This is my function
var('h,r')
f=r^2*arccos((r-h)/r)-(r-h)*sqrt(2*r*h-h^2)
taylor(f,h,0,3)
Result:
-1/5*sqrt(2)*h^(5/2)/sqrt(r) + 4/3*sqrt(2)*h^(3/2)*sqrt(r)
I expected an expression of the form ax^3+bx^2+cx+d but I got 5/2 and 3/2 as exponents for h. Why is that?
This is essentially directly using Maxima, so
(%i11) display2d: false;
(%o11) false
(%i12) f:r^2*acos((r-h)/r)-(r-h)*sqrt(2*r*h-h^2);
(%o12) r^2*acos((r-h)/r)-(r-h)*sqrt(2*h*r-h^2)
(%i13) taylor(f,h,0,3);
(%o13) 4*sqrt(r)*sqrt(2)*h^(3/2)/3-sqrt(r)*sqrt(2)*h^(5/2)/(5*r)
Expanding around other points gives what we expect, so I guess this is some kind of bug (or undocumented feature) in Maxima.
(%i22) taylor(sqrt(x),x,0,5);
(%o22) +sqrt(x)
(%i23) powerseries(sqrt(x),x,0);
(%o23) sqrt(x)
Maybe they like Puiseux series? I've reported this at https://sourceforge.net/p/maxima/bugs/2850/
Edit: Of course, there is the problem that the square root function is not particularly well-behaved at zero! But still one would expect something else, I think.