Calculating determinants for various values of n?

Questions
1

Hello, I have prepared the determinant for a matrix, where its elements are Bessel functions of n’th degree. The determinant is given by:

Table[-I^-n \[Zeta] (BesselJ[
      n, \[Kappa]\[Zeta]] BesselY[-1 + n, \[Kappa]\[Zeta]] - 
    BesselJ[-1 + n, \[Kappa]\[Zeta]] BesselY[
      n, \[Kappa]\[Zeta]]) (BesselJ[
      n, \[Lambda]] HankelH1[-1 + n, \[Lambda]] - 
    BesselJ[-1 + n, \[Lambda]] HankelH1[n, \[Lambda]]), {n, 1, 20}]

Here, zeta, kappa and lambda are coefficients, respectively kappa=0.5, lambda=1 and zeta=3.

How can I automatically calculate the 40 determinants for n= 0 up to ±20 ?

Thanks!

All the best.

2

\[Kappa]\[Zeta] should be \[Kappa] \[Zeta] or \[Kappa]*\[Zeta].

Determinant operates on square matrices and there are no square matrices defined in the code so the question is not clear.

3

This is not the matrix, its the determinant.

4

You can use replacement rules to plug in coefficient values and Table to iterate over values of n from -20 to 20

Table[ 
    -I^-n*\[Zeta] 
        * ( BesselJ[n,\[Kappa]*\[Zeta]] * BesselY[-1+n,\[Kappa]*\[Zeta]] 
            - BesselJ[-1 + n, \[Kappa]*\[Zeta]] * BesselY[ n, \[Kappa]*\[Zeta]]) 
        * (BesselJ[ n, \[Lambda]] * HankelH1[-1 + n, \[Lambda]] 
            - BesselJ[-1+n, \[Lambda]] * HankelH1[n, \[Lambda]]),
    {n, -20, 20}
]  /. {\[Kappa]->0.5,\[Lambda]->1,\[Zeta]->3}

One other issue here is that when you put two coefficients beside each other without a space, the system assumes it’s a a single variable made up a two Greek letters, rather than two variable multiplied by each other. I’ve inserted * symbols to make the multiplication explicit, but you could use spaces as well.

5

Thanks for this. It is exactly what I looked for.

6

PS: I got this long list:

