A Cross Product of Bessel Functions that I am interested in is of the form
Jnu(z)Ynu(kz)-Jnu(kz)Ynu(z), where Jnu(z) and Ynu(z) are Bessel functions of the first and second kinds of order nu, and z is a real number.
There exist asymptotic expansions for this cross product when k is close to 1.
Is there an asymptotic expression for this cross product that works well when k is not close to 1?
Any information or references that answer this question will be very much appreciated.
What I meant to say is that there exists an asymptotic expansion for the values of z that make this cross product of Bessel functions equal to zero when k is close to 1.
Does there exist an asymptotic expressions for the zeros (z) of this cross product when k is not close to 1?
Yes, you can try Taylor Series expansion or Hypergeometric series of some kind expansion and may recourse to approximation. See for example https://arxiv.org/pdf/1604.05163.pdf
Maple will handle this and can produce an expansion. I just typed in implicitplot3d(BesselJ(nu,z)*BesselY(nu,k*z)-BesselJ(nu,k*z)*BesselY(nu,z),z=0..3,k=0..3,nu=0..3,grid=[15,15,15]); and got the following plot. I also plotted it in Excel. I attached the figures and the spreadsheet.
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