I want a continous anti-derivate (cos(2*u*m)-alpha*u*sin(2*u*m))/(4*u^2+1)
w.r.t. u, for alpha and m Reals, 0 < u.
What I get with Maple in terms of Ci and Si jumps in m=0 (and for m=0 it
is not defined, due to Ci).
The best I got was
Si(-conjugate(v))-Si(v); eval(%, v=m*(2*u+I));
psi:= unapply(%, u,m);
chi:= (u,m) -> piecewise(0 <= m, -Ci((2*u-I)*m)+Ci(m*(2*u+I)),
m < 0, -Ci((2*u-I)*m)+Ci(m*(2*u+I)) + 2*I*Pi);
as one can see from examples like
plot([Re(%), Im(%)], m=-1..1);
To see that L solves the task one differentiates it w.r.t. and uses
L assuming (0 <m); # or the converse
diff(%,u); convert(%, expln): evalc(%): collect(%,exp): combine(%,exp): sort(%)
Also one can add a constant (w.r.t. u) to chi.
Any nicer idea for that? My special concern is Numerics to cover m ~ 0, for
which I am not sure to switch to Ci = Sum(...) + ln() + gamma [ according to
the FunctionAdvisor for Ci, ln can be treated similar by piecewise].