{-0.63662 (BesselJ[21, \[Lambda]] HankelH1[20, \[Lambda]] - 
    BesselJ[20, \[Lambda]] HankelH1[21, \[Lambda]]), (0. + 
    0.63662 I) (BesselJ[20, \[Lambda]] HankelH1[19, \[Lambda]] - 
    BesselJ[19, \[Lambda]] HankelH1[20, \[Lambda]]), 
 0.63662 (BesselJ[19, \[Lambda]] HankelH1[18, \[Lambda]] - 
    BesselJ[18, \[Lambda]] HankelH1[19, \[Lambda]]), (0. - 
    0.63662 I) (BesselJ[18, \[Lambda]] HankelH1[17, \[Lambda]] - 
    BesselJ[17, \[Lambda]] HankelH1[
      18, \[Lambda]]), -0.63662 (BesselJ[17, \[Lambda]] HankelH1[
      16, \[Lambda]] - 
    BesselJ[16, \[Lambda]] HankelH1[17, \[Lambda]]), (0. + 
    0.63662 I) (BesselJ[16, \[Lambda]] HankelH1[15, \[Lambda]] - 
    BesselJ[15, \[Lambda]] HankelH1[16, \[Lambda]]), 
 0.63662 (BesselJ[15, \[Lambda]] HankelH1[14, \[Lambda]] - 
    BesselJ[14, \[Lambda]] HankelH1[15, \[Lambda]]), (0. - 
    0.63662 I) (BesselJ[14, \[Lambda]] HankelH1[13, \[Lambda]] - 
    BesselJ[13, \[Lambda]] HankelH1[
      14, \[Lambda]]), -0.63662 (BesselJ[13, \[Lambda]] HankelH1[
      12, \[Lambda]] - 
    BesselJ[12, \[Lambda]] HankelH1[13, \[Lambda]]), (0. + 
    0.63662 I) (BesselJ[12, \[Lambda]] HankelH1[11, \[Lambda]] - 
    BesselJ[11, \[Lambda]] HankelH1[12, \[Lambda]]), 
 0.63662 (BesselJ[11, \[Lambda]] HankelH1[10, \[Lambda]] - 
    BesselJ[10, \[Lambda]] HankelH1[11, \[Lambda]]), (0. - 
    0.63662 I) (BesselJ[10, \[Lambda]] HankelH1[9, \[Lambda]] - 
    BesselJ[9, \[Lambda]] HankelH1[10, \[Lambda]]), -0.63662 (BesselJ[
      9, \[Lambda]] HankelH1[8, \[Lambda]] - 
    BesselJ[8, \[Lambda]] HankelH1[9, \[Lambda]]), (0. + 
    0.63662 I) (BesselJ[8, \[Lambda]] HankelH1[7, \[Lambda]] - 
    BesselJ[7, \[Lambda]] HankelH1[8, \[Lambda]]), 
 0.63662 (BesselJ[7, \[Lambda]] HankelH1[6, \[Lambda]] - 
    BesselJ[6, \[Lambda]] HankelH1[7, \[Lambda]]), (0. - 
    0.63662 I) (BesselJ[6, \[Lambda]] HankelH1[5, \[Lambda]] - 
    BesselJ[5, \[Lambda]] HankelH1[6, \[Lambda]]), -0.63662 (BesselJ[
      5, \[Lambda]] HankelH1[4, \[Lambda]] - 
    BesselJ[4, \[Lambda]] HankelH1[5, \[Lambda]]), (0. + 
    0.63662 I) (BesselJ[4, \[Lambda]] HankelH1[3, \[Lambda]] - 
    BesselJ[3, \[Lambda]] HankelH1[4, \[Lambda]]), 
 0.63662 (BesselJ[3, \[Lambda]] HankelH1[2, \[Lambda]] - 
    BesselJ[2, \[Lambda]] HankelH1[3, \[Lambda]]), (0. - 
    0.63662 I) (BesselJ[2, \[Lambda]] HankelH1[1, \[Lambda]] - 
    BesselJ[1, \[Lambda]] HankelH1[2, \[Lambda]]), -0.63662 (BesselJ[
      1, \[Lambda]] HankelH1[0, \[Lambda]] - 
    BesselJ[0, \[Lambda]] HankelH1[1, \[Lambda]]), (0. + 
    0.63662 I) (BesselJ[1, \[Lambda]] HankelH1[0, \[Lambda]] - 
    BesselJ[0, \[Lambda]] HankelH1[1, \[Lambda]]), 
 0.63662 (BesselJ[2, \[Lambda]] HankelH1[1, \[Lambda]] - 
    BesselJ[1, \[Lambda]] HankelH1[2, \[Lambda]]), (0. - 
    0.63662 I) (BesselJ[3, \[Lambda]] HankelH1[2, \[Lambda]] - 
    BesselJ[2, \[Lambda]] HankelH1[3, \[Lambda]]), -0.63662 (BesselJ[
      4, \[Lambda]] HankelH1[3, \[Lambda]] - 
    BesselJ[3, \[Lambda]] HankelH1[4, \[Lambda]]), (0. + 
    0.63662 I) (BesselJ[5, \[Lambda]] HankelH1[4, \[Lambda]] - 
    BesselJ[4, \[Lambda]] HankelH1[5, \[Lambda]]), 
 0.63662 (BesselJ[6, \[Lambda]] HankelH1[5, \[Lambda]] - 
    BesselJ[5, \[Lambda]] HankelH1[6, \[Lambda]]), (0. - 
    0.63662 I) (BesselJ[7, \[Lambda]] HankelH1[6, \[Lambda]] - 
    BesselJ[6, \[Lambda]] HankelH1[7, \[Lambda]]), -0.63662 (BesselJ[
      8, \[Lambda]] HankelH1[7, \[Lambda]] - 
    BesselJ[7, \[Lambda]] HankelH1[8, \[Lambda]]), (0. + 
    0.63662 I) (BesselJ[9, \[Lambda]] HankelH1[8, \[Lambda]] - 
    BesselJ[8, \[Lambda]] HankelH1[9, \[Lambda]]), 
 0.63662 (BesselJ[10, \[Lambda]] HankelH1[9, \[Lambda]] - 
    BesselJ[9, \[Lambda]] HankelH1[10, \[Lambda]]), (0. - 
    0.63662 I) (BesselJ[11, \[Lambda]] HankelH1[10, \[Lambda]] - 
    BesselJ[10, \[Lambda]] HankelH1[
      11, \[Lambda]]), -0.63662 (BesselJ[12, \[Lambda]] HankelH1[
      11, \[Lambda]] - 
    BesselJ[11, \[Lambda]] HankelH1[12, \[Lambda]]), (0. + 
    0.63662 I) (BesselJ[13, \[Lambda]] HankelH1[12, \[Lambda]] - 
    BesselJ[12, \[Lambda]] HankelH1[13, \[Lambda]]), 
 0.63662 (BesselJ[14, \[Lambda]] HankelH1[13, \[Lambda]] - 
    BesselJ[13, \[Lambda]] HankelH1[14, \[Lambda]]), (0. - 
    0.63662 I) (BesselJ[15, \[Lambda]] HankelH1[14, \[Lambda]] - 
    BesselJ[14, \[Lambda]] HankelH1[
      15, \[Lambda]]), -0.63662 (BesselJ[16, \[Lambda]] HankelH1[
      15, \[Lambda]] - 
    BesselJ[15, \[Lambda]] HankelH1[16, \[Lambda]]), (0. + 
    0.63662 I) (BesselJ[17, \[Lambda]] HankelH1[16, \[Lambda]] - 
    BesselJ[16, \[Lambda]] HankelH1[17, \[Lambda]]), 
 0.63662 (BesselJ[18, \[Lambda]] HankelH1[17, \[Lambda]] - 
    BesselJ[17, \[Lambda]] HankelH1[18, \[Lambda]]), (0. - 
    0.63662 I) (BesselJ[19, \[Lambda]] HankelH1[18, \[Lambda]] - 
    BesselJ[18, \[Lambda]] HankelH1[
      19, \[Lambda]]), -0.63662 (BesselJ[20, \[Lambda]] HankelH1[
      19, \[Lambda]] - 
    BesselJ[19, \[Lambda]] HankelH1[20, \[Lambda]])}

Is it possible to sort it by value of N?

Thanks!

7

There is no N in the list, so not clear what you want to sort by.

8

Have a look at the original code by
Nicholas Engelking

Table[ 
    -I^-n*\[Zeta] 
        * ( BesselJ[n,\[Kappa]*\[Zeta]] * BesselY[-1+n,\[Kappa]*\[Zeta]] 
            - BesselJ[-1 + n, \[Kappa]*\[Zeta]] * BesselY[ n, \[Kappa]*\[Zeta]]) 
        * (BesselJ[ n, \[Lambda]] * HankelH1[-1 + n, \[Lambda]] 
            - BesselJ[-1+n, \[Lambda]] * HankelH1[n, \[Lambda]]),
    {n, -20, 20}
]  /. {\[Kappa]->0.5,\[Lambda]->1,\[Zeta]->3}

There are certainly values of N.

9

WL is case sensitive so you need to be precise. It is n, not N.

n appears in the expression but it takes on values from -20 to 20 from Table. There is no n in the evaluated expression. Table does not rearrange items so they are in the order of n ranging from -20 to 20.

10

Fine, but do you have an answer to the original question to Nicholas Engelking?

11

When I evaluate the code from Nicholas I get

{0. - 0.810569 I, -0.810569 + 0. I, 0. + 0.810569 I, 0.810569 + 0. I, 
 0. - 0.810569 I, -0.810569 + 0. I, 0. + 0.810569 I, 0.810569 + 0. I, 
 0. - 0.810569 I, -0.810569 + 0. I, 0. + 0.810569 I, 0.810569 + 0. I, 
 0. - 0.810569 I, -0.810569 + 0. I, 0. + 0.810569 I, 0.810569 + 0. I, 
 0. - 0.810569 I, -0.810569 + 0. I, 0. + 0.810569 I, 0.810569 + 0. I, 
 0. - 0.810569 I, -0.810569 + 0. I, 0. + 0.810569 I, 0.810569 + 0. I, 
 0. - 0.810569 I, -0.810569 + 0. I, 0. + 0.810569 I, 0.810569 + 0. I, 
 0. - 0.810569 I, -0.810569 + 0. I, 0. + 0.810569 I, 0.810569 + 0. I, 
 0. - 0.810569 I, -0.810569 + 0. I, 0. + 0.810569 I, 0.810569 + 0. I, 
 0. - 0.810569 I, -0.810569 + 0. I, 0. + 0.810569 I, 0.810569 + 0. I, 
 0. - 0.810569 I}
12

I get the same. But which value is related to value of n?

13

Include n in your Table

Table[{n, -I^-n*\[Zeta]*(BesselJ[n, \[Kappa]*\[Zeta]]*
       BesselY[-1 + n, \[Kappa]*\[Zeta]] - 
      BesselJ[-1 + n, \[Kappa]*\[Zeta]]*
       BesselY[n, \[Kappa]*\[Zeta]])*(BesselJ[n, \[Lambda]]*
       HankelH1[-1 + n, \[Lambda]] - 
      BesselJ[-1 + n, \[Lambda]]*HankelH1[n, \[Lambda]])}, {n, -20, 
   20}] /. {\[Kappa] -> 0.5, \[Lambda] -> 1, \[Zeta] -> 3}