In[]:=
g[x_]=x^(1/x);Timing[N[Quiet[M2=NIntegrate[Exp[IPit](g[t]-1),{t,1,InfinityI},WorkingPrecision1000,Method"Trapezoidal",MaxRecursion10]]]]
Out[]=
{6.45313,0.070776-0.0473806}
In[]:=
g[x_]=x^(1/x);Timing[N[Quiet[M2-NIntegrate[Exp[IPit](g[t]-1),{t,1,InfinityI},WorkingPrecision1000,Method"MultiPeriodic",MaxRecursion10]]]]
Out[]=
{2.375,1.70529×
-24
10
+1.11456×
-45
10
}
In[]:=
g[x_]=x^(1/x);Timing[N[Quiet[M2-NIntegrate[Exp[IPit](g[t]-1),{t,1,InfinityI},WorkingPrecision1000,Method"GlobalAdaptive",MaxRecursion10]]]]
General
:5.7603243575997933796052033866244616064364321745436694793865078551979870023907×
-924
10
+1.4517126488394011543506760668111490943611568612124261609613399387685386102159×
-924
10
 is too small to represent as a normalized machine number; precision may be lost.
Out[]=
{24.75,0.+0.}
In[]:=
g[x_]=x^(1/x);Timing[N[Quiet[M2-NIntegrate[Exp[IPit](g[t]-1),{t,1,InfinityI},WorkingPrecision1000,Method"LocalAdaptive",MaxRecursion10]]]]
Out[]=
$Aborted
In[]:=
g[x_]=x^(1/x);Timing[N[Quiet[M2-NIntegrate[Exp[IPit](g[t]-1),{t,1,InfinityI},WorkingPrecision1000,Method"DoubleExponential",MaxRecursion10]]]]
Out[]=
{6.60938,0.+0.}
(∞
∫
1
exp(πt)(g(t)-1)t=M2
In[]:=
g[x_]=x^(1/x);Timing[N[Quiet[M2-NIntegrate[Exp[IPit](g[t]-1),{t,1,InfinityI},WorkingPrecision1000,Method"DoubleExponential",MaxRecursion10]]]]
Out[]=
{6.01563,0.+0.}
∞
∫
1
exp(πt)(g(t)-t)t=M2-
1
2
π
In[]:=
g[x_]=x^(1/x);Timing[N[Quiet[M2-(1/Pi^2+NIntegrate[Exp[IPit](g[t]-t),{t,1,InfinityI},WorkingPrecision100,Method"DoubleExponential",MaxRecursion10])],40]]
Out[]=
{0.046875,0.×
-102
10
+0.×
-102
10
}
∞
∫
1
exp(πt)(g(t)-
2
t
)t=M2-
2
2
π
+
2
3
π
In[]:=
g[x_]=x^(1/x);Timing[N[Quiet[M2-(2/Pi^2+2/Pi^3I+NIntegrate[Exp[IPit](g[t]-t^2),{t,1,InfinityI},WorkingPrecision100,Method"DoubleExponential",MaxRecursion10])],40]]
Out[]=
{0.046875,0.×
-101
10
+0.×
-101
10
}
In[]:=
g[x_]=x^(1/x);TimingNQuietM2-
3
2
π
-
6
4
π
+6Pi^3I+NIntegrate[Exp[IPit](g[t]-t^3),{t,1,InfinityI},WorkingPrecision100,Method"DoubleExponential",MaxRecursion10],40
Out[]=
{0.046875,0.×
-101
10
+0.×
-101
10
}
In[]:=
N[(Timing[test=(NIntegrate[(Exp[Log[t]/t-Pit/I]),{t,1,InfinityI},WorkingPrecision4453,Method"Trapezoidal",MaxRecursion11]+I/Pi)])]​​
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 11 iterated refinements in t in the region {{1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000,∞}}. NIntegrate obtained 4462-4461 and 4466 for the integral and error estimates.
Out[]=
{157.938,0.070776-0.0473806}
200018​​300038​​Method(A,BorC)​​sec​​digitsABC​​2000232018​​3000968036​​400016514164​​5000442418386​​6000623591544​​10000325030702000​​40000175,551164,005​​MethodA:g[x_]=x^(1/x);t=(Timing[test3=-(INIntegrate[(g[(1+tI)])/(Exp[Pit]),{t,0,Infinity},WorkingPrecision40000,Method"Trapezoidal",MaxRecursion15]+I/Pi)])[[1]];​​​​MethodB:Timing[NIntegrate[I(f[It]),{t,0,Infinity},WorkingPrecision40000,Method"Trapezoidal",MaxRecursion15]]​​​​MethodC:N[(Timing[(NIntegrate[(Exp[Log[t]/t-Pit/I]),{t,1,InfinityI},WorkingPrecision40000,Method"Trapezoidal",MaxRecursion15]+I/Pi)])]​​​​​​64000784,937=218hours,halfadaylongerthanthe35,000usingthelongcode.
In[]:=
g[x_]=x^(1/x);Timing[N[Quiet[M2=NIntegrate[Exp[IPit](g[t]-1),{t,1,InfinityI},WorkingPrecision6000,Method"Trapezoidal",MaxRecursion12]]]]
Out[]=
{596.609,0.070776-0.0473806}
In[]:=
N[(Timing[M2-(NIntegrate[(Exp[Log[t]/t-Pit/I]),{t,1,InfinityI},WorkingPrecision5000,Method"Trapezoidal",MaxRecursion12]+I/Pi)]),20]
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 12 iterated refinements in t in the region {{1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000