This notebook is a sample file for the Mathematica package "weakrel.m" by Tadas K. Nakamura.  1998/10/12

Example

The following cells are intended to compare the result of WeaklyRelativisticZ1 with  numerical integration. Here Im(ζ) must be positive because the numerical integration NumInt does not handle the analytical continuation properly.

Note:  The numerical integration used in NumInt requires a considerable amount of computation. It took about 50 seconds to calculate this example on my machine (435MHz PowerPC G3).

First Load the package weakrel.m (the directory that weakrel.m is in must be on the search path, $Path):

<< "weakrel.m"

Define a bound for numerical integration

largeNumber = 4 ;

This is  a Numerical Integration approach for WeaklyRelativisticZ1:

\(\(
\(NumInt[ζ_, η_, b_, ν_, m_] := \ 
	
      NIntegrate[
        NIntegrate[
          \(\ x\ \ y\^ν\ \ \(BesselJ[m, b\ x]\^2\) 
              Exp[\(-x\^2\) - y\^2]\)\/\(y - ζ - 
              η \((y\^2 + x\^2)\)\), {y, \(-largeNumber\), 
            largeNumber}], {x, 0, largeNumber}]; \)\)\)

Specify the arguments: η must be small (η << 1)  and Im(ζ) must be positive.

\(\(ζ = \(-1\) + 0.1  I; η = 0.1; b = 3.0; ν = 1; 
    m = \(-1\); 
\)\)

Compare results:

NumInt[ζ, η, b, ν, m]

WeaklyRelativisticZ1[ζ, η, b, ν, m]

Discussion

The package weakrel.m is used to calculate three generalized plasma dispersion functions defined as,

\(
\ 
      WeaklyRelativisticZ1[ζ, η, b, ν, m]
	\ 	 = 	
        ∫\_0\%∞dx\ x\ \(J\_m\^2\) \((bx)\) 
            \(∫\_\(-∞\)\%∞dy\ 
                \(\(\ \(y\^ν\) e\^\(-y\^2\)\)\/\(y - ζ + 
                      η \((x\^2 + y\^2)\)\)\) x\)\)

\(			
\ 
      WeaklyRelativisticZ2[ζ, η, b, ν, m]
\ 	 = 	
        ∫\_0\%∞dx\ x\^2\ \(J\_m\) \((bx)\) 
            J\_m' \((bx)\) 
            \(∫\_\(-∞\)\%∞dy\ 
                \(\(\(y\^ν\) e\^\(-y\^2\)\)\/\(y - ζ + 
                      η \((x\^2 + y\^2)\)\)\) x\)\)

\(			

      WeaklyRelativisticZ2[ζ, η, b, ν, m]
		 = 	
        ∫\_0\%∞dx\ x\^2\ \(J\_m\) \((bx)\) 
            J\_m' \((bx)\) 
            \(∫\_\(-∞\)\%∞dy\ 
                \(\(\(y\^ν\) e\^\(-y\^2\)\)\/\(y - ζ + 
                      η \((x\^2 + y\^2)\)\)\) x\)\)

where J is a Bessel Function.

These functions are necessary to calculate electromagnetic wave propagation in a weakly relativistic magnetized plasma. The integration in the above expressions cannot be carried out directly; this package makes use of a rational approximation to the Gauss function, which is

exp(-(x^2)) = ((Subscript[C, 0])/((∏_(k = 1)^N)(x - (Subscript[c, k])) 
                  (x + (Subscript[c, k]))))

where N is the order of the approximation. Larger N gives more precise results. One can choose N=5 or 3 (variable name is "nmax") at the top of the package.

With this approximation the integration can be carried out analytically with the aid of the residue theorem. For example, WeaklyRelativisticZ1 can be written as,

WeaklyRelativisticZ1[ζ, η, b, ν, m]
		 = ((((Subscript[C, 0])^2)(π^2))/4)(\(∑\+\(i, j
                  \)\(\(\(c\_j\^\(ν - 1\)\) \(H\_n\^+\) \((\(c\_i\) b)\) 
                        \(Z\_n\) \((\(c\_i\) b)\)\)\/\(c\_j - ζ - 
                        η \((c\_i\^2 + c\_j\^2)\)\)\) 
                  1\/\(∏\+\(k ≠ i\)\((c\_i\^2 - c\_k\^2)
                        \)\)\ 1
                    \/\(∏\+\(k ≠ j\)\((c\_j\^2 - c\_k\^2)
                        \)\)\)
						   -\(∑\+\(i, j
                  \)\(\(\(c\_j\^\(ν - 1\)\) \(H\_n\^-\) \((\(-c\_i\) b)\) 
                        \(Z\_n\) \((\(-c\_i\) b)\)\)\/\(c\_j - ζ - 
                        η \((c\_i\^2 + c\_j\^2)\)\)\) 
                  1\/\(∏\+\(k ≠ i\)\((c\_i\^2 - c\_k\^2)
                        \)\)\ 1
                    \/\(∏\+\(k ≠ j\)\((c\_j\^2 - c\_k\^2)
                        \)\)\))

where \(TraditionalForm\`\(H\^+\)\) and \(TraditionalForm\`\(H\^-\)\) are Hankel Functions of first and second kind. This package  calculates the dispersion functions using the above formulae.

For more details, see:

Nakamura, T. K., and M. Hoshino, "One-over-polynomial approximation for linear kinetic dispersion and its application to relativistic cyclotron resonance,"  Phys of Plasmas, 5, 3547, 1998.

This paper is available on-line at:  http://www.s.fpu.ac.jp/home/u-tadas/www/on-line-paper.html

The package "weakrel" is absolutely free and you do not need to acknowledge the authors' name in publication or in any other use. However, please let me (tadas@fpu.ac.jp) know your address so that I can reach you for further information, such as bug fixes. Your suggestions and comments are also welcome.