CMRB=0.18785964246206712024...
n=1or25.65665403510586285599...
g(x_)=
n/x
x
;
-1
2
Im
∞
∫
(-∞)
g(1-t)
sin(πt)
t=CMRB.
g(x_)=
n/x
x
;
1
2
Im
∞
∫
(-∞)
g(1+t)
sin(πt)
t=CMRB.
g(x_)=
n/x
x
;​​
1
4
Im
∞
∫
(-∞)
g(t+1)-g(1-t)
sin(πt)
t=CMRB.
In[]:=
n={1,25.6566540351058628559907};
In[]:=
g[x_]=x^(n/x);​​-1/2Im[N[NIntegrate[(g[(1-t)])/(Sin[πt]),{t,-InfinityI,InfinityI},WorkingPrecision60],20]]
Out[]=
{0.18785964246206712025,0.18785964246206712025}
In[]:=
g[x_]=x^(n/x);​​1/2Im[N[NIntegrate[(g[(1+t)])/(Sin[πt]),{t,-InfinityI,InfinityI},WorkingPrecision60],20]]
Out[]=
{0.18785964246206712025,0.18785964246206712025}
In[]:=
g[x_]=x^(n/x);​​1/4Im[N[NIntegrate[(g[(1+t)]-(g[(1-t)]))/(Sin[πt]),{t,-InfinityI,InfinityI},WorkingPrecision60],20]]
Out[]=
{0.18785964246206712025,0.18785964246206712025}