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Calculate this expression with Bessel functions?

Posted 5 years ago

Consider the following code:

eq20 = (BesselI[0, k/\[Lambda] r] BesselK[1, k/\[Lambda] \[Rho]] + 
          BesselK[0, k/\[Lambda] r] BesselI[1, 
            k/\[Lambda] \[Rho]])/(BesselI[1, k/\[Lambda]] BesselK[1, 
            k/\[Lambda] \[Rho]] - 
          BesselI[1, k/\[Lambda] \[Rho]] BesselK[1, k/\[Lambda]] ) // 
       Simplify // Normal

where [Lambda]=800, [Rho]=10, and k is an arbitrary integer. The object of calculation is to estimate the order of k or the asympt

POSTED BY: Jacques Ou
3 Replies

Maybe something like this will help.

neq20 = (BesselI[0, k/\[Lambda] r] BesselK[1, 
       k/\[Lambda] \[Rho]] + 
     BesselK[0, k/\[Lambda] r] BesselI[1, 
       k/\[Lambda] \[Rho]])/(BesselI[1, k/\[Lambda]] BesselK[1, 
       k/\[Lambda] \[Rho]] - 
     BesselI[1, k/\[Lambda] \[Rho]] BesselK[1, 
       k/\[Lambda]]) /. {\[Lambda] -> 800, \[Rho] -> 10}

(* Out[151]= (
BesselI[1, k/80] BesselK[0, (k r)/800] + 
 BesselI[0, (k r)/800] BesselK[1, k/80])/(-BesselI[1, k/80] BesselK[1,
    k/800] + BesselI[1, k/800] BesselK[1, k/80]) *)

TrigToExp[Normal[Series[neq20, {k, Infinity, 1}]]]

(* Out[153]= -((
  40 I Sqrt[10] E^(
   k (-(1/80) - r/800)))/(((40 Sqrt[10] E^(-9 k/800))/k - (
     40 Sqrt[10] E^(9 k/800))/k) k Sqrt[r])) + (
 40 Sqrt[10] E^(
  k (1/80 - r/800)))/(((40 Sqrt[10] E^(-9 k/800))/k - (
    40 Sqrt[10] E^(9 k/800))/k) k Sqrt[r]) + (
 20 Sqrt[10]
   E^(-k/80) (E^((I \[Pi])/4 - (k r)/800) + 
    E^(-((I \[Pi])/4) + (k r)/800)) (r + Sqrt[r^2]))/(((
    40 Sqrt[10] E^(-9 k/800))/k - (40 Sqrt[10] E^(9 k/800))/
    k) k Sqrt[-I r] r) + (
 20 Sqrt[10]
   E^(-k/80) (E^(-((I \[Pi])/4) - (k r)/800) + 
    E^((I \[Pi])/4 + (k r)/800)) (1 - Sqrt[r^2]/r))/(((
    40 Sqrt[10] E^(-9 k/800))/k - (40 Sqrt[10] E^(9 k/800))/k) k Sqrt[
  I r])*)
POSTED BY: Daniel Lichtblau
Posted 5 years ago

r is variable and \Rho is a constant.

POSTED BY: Jacques Ou

Is BesslK respectively BesslI supposed to be BesselK resp. BesselI? Is r supposed to be \[Rho]?

POSTED BY: Daniel Lichtblau
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