Boole[] returns non-numerical value?

Question

I am trying to find the maximum value $t$ such that my function $h$ is bigger than $t$ on (at least) half of the interval $[0,2\pi]$ and smaller than $t$ on the other half, i.e.: $$ \text{sup}\{t\in \mathbb{R} : \lambda(\{ x\in [0, 2\pi]: h(x)\geq t\})>\pi\}.$$

In my case, the function $h$ is the integral of the difference between two other functions, $f$ and $g$. I am trying to calculate the length of the interval where $h(x) \geq t$ by integrating a Boolean. I have tried the following:

Clear[f, g, h, q];
f[x_?NumericQ] = Sum[(1 - Abs[k]/6)*E^(I*k*(x - (Pi/3))), {k, -5, 5}];
g[x_?NumericQ] = Sum[(1 - Abs[k]/6)*E^(I*k*(x - (5*Pi/3))), {k, -5, 5}];
h[t_?NumericQ] = Integrate[f[x] - g[x], {x, 0, t}]/(2*Pi);
q[z_?NumericQ, v_?NumericQ] = Boole[Chop[h[v]] <= Chop[h[z]]];
FindMaxValue[{Chop[h[z]], (Integrate[q[z, v], {v, 0, 2*Pi}]) >= Pi}, z]

Now I am getting the following errors:

NIntegrate::inumr: The integrand q[1.,v] has evaluated to non-numerical values for all sampling points in the region with boundaries {{0,6.28319}}.

General::stop: Further output of NIntegrate::inumr will be suppressed during this calculation.

FindMaxValue::nrlnum: The function value {-[Pi]+!(*SubsuperscriptBox[([Integral]), (0), (2\ [Pi])](q[1.`, v] [DifferentialD]v))} is not a list of real numbers with dimensions {1} at {z} = {1.}.

I guess that the problem is that Mathematica is trying to evaluate q[1,v] which indeed is not a number. How do I get Mathematica to integrate this right?

Answer 1

In this case,

• ?NumericQ condition is not necessary in function definitions, but
• ?NumericQ is needed for NIntegrate (ref. the documentation for message), so the integral should be split off, here to lambda[z_?NumericQ]:
  • using := for evaluation to happen only when arguments are passed in.

ClearAll[f, g, h, q, lambda];
f[x_] = Sum[(1 - Abs[k]/6)*E^(I*k*(x - (Pi/3))), {k, -5, 5}];
g[x_] = Sum[(1 - Abs[k]/6)*E^(I*k*(x - (5*Pi/3))), {k, -5, 5}];
h[t_] = Integrate[f[x] - g[x], {x, 0, t}]/(2*Pi) // FullSimplify // N;
q[z_, v_] := Boole[Chop[h[v]] <= Chop[h[z]]]
lambda[z_?NumericQ] := NIntegrate[q[z, v], {v, 0, 2 Pi}]
FindMaximum[{Chop[h[z]], 0 <= z <= 2 Pi && lambda[z] > Pi}, {z, Pi}]

{0.8821262327, {z -> 3.141592654}}

Answer 2

Is this what you want?

I interpreted: "I am trying to find the maximum value h[ts] such that my function h is bigger than h[ts] on half of the interval [0,2π] and smaller than h[ts] on the other half, i.e.:"

f[x_] = Sum[(1 - Abs[k]/6)*E^(I*k*(x - (Pi/3))), {k, -5, 5}];
g[x_] = Sum[(1 - Abs[k]/6)*E^(I*k*(x - (5*Pi/3))), {k, -5, 5}];
h[t_] = Integrate[f[x] - g[x], {x, 0, t}]/(2*Pi) // FullSimplify

(*   (4*(35 + 40*Cos[t] + 13*Cos[2*t] + 2*Cos[3*t])*
  Sin[t/2]^4)/(5*Sqrt[3]*Pi)   *)

{p1, p2} = {t1, t2} /. 
  First@Solve[{h[t2] == h[t1], t2 - t1 == Pi, 0 < t1 < Pi}, {t1, t2},
  Reals] // FullSimplify

(*   {Pi/2, (3*Pi)/2}   *)

Plot[{h[Pi/2], h[t]}, {t, 0, 2 Pi}, 
  Epilog -> Point@{{p1, h[p1]}, {p2, h[p2]}}
]

Edit Extension for functions with more humps.

Use UnitStep and provide the exact zeros of h[t]- hs (hs searched for) to get better integration results.

nsol[hs_?NumericQ] := 
  t /. NSolve[
  0 < t < 2 Pi &&   TrigExpand[h[t] - hs] == 0, t
]

nint[hs_?NumericQ] := 
 NIntegrate[UnitStep[h[t] - hs], 
  Evaluate[Flatten[{t, 0, nsol[hs], 2 Pi}]]]

FindRoot[nint[hs] == Pi, {hs, .7}]

(*   {hs -> 0.808616}   *)

h[Pi/2] // N[#, 15] &

(*   0.808615713285295   *)