# Asymptotic behaviour of sums of special functions: StruveH and BesselY

In my research I obtained as result for a real function the following expression:

$$u(r) = H_0\left((-1)^{1/3} \, r \right) - Y_0\left((-1)^{1/3} \, r \right) + H_0\left(- (-1)^{2/3}\, r \right)- Y_0\left(- (-1)^{2/3}\, r \right)- H_0\left(r \right)- Y_0\left(r \right)$$

Where $H_0$ is the zeroth order StruveH function and $Y_0$ is the zeroth order Bessel function of the second kind, and $r$ a real number greater than zero. I was quite happy about my result so i decided to plot it with mathematica and the result is the following:

The solution starts to behave strangely for large $r$, but i am not sure if this is just Mathematica not properly plotting, or if actually the solution is diverging at large $r$. Is there a way of evaluate such asymptotic behaviour? do you have any reference for that?

• Recall that $K=H-Y$ and use this asymptotic expansions: dlmf.nist.gov/11.6 – tired Apr 20 '17 at 17:55
• It definitely looks like numerical error. You can probably show it rigorously by using the known asymptotics for the Bessel functions and Struve functions. – Antonio Vargas Apr 20 '17 at 17:56
• how are the third-roots of negatives defined? – mathreadler Apr 20 '17 at 18:10
• @tired in the summation you also have a -H -Y, how do you treat that? – SSC Napoli Apr 20 '17 at 18:39
• If it is numerical error, do Bessel and Struve have any "move" relations like sin and cos families do? Say for example being able to calculate $\cos(x)/x$ from $\cos(x+2\pi)/(x+2\pi)$. If they do maybe that could reduce the numerical error for expansions. – mathreadler Apr 20 '17 at 18:46