In[]:=
In[]:=
In[]:=
NIntegrateExpPiIx+
Log[x]
x
,{x,1,InfinityI}
Out[]=
0.070776-0.365691
In[]:=
MKB=re+imI;
In[]:=
​​digits=1000;​​wp=digits+50;​​T=Ceiling[(digits+20)Log[10]/Pi];AbsoluteTimingNIntegrateExpPiIx+
Log[x]
x
,{x,1,InfinityI},WorkingPrecisionwp,Method"DoubleExponential",MaxRecursionFloor[Log2[T]]-IPi-MKB//Quiet
Out[]=
{1.74661,0.×
-1051
10
+0.×
-1051
10
}
In[]:=
​​ClearAll[x,MKBContour];​​​​digits=1000;​​guard=10;​​wp=digits+guard;​​​​T=Ceiling[(digits+20)Log[10]/Pi];​​mr=Floor[Log2[T]];​​​​Print[AbsoluteTiming[MKBContour=Quiet[NIntegrate[Exp[PiIx+Log[x]/x],{x,1,IInfinity},WorkingPrecisionwp,Method"DoubleExponential",MaxRecursionmr]-I/Pi];][[1]]," ",N[MKBContour-MKB,20]]​​```​​
1.70933 0.×
-1011
10
+0.×
-1011
10

In[]:=
​​digits=2000;​​wp=digits+50;​​T=Ceiling[(digits+20)Log[10]/Pi];AbsoluteTimingNIntegrateExpPiIx+
Log[x]
x
,{x,1,InfinityI},WorkingPrecisionwp,Method"DoubleExponential",MaxRecursionFloor[Log2[T]]-IPi-MKB//Quiet
Out[]=
{10.801,0.×
-2051
10
+0.×
-2051
10
}
In[]:=
​​ClearAll[x,MKBContour];​​​​digits=2000;​​guard=10;​​wp=digits+guard;​​​​T=Ceiling[(digits+20)Log[10]/Pi];​​mr=Floor[Log2[T]];​​​​Print[AbsoluteTiming[MKBContour=Quiet[NIntegrate[Exp[PiIx+Log[x]/x],{x,1,IInfinity},WorkingPrecisionwp,Method"DoubleExponential",MaxRecursionmr]-I/Pi];][[1]]," ",N[MKBContour-MKB,20]]​​```​​
10.5149 0.×
-2011
10
+0.×
-2011
10

In[]:=
​​digits=3000;​​wp=digits+50;​​T=Ceiling[(digits+20)Log[10]/Pi];AbsoluteTimingNIntegrateExpPiIx+
Log[x]
x
,{x,1,InfinityI},WorkingPrecisionwp,Method"DoubleExponential",MaxRecursionFloor[Log2[T]]-IPi-MKB//Quiet
Out[]=
{45.4401,0.×
-3051
10
+0.×
-3051
10
}
In[]:=
​​ClearAll[x,MKBContour];​​​​digits=3000;​​guard=10;​​wp=digits+guard;​​​​T=Ceiling[(digits+20)Log[10]/Pi];​​mr=Floor[Log2[T]];​​​​Print[AbsoluteTiming[MKBContour=Quiet[NIntegrate[Exp[PiIx+Log[x]/x],{x,1,IInfinity},WorkingPrecisionwp,Method"DoubleExponential",MaxRecursionmr]-I/Pi];][[1]]," ",N[MKBContour-MKB,20]]​​```​​
44.6987 0.×
-3011
10
+0.×
-3011
10

In[]:=
​​digits=4000;​​wp=digits+50;​​T=Ceiling[(digits+20)Log[10]/Pi];AbsoluteTimingNIntegrateExpPiIx+
Log[x]
x
,{x,1,InfinityI},WorkingPrecisionwp,Method"DoubleExponential",MaxRecursionFloor[Log2[T]]-IPi-MKB//Quiet
Out[]=
{79.2178,0.×
-4051
10
+0.×
-4051
10
}
In[]:=
​​ClearAll[x,MKBContour];​​​​digits=4000;​​guard=10;​​wp=digits+guard;​​​​T=Ceiling[(digits+20)Log[10]/Pi];​​mr=Floor[Log2[T]];​​​​Print[AbsoluteTiming[MKBContour=Quiet[NIntegrate[Exp[PiIx+Log[x]/x],{x,1,IInfinity},WorkingPrecisionwp,Method"DoubleExponential",MaxRecursionmr]-I/Pi];][[1]]," ",N[MKBContour-MKB,20]]​​```​​
78.1489 0.×
-4011
10
+0.×
-4011
10

​​​​
In[]:=
ClearAll[t,MKB1,integrand];​​​​prec=5000;​​​​LaunchKernels[];​​nparts=Max[1,$KernelCount/2];​​​​integrand[t_]:=Exp[N[(πt)/(-1+t),prec]+((-1+t)N[Log[1-(It)/(-1+t)],prec])/(-1+(1-I)t)]/((1-t)^2);​​​​parts=N[Subdivide[0,1,nparts],prec];​​​​Print[AbsoluteTiming[MKB1=-ITotal[ParallelTable[Quiet[NIntegrate[integrand[t],{t,parts[[m]],parts[[m+1]]},WorkingPrecision->prec,Method->"DoubleExponential",MaxRecursion->Floor[Log2[prec]-1]]],{m,1,nparts},DistributedContexts->Automatic]]-I/Pi][[1]]," ",N[re-Re[MKB1],20]]