I have generated various determinants for a system of differential equations, with the Bessel functions, which generate complex values for apparently all levels of n. I show here one of the determinants. What can I do to identify a level of n that yields real values?
For instance, one of the Bessel-function determinants is given by:
$\begin{equation} \begin{array}{cc} B_n=-\frac{1}{3} i^{-n} \left(2.25 \left(J_{n+1}\left(\frac{1}{10}\right)-J_{n-1}\left(\frac{1}{10}\right)\right) Y_n\left(\frac{3}{2}\right)+6 J_n\left(\frac{1}{10}\right) \left(\frac{3}{2} Y_{n-1}\left(\frac{3}{2}\right)-\\n Y_n\left(\frac{3}{2}\right)\right)\right) (J_n(1) H_{n-1}^{(1)}(1)-J_{n-1}(1) H_n^{(1)}(1)) \end{array} \end{equation}$
Here $J_{n}$ are $Y_{n}$ the Bessel function of n degree of the first and second kind respectively. I have quadruple-checked all matrices, and they seem to be fine. So I don't know what to do about these matrices in order to find real-valued determinants.
What can I do to find the right levels of n that give real values?
Any ideas appreciated!
I tried your code and I get these determinant values for the first 20 levels of n:
$**\left\{3 i \left(J_1\left(\frac{3}{2}\right) Y_0\left(\frac{3}{2}\right)-J_0\left(\frac{3}{2}\right) Y_1\left(\frac{3}{2}\right)\right) (J_1(1) H_0^{(1)}(1)-J_0(1) H_1^{(1)}(1)),3 \left(J_2\left(\frac{3}{2}\right) Y_1\left(\frac{3}{2}\right)-J_1\left(\frac{3}{2}\right) Y_2\left(\frac{3}{2}\right)\right) (J_2(1) H_1^{(1)}(1)-J_1(1) H_2^{(1)}(1)),-3 i \left(J_3\left(\frac{3}{2}\right) Y_2\left(\frac{3}{2}\right)-J_2\left(\frac{3}{2}\right) Y_3\left(\frac{3}{2}\right)\right) (J_3(1) H_2^{(1)}(1)-J_2(1) H_3^{(1)}(1)),-3 \left(J_4\left(\frac{3}{2}\right) Y_3\left(\frac{3}{2}\right)-J_3\left(\frac{3}{2}\right) Y_4\left(\frac{3}{2}\right)\right) (J_4(1) H_3^{(1)}(1)-J_3(1) H_4^{(1)}(1)),3 i \left(J_5\left(\frac{3}{2}\right) Y_4\left(\frac{3}{2}\right)-J_4\left(\frac{3}{2}\right) Y_5\left(\frac{3}{2}\right)\right) (J_5(1) H_4^{(1)}(1)-J_4(1) H_5^{(1)}(1)),3 \left(J_6\left(\frac{3}{2}\right) Y_5\left(\frac{3}{2}\right)-J_5\left(\frac{3}{2}\right) Y_6\left(\frac{3}{2}\right)\right) (J_6(1) H_5^{(1)}(1)-J_5(1) H_6^{(1)}(1)),-3 i \left(J_7\left(\frac{3}{2}\right) Y_6\left(\frac{3}{2}\right)-J_6\left(\frac{3}{2}\right) Y_7\left(\frac{3}{2}\right)\right) (J_7(1) H_6^{(1)}(1)-J_6(1) H_7^{(1)}(1)),-3 \left(J_8\left(\frac{3}{2}\right) Y_7\left(\frac{3}{2}\right)-J_7\left(\frac{3}{2}\right) Y_8\left(\frac{3}{2}\right)\right) (J_8(1) H_7^{(1)}(1)-J_7(1) H_8^{(1)}(1)),3 i \left(J_9\left(\frac{3}{2}\right) Y_8\left(\frac{3}{2}\right)-J_8\left(\frac{3}{2}\right) Y_9\left(\frac{3}{2}\right)\right) (J_9(1) H_8^{(1)}(1)-J_8(1) H_9^{(1)}(1)),3 \left(J_{10}\left(\frac{3}{2}\right) Y_9\left(\frac{3}{2}\right)-J_9\left(\frac{3}{2}\right) Y_{10}\left(\frac{3}{2}\right)\right) (J_{10}(1) H_9^{(1)}(1)-J_9(1) H_{10}^{(1)}(1)),-3 i \left(J_{11}\left(\frac{3}{2}\right) Y_{10}\left(\frac{3}{2}\right)-J_{10}\left(\frac{3}{2}\right) Y_{11}\left(\frac{3}{2}\right)\right) (J_{11}(1) H_{10}^{(1)}(1)-J_{10}(1) H_{11}^{(1)}(1)),-3 \left(J_{12}\left(\frac{3}{2}\right) Y_{11}\left(\frac{3}{2}\right)-J_{11}\left(\frac{3}{2}\right) Y_{12}\left(\frac{3}{2}\right)\right) (J_{12}(1) H_{11}^{(1)}(1)-J_{11}(1) H_{12}^{(1)}(1)),3 i \left(J_{13}\left(\frac{3}{2}\right) Y_{12}\left(\frac{3}{2}\right)-J_{12}\left(\frac{3}{2}\right) Y_{13}\left(\frac{3}{2}\right)\right) (J_{13}(1) H_{12}^{(1)}(1)-J_{12}(1) H_{13}^{(1)}(1)),3 \left(J_{14}\left(\frac{3}{2}\right) Y_{13}\left(\frac{3}{2}\right)-J_{13}\left(\frac{3}{2}\right) Y_{14}\left(\frac{3}{2}\right)\right) (J_{14}(1) H_{13}^{(1)}(1)-J_{13}(1) H_{14}^{(1)}(1)),-3 i \left(J_{15}\left(\frac{3}{2}\right) Y_{14}\left(\frac{3}{2}\right)-J_{14}\left(\frac{3}{2}\right) Y_{15}\left(\frac{3}{2}\right)\right) (J_{15}(1) H_{14}^{(1)}(1)-J_{14}(1) H_{15}^{(1)}(1)),-3 \left(J_{16}\left(\frac{3}{2}\right) Y_{15}\left(\frac{3}{2}\right)-J_{15}\left(\frac{3}{2}\right) Y_{16}\left(\frac{3}{2}\right)\right) (J_{16}(1) H_{15}^{(1)}(1)-J_{15}(1) H_{16}^{(1)}(1)),3 i \left(J_{17}\left(\frac{3}{2}\right) Y_{16}\left(\frac{3}{2}\right)-J_{16}\left(\frac{3}{2}\right) Y_{17}\left(\frac{3}{2}\right)\right) (J_{17}(1) H_{16}^{(1)}(1)-J_{16}(1) H_{17}^{(1)}(1)),3 \left(J_{18}\left(\frac{3}{2}\right) Y_{17}\left(\frac{3}{2}\right)-J_{17}\left(\frac{3}{2}\right) Y_{18}\left(\frac{3}{2}\right)\right) (J_{18}(1) H_{17}^{(1)}(1)-J_{17}(1) H_{18}^{(1)}(1)),-3 i \left(J_{19}\left(\frac{3}{2}\right) Y_{18}\left(\frac{3}{2}\right)-J_{18}\left(\frac{3}{2}\right) Y_{19}\left(\frac{3}{2}\right)\right) (J_{19}(1) H_{18}^{(1)}(1)-J_{18}(1) H_{19}^{(1)}(1)),-3 \left(J_{20}\left(\frac{3}{2}\right) Y_{19}\left(\frac{3}{2}\right)-J_{19}\left(\frac{3}{2}\right) Y_{20}\left(\frac{3}{2}\right)\right) (J_{20}(1) H_{19}^{(1)}(1)-J_{19}(1) H_{20}^{(1)}(1))\right\}**$
Seemingly you have some real values there.