In[]:=
f[x_]=E^(IPix)1-
1
1+x
(1+x)
;
​
Re
∞
∫
0
f(t)t=Im
∞
∫
0
f(-t)+f(t)
exp(2πt)-1
t
Im
∞
∫
0
f(t)t=Re
∞
∫
0
f(-t)-f(t)
exp(2πt)-1
t
Im
∞
∫
0
f(t)t=Re
∞
∫
0
f(-t)+f(t)
exp(2πt)-1
t+a​​
a=2Re
∞
∫
0
f(t)
exp(2πt)-1
t.​
​a=-
1
2
Re
7!300
∑
n=0
f(n)
∞
∫
0
f(t)t±7*
-12
10
.​
​a=-
1
2
Rei
7!300
∑
n=0
f(n)
7
∫
0
f(t)t±9.8*
-12
10
.
​​
Re
∞
∫
0
f(t)t=Im
∞
∫
0
f(t)-f(-t)
exp(2πt)-1
t+
∞
∑
n=0
f(n)=2Im
∞
∫
0
f(-t)
2πt

-1
t
​​
Whyforf[x_]=E^(IPix)1-
1
1+x
(1+x)
,
∞
∫
0
f(t)t=
∞
∫
0
f(t)t?
In[]:=
NIntegrate[(f[t]),{t,0,Infinity},WorkingPrecision100]-INIntegrate[(f[tI]),{t,0,Infinity},WorkingPrecision100]
Out[]=
9.48811918461279823105578121961062159724090948521×
-53
10
-9.94031886872260377157400403878717967993835295319×
-53
10

∞
∫
0
f(t)t=
7
∫
0
f(t)t±
-11
10
In[]:=
NIntegrate[(f[t]),{t,0,Infinity},WorkingPrecision37]-INIntegrate[(f[tI]),{t,0,7},WorkingPrecision37]
Out[]=
-2.69997169907793203150972153×
-11
10
-1.98306568448535333130262161×
-11
10

In[]:=
N[f[t]-If[It]]/.t4
Out[]=
-0.379728+1.52283×
-6
10

In[]:=
-Re[NSum[f[n],{n,0,1512000},Method"AlternatingSigns",WorkingPrecision20]INIntegrate[(f[tI]),{t,0,7},WorkingPrecision37]]/2-2ReNIntegrate
f[t]
Exp[2πt]-1
,{t,0,Infinity},WorkingPrecision15
Out[]=
-9.88200×
-12
10
In[]:=
-Re[NSum[f[n],{n,0,1512000},Method"AlternatingSigns",WorkingPrecision20]INIntegrate[(f[tI]),{t,0,Infinity},WorkingPrecision37]]/2-2ReNIntegrate
f[t]
Exp[2πt]-1
,{t,0,Infinity},WorkingPrecision15
Out[]=
-7.34598×
-12
10
In[]:=
Re[NIntegrate[(f[t]),{t,0,Infinity},WorkingPrecision30]]
Out[]=
0.0707760393115288238343623560886
In[]:=
ImNIntegrate
f[t]-f[-t]
Exp[2πt]-1
,{t,0,∞},WorkingPrecision30+NSum[f[n],{n,0,∞},Method"AlternatingSigns",WorkingPrecision37]
Out[]=
0.070776039311528803539528021826
2Im
∞
∫
0
f(-t)
2πt

-1
t=
∞
∑
n=0
f(n)
In[]:=
2ImNIntegrate(f[-It])(
2Pit
E
-1),{t,0,Infinity},WorkingPrecision30-NSum[f[n],{n,0,Infinity},Method"AlternatingSigns",WorkingPrecision37]
Out[]=
5.×
-30
10
In[]:=
im=Im[NIntegrate[(f[t]),{t,0,Infinity},WorkingPrecision30]]
Out[]=
-0.0473806170703507906668225704847
In[]:=
ReNIntegrate
f[-t]+f[t]
Exp[2πt]-1
,{t,0,∞},WorkingPrecision30+0.006647814199317396773815965
Out[]=
-0.04738061707035079066682257
In[]:=
ReNIntegrate
f[-t]+f[t]
Exp[2πt]-1
,{t,0,∞},WorkingPrecision30+ReNIntegrate
-2f[t]
Exp[2πt]-1
,{t,0,∞},WorkingPrecision30
Out[]=
-0.0473806170703507861072094065026
2Re
∞
∫
0
f(t)
exp(2πt)-1
t
In[]:=
2ReNIntegrate
f[t]
Exp[2πt]-1
,{t,0,∞},WorkingPrecision30
Out[]=
-0.00664781419931740133342912906003
Re
∞
∫
0
2f(t)
exp(2πt)-1
t
In[]:=
ReNIntegrate
f[-t]-f[t]
Exp[2πt]-1
,{t,0,∞},WorkingPrecision30
Out[]=
-0.0473806170703507861072094065026
In[]:=
-NSum[f[n],{n,0,∞},Method"AlternatingSigns",WorkingPrecision37]/2Re[NIntegrate[(f[t]),{t,0,Infinity},WorkingPrecision37]]-a
Out[]=
-1.6652065510275311121474925×
-7
10
-
1
2
Re
∞
∑
n=0
f(n)
∞
∫
0
f(t)t
In[]:=
-Re[NSum[f[n],{n,0,∞},Method"AlternatingSigns",WorkingPrecision37]NIntegrate[(f[t]),{t,0,Infinity},WorkingPrecision37]]/2-a
Out[]=
-1.6652065510275311121474925×
-7
10
In[]:=
2^21
Out[]=
2097152
b=10lcm(1,2,3,4,5,6)lcm(1,2,3,4,5,6,7,8,9)
In[]:=
b=10LCM[1,2,3,4,5,6]LCM[1,2,3,4,5,6,7,8,9]
1512000/100(7!)
7!100
Out[]=
300
In[]:=
-Re[NSum[f[n],{n,0,b},Method"AlternatingSigns",WorkingPrecision37]NIntegrate[(f[t]),{t,0,Infinity},WorkingPrecision37]]/2-a
Out[]=
-7.41138228663206934567×
-12
10
-
1
2
Re
∞
∑
n=0
f(n)
∞
∫
0
f(t)t<a<f(1050)
In[]:=
-Re[NSum[f[n],{n,0,∞},Method"AlternatingSigns",WorkingPrecision37]NIntegrate[(f[t]),{t,0,Infinity},WorkingPrecision37]]/2<a<N[f[1050]]
Out[]=
True