How to use Working Precision to get good result and plot?

Question

I have a function f(x) which I would like to see going up for very low values of x: from an asymptotic study, I expect it to go to Infinity for low x; however, I can see an elbow and many oscillations. Are they due to numerical precision? How can I fix it? The function f(x) of interest is fon(beta).

 firstk = -8;
 lastk = 0;
 fon = 0.05379*beta/IntegralExact;
 IntegrandON = 
 Simplify[Abs[
  1 - (1 - I)/2*Sqrt[(beta/2)]*
    Cosh[(1 - I)*z*
       Sqrt[(beta/2)]]/(Sinh[(1 - I)/2*Sqrt[(beta/2)]])] // 
 ComplexExpand, beta > 0]^2;
 IntegralExact = FullSimplify[Integrate[IntegrandON, {z, -1/2, 1/2}]];

 Show[Flatten[{LogLogPlot[{fon}, {beta, 10^firstk, 10^lastk}, 
PlotLegends -> {"On-axis"}, PlotStyle -> Black, 
PlotRange -> {{10^firstk, 10^lastk}, {10^-6, 10^(8)}}, 
GridLines -> {{46}, All}]}], ImageSize -> Large,  
Frame -> True]

Answer 1

I'd suggest doing all the heavy lifting numerically when you actually assign a value to beta, i.e. during plotting. This allows you to use NIntegrate at an appropriate working precision:

firstk = -20; (*Notice the even lower limit*)
lastk = 0;

Clear[IntegrandON]
(* Avoiding simplification and expansion significantly speeds up the calculation*)
IntegrandON[beta_] = 
  Abs[1 - (1 - I)/2*Sqrt[(beta/2)]*
      Cosh[(1 - I)*z*
         Sqrt[(beta/2)]]/(Sinh[(1 - I)/2*Sqrt[(beta/2)]])]^2;

ClearAll[fon]
fon[beta_?NumericQ] := 5379/100000*beta/NIntegrate[IntegrandON[beta], {z, -1/2, 1/2}, WorkingPrecision -> 15];

LogLogPlot[
  fon[beta],
  {beta, 10^firstk, 10^lastk},
  PlotStyle -> Black, Frame -> True,
  WorkingPrecision -> 30
]

Answer 2

IntegrandON = 
  Simplify[Abs[
      1 - (1 - I)/2*Sqrt[(beta/2)]*
        Cosh[(1 - I)*z*
           Sqrt[(beta/2)]]/(Sinh[(1 - I)/2*Sqrt[(beta/2)]])] // 
     ComplexExpand, beta > 0]^2;

IntegralExact = FullSimplify[Integrate[IntegrandON, {z, -1/2, 1/2}]]

firstk = -8;
lastk = 0;
fon = 5379*10^-5*beta/IntegralExact;

LogLogPlot[fon, {beta, 10^firstk, 10^lastk},
 PlotLegends -> Placed[{"On-axis"}, {.25, .7}],
 PlotStyle -> Black,
 PlotRange -> {{10^firstk, 10^lastk}, {1, 10^8}},
 GridLines -> {10^Range[-7, -1, 2], All},
 Frame -> True,
 ImageSize -> Large,
 WorkingPrecision -> 20,
 PlotPoints -> 150]