A simpler example might be Plot[ArcCos[x]^2,{x,-1,5}], where the plotter
needed to call a complex-valued function to get real results. Should we put acos^2 on pocket calculators? --rwg
We should put complex numbers on pocket calculators.
Exercise: Sketch the graph of asin(sqrt(sin(x)))^2. Now compare with Mathemaplema. For a mild start, plot the derivative. --rwg
Shifting by pi/2, asin(sqrt(cos(x)))^2 is an even, periodic fn with a
cosine all-positive Fourier series, with a5F4[p,...] constant term and 4F3 higher terms (http://tweedledum.com/rwg/asincos.png). Mma was little
help:
SeriesCoefficient flunks ArcSin^2, and after you tell it, Assuming[k > 0 && n > 0 && k \[Element] Integers && n \[Element] Integers, Integrate[Cos[k*x]*Cos[x]^n, {x, -Pi, Pi}]] is a mess including Gamma[-k]. --rwg