In[]:=
ClearAll["`*"]
In[]:=
L=200;​​λ=5/100;​​Ω=3/100;​​σ=2/100;​​​​​​​​I0[a_?NumericQ,m_?NumericQ]:=4/Sqrt[4πm]*NIntegrate[Exp[-(τ^2)/(2σ^2)]*Cos[Ωτ+(2πm)/(La)Sinh[aτ]]*Cos[(2πm)/(La)(Cosh[aτ]-1)],{τ,0,Infinity},Method{Automatic,"SymbolicProcessing"0}];​​​​I1[a_?NumericQ,m_?NumericQ]:=4/Sqrt[4πm]*NIntegrate[Exp[-(τ^2)/(2σ^2)]*Cos[-Ωτ+(2πm)/(La)Sinh[aτ]]*Cos[(2πm)/(La)(Cosh[aτ]-1)],{τ,0,Infinity},Method{Automatic,"SymbolicProcessing"0}];​​​​​​η0[a_?NumericQ]:=λ*Sum[I0[a,m],{m,1,1000}];​​η1[a_?NumericQ]:=λ*Sum[I1[a,m],{m,1,1000}];​​​​​​F[a_?NumericQ]:=(1-η0[a]η1[a])/Sqrt[(1+η0[a]^2)(1+η1[a]^2)]
In[]:=
F[50]
Out[]=
0.987101
In[]:=
ListPlot[Table[{a,F[a]},{a,1,50,1}],Frame->True,FrameStyle->Directive[Black,12,Bold],FrameLabel->{Style["a",Italic,13,Bold,Black],Style["F(a)",Italic,13,Bold,Black]},DataRange->{1,50},PlotRangePadding->0]
Out[]=
​
In[]:=
ClearAll["`*"]
In[]:=
L=200;​​λ=5/100;​​Ω=3/100;​​σ=2/100;
In[]:=
η0[a_?NumericQ]:=NIntegrate
a
-
1
2
τ
τ
2
σ
+2Ω

Lλ
aτ

(-2+2
-aτ

)π
aL
+x
HeavisideTheta
(-2+2
-aτ

)π
aL
+x
-2(-1+
aτ

)π+a
aτ

Lx
+
HeavisideTheta-
2(-1+
aτ

)π
aL
+x
aL
-
2(-1+
aτ

)π
aL
+x
+
2τΩ

HeavisideTheta
2(-1+
aτ

)π
aL
+x
aL
2(-1+
aτ

)π
aL
+x
+
τ(a+2Ω)

2π-2
-aτ

π+aLx
aL
HeavisideTheta
2π-2
-aτ

π+aLx
aL

2(-1+
aτ

)π+a
aτ

Lx
2(-1+
x

)π
,{x,0,Infinity},{τ,0,Infinity}​​η1[a_?NumericQ]:=NIntegrate
a
-
1
2
τ
τ
2
σ
+2Ω

Lλ
τ(a+2Ω)

(-2+2
-aτ

)π
aL
+x
HeavisideTheta
(-2+2
-aτ

)π
aL
+x
-2(-1+
aτ

)π+a
aτ

Lx
+
2τΩ

HeavisideTheta-
2(-1+
aτ

)π
aL
+x
aL
-
2(-1+
aτ

)π
aL
+x
+
HeavisideTheta
2(-1+
aτ

)π
aL
+x
aL
2(-1+
aτ

)π
aL
+x
+
aτ

2π-2
-aτ

π+aLx
aL
HeavisideTheta
2π-2
-aτ

π+aLx
aL

2(-1+
aτ

)π+a
aτ

Lx
2(-1+
x

)π
,{x,0,Infinity},{τ,0,Infinity};
In[]:=
F[a_?NumericQ]:=(1-η0[a]η1[a])/Sqrt[(1+η0[a]^2)(1+η1[a]^2)]
In[]:=
F[50]
NIntegrate
:Integral and error estimates are 0 on all integration subregions. Try increasing the value of the MinRecursion option. If value of integral may be 0, specify a finite value for the AccuracyGoal option.
NIntegrate
:Integral and error estimates are 0 on all integration subregions. Try increasing the value of the MinRecursion option. If value of integral may be 0, specify a finite value for the AccuracyGoal option.
NIntegrate
:Integral and error estimates are 0 on all integration subregions. Try increasing the value of the MinRecursion option. If value of integral may be 0, specify a finite value for the AccuracyGoal option.
General
:Further output of NIntegrate::izero will be suppressed during this calculation.
Out[]=
1.+0.
In[]:=
Table[{a,F[a]},{a,1,50,1}]//Quiet
Out[]=
{{1,1.+0.},{2,1.+0.},{3,1.+0.},{4,1.+0.},{5,1.+0.},{6,1.+0.},{7,1.+0.},{8,1.+0.},{9,1.+0.},{10,1.+0.},{11,1.+0.},{12,1.+0.},{13,1.+0.},{14,1.+0.},{15,1.+0.},{16,1.+0.},{17,1.+0.},{18,1.+0.},{19,1.+0.},{20,1.+0.},{21,1.+0.},{22,1.+0.},{23,1.+0.},{24,1.+0.},{25,1.+0.},{26,1.+0.},{27,1.+0.},{28,1.+0.},{29,1.+0.},{30,1.+0.},{31,1.+0.},{32,1.+0.},{33,1.+0.},{34,1.+0.},{35,1.+0.},{36,1.+0.},{37,1.+0.},{38,1.+0.},{39,1.+0.},{40,1.+0.},{41,1.+0.},{42,1.+0.},{43,1.+0.},{44,1.+0.},{45,1.+0.},{46,1.+0.},{47,1.+0.},{48,1.+0.},{49,1.+0.},{50,1.+0.}}
In[]:=
ListPlot[Re[%],Frame->True,FrameStyle->Directive[Black,12,Bold],FrameLabel->{Style["a",Italic,13,Bold,Black],Style["F(a)",Italic,13,Bold,Black]},DataRange->{1,50},PlotRangePadding->0]
Out[]=