0,Infinity},WorkingPrecision50]
Out[]=
-2.×
-50
10
g[x_]=x^(1/x);CMRB-NIntegrate[Im[g[1+It]]/(Sinh[Pit]),{t,0,Infinity},WorkingPrecision50]
Out[]=
-2.×
-50
10
f[x_]=(-1)^x(x^(1/x)-1);Timing[CMRB-Im[NIntegrate[(2f[(1-It)])/(Exp[2Pit]-1),{t,0,Infinity},Method"Trapezoidal",WorkingPrecision50]]]
Out[]=
{0.09375,-2.×
-50
10
}
∞
∫
0
2f(1-t)
exp(2πt)-1
t
ReMKB=Im[N[NIntegrate[(f[(1-It)]-f[(1+It)])/(Exp[2Pit]+1),{t,​​0,Infinity},WorkingPrecision100],50]]
Out[]=
0.070776039311528803539528021830282001365754696203363
ReMKB-Im[N[NIntegrate[(f[(1-It)]+f[(1+It)])/(Exp[2Pit]-1),{t,​​0,Infinity},WorkingPrecision100],50]]
Out[]=
0.×
-51
10
ReMKB-N[NIntegrate[Cos[Pit](t^(1/t)),{t,​​1,Infinity},WorkingPrecision100,MaxRecursion20],50]
Out[]=
0.×
-51
10
NoMKB=Im[N[NIntegrate[(f[(1-It)]-f[(1+It)])/(Exp[2Pit]-1),{t,​​0,Infinity},WorkingPrecision100],50]]
Out[]=
0.117083603150538316708989912223991228690148398696776
ReMKB+NoMKB-CMRB
Out[]=
0.×
-51
10
In[]:=
g[x_]=x^(1/x);
MKB=N[NIntegrate[Exp[IPit]g[t],{t,1,1Infinity},WorkingPrecision100],50]-I/Pi
Out[]=
0.07077603931152880353952802183028200136575469620336-0.68400038943793212918274445999266112671099148265500
x=SetPrecision[x,pr];​​y=x^n;z=(n-y)/y;​​t=2n-1;t2=t^2;​​x=x*(1+SetPrecision[4.5,pr](n-1)/t2+(n+1)z/(2nt)-SetPrecision[13.5,pr]n(n-1)1/(3nt2+t^3z));​​(*N[Exp[Log[n]/n],pr]*)
In[]:=
x=N[n^(1/n),100];​​(*xstartsoutasarelativelysmallprecisionapproximationton^(1/n)*)​​pc=Precision[x];​​(*pristhedesiredprecisionofyourn^(1/n)*)​​t0[pr_]=Module[{},While[pc<pr,pc=Min[4pc,pr];​​x=SetPrecision[x,pc];​​y=x^n;z=(n-y)/y;​​t=2n-1;t2=t^2;​​x=x*(1+SetPrecision[4.5,pc](n-1)/t2+(n+1)z/(2nt)-SetPrecision[13.5,pc]n(n-1)/(3nt2+t^3z))]];
Letg(x)=
1/x
x
.
CMRB=
lim
N∞
2N
∑
n=1
n
(-1)
g(n)=I

N∞
2N
∫
0
g(1+t)
sinh(πt)
t.
​​​​
g(x_)=
1/x
x
;u=
t
1-t
;​​MKB=

N∞
2N
∫
1
exp(πt)g(x)x​​=(-)
∞
∫
0
g(1+t)
exp(πt)
t-

π
​​=(-)
1
∫
0
g(1+u)
2
(1-t)
exp(πu)
t-

π
In[]:=
g[x_]=x^(1/x);Timing[MKB=NIntegrate[Exp[IPit](g[t]),{t,1,Infinity},WorkingPrecision500]​​-I/Pi];
In[]:=
u:=(t/(1-t));
In[]:=
Timing[MKB=(-INIntegrate[(g[(1+tI)])/(Exp[Pit]),{t,0,Infinity},WorkingPrecision2000,Method"Trapezoidal",MaxRecursion11]-I/Pi)][[1]]
Out[]=
68.9219
​​​​​​​​​​
Although
In[]:=
g[x_]=x^(1/x);(N[NIntegrate[Exp[IPit]g[t],{t,1,1Infinity},WorkingPrecision100],50]-I/Pi)
Out[]=
0.07077603931152880353952802183028200136575469620336-0.68400038943793212918274445999266112671099148265500
is too slow and unreliable, we have the following:
​
Then ReMKB + NoMKB - CMRB = 0.
Then ImMKB + ReMKB - MKB = 0.