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MathGroup Archive 2006

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Distinguished logarithm, branch cuts, etc.

  • To: mathgroup at smc.vnet.net
  • Subject: [mg64878] Distinguished logarithm, branch cuts, etc.
  • From: "Alan" <info at optioncity.REMOVETHIS.net>
  • Date: Mon, 6 Mar 2006 05:01:38 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Imagine that you have a probability density p(x)
on the real x-axis and you compute a characteristic function:

f(u) = \integral e^(I u x) p(x) dx,  where u is real.

It is well known that f(u) must be a continuous (complex-valued)
function of u, at every real u.

However, an analytically derived expression for f(u)
may often involve logarithms, powers, and other
multi-valued functions. These analytic expressions,
evaluated in Mathematica, will often be discontinuous at