Definite integraton f(x) / sqrt (-x^2-bx-c)

[ianbell]

I am interested in the definite integration of
f(x) / sqrt(-Q(x))
over the range of x for which quadratic
Q(x) = x^2 + b x + c is negative .

f(x) = sin(a x)/x in particular but others too.

Can anybody point me at any known formulae?

TIA.

 

[Crosson]

This is a case for contour integration!

 

[ianbell]

Can you elaborate?

Going complex, we might consider the imaginary part of
exp ( i a z) / ( z sqrt(-(z-q1)(z-q2) )
or more generally f(z) / sqrt(-(z-q1)(z-q2))
as z moves along the real axis from q1 to q2 , the two real roots of quadratic Q(x)=x^2+bx+c.