L=200;λ=1/100;M=100;Ω=1;m=100;(*fixedm*)I1[a_]:=2/Sqrt[4πAbs[m]]*NIntegrate[Exp[-(25τ^2)/8]*Cos[Ωτ+(2πm)/(La)Sinh[aτ]]*Cos[(2πm)/(La)(Cosh[aτ]-1)],{τ,-30,30},Method"GlobalAdaptive"]I2[a_]:=1/Sqrt[4πAbs[m]]*NIntegrate[Exp[-(25τ^2)/8]*Cos[Ωτ+(2π)/(La)(-m(Cosh[aτ]-1)+Abs[m]Sinh[aτ])]+Exp[-(25τ^2)/8]*Cos[Ωτ+(2π)/(La)(m(Cosh[aτ]-1)+Abs[m]Sinh[aτ])],{τ,-30,30},Method"GlobalAdaptive"]I3[a_]:=NIntegrate[1/Sqrt[4πAbs[m]]*(Exp[IΩτ-(25τ^2)/8+(2Iπ)/(La)(-m(Cosh[aτ]-1)+Abs[m]Sinh[aτ])]+Exp[IΩτ-(25τ^2)/8+(2Iπ)/(La)(m(Cosh[aτ]-1)+Abs[m]Sinh[aτ])]),{τ,-30,30},Method"GlobalAdaptive"]Plot[{I1[a],I2[a],Re[I3[a]]},{a,0.01,3},PlotRangeAll,PlotStyle{Blue,Green,Red},PlotLegends{"I1(a)","I2(a)","Re[I3(a)]"},AxesLabel{"a","Value"},GridLinesAutomatic]