Healthy variety in Mathematica calculus

[gathercoledt (David Gathercole)]

Take three definitions of the same function ArcTan[Tan[k]]. Can these be integrated? Can their integrals be differentiated?

sf[1, k_] := ArcTan[Tan[k]];
sf[2, k_] := Mod[k + \[Pi]/2, \[Pi]] - \[Pi]/2
sf[3, k_] := \[Pi] SawtoothWave[k/\[Pi] - 1/2] - \[Pi]/2

The integrals Mathematica computes take varied form. Sawtooth[] does not integrate, which is fine.

The other two look quite distinct, a plot comparison confirms.

It appears that the Mod form of the equation is the cleanest to work with, the trigonometric integral clearly being erroneous. Differentiating these integrals we hope to return to the original equation.

It is now the Mod form that produces odd results.

It seems no form of ArcTan[Tan[k]] is suitable for automatic calculus; stemming presumably from tricky issues with regular discontinuity. It’s concerning that so many of these procedures fail silently. If I integrate similar functions unexpectedly in the body of a volume of work, I likely wont observe these complications.

This exploration was prompted by @Marco Thiel’s work computing a convergent integral here .

[danl (Daniel Lichtblau)]

The original code is probably fine but there is some confusing (to me) usage of x vs. k, possibly to emulate an indefinite integral. I made a few changes both to separate these uses and to give the definite integrals a fighting chance by understanding they are integrating along the positive real axis.

 sf[1, x_] := ArcTan[Tan[x]]
sf[2, x_] := Mod[x + \[Pi]/2, \[Pi]] - \[Pi]/2
sf[3, x_] := \[Pi] SawtoothWave[x/\[Pi] - 1/2] - \[Pi]/2

ci[1, k_] = Integrate[sf[1, x], {x, 0, k}, Assumptions -> k > 0]

(* ConditionalExpression[
 Piecewise[{{(1/2)*Pi^2*FractionalPart[k/Pi]^2, -(1/2) < 
     FractionalPart[k/Pi] < 1/2}}, 
     (-(1/2))*
   Pi*(-Pi - 2*ArcTan[Tan[k]]*FractionalPart[k/Pi] + 
     Pi*FractionalPart[k/Pi]^2)], 2*k >= Pi] *)

ci[2, k_] = Integrate[sf[2, x], {x, 0, k}, Assumptions -> k > 0]

(* -((k*Pi)/2) + (1/2)*Pi^2*IntegerPart[k/Pi] + 
 Piecewise[{{-(Pi^2/8), FractionalPart[k/Pi] == 0}, 
       {(1/2)*
     Pi^2*(1 - FractionalPart[k/Pi] + FractionalPart[k/Pi]^2), 
    2*FractionalPart[k/Pi] > 1}}, 
     (1/2)*Pi^2*FractionalPart[k/Pi]*(1 + FractionalPart[k/Pi])] *)

ci[3, k_] = Integrate[sf[3, x], {x, 0, k}, Assumptions -> k > 0]

(* Out[869]= Integrate[-([Pi]/
2) + [Pi] SawtoothWave[1/2 + x/[Pi]], {x, 0, k},
Assumptions → k > 0] *)

(Two out of three ain’t bad, as a song famously put it.)

Let’s do some numeric checks.

In[870]:= ci[1, 5.5]

Out[870]= 0.30669

In[871]:= ci[2, 5.5]

Out[871]= 0.30669

In[872]:= ci[3, 5.5]

Out[872]= 0.30669

In[873]:= NIntegrate[sf[1, x], {x, 0, 5.5}]

Out[873]= 0.305746

[gathercoledt (David Gathercole)]

Interestingly your ‘k non-negative’ assumptions (which at first glance appear redundant) make all the difference. Asserting k is real also works here. Unfortunately similar assumptions I’ve tried do not recover the derivative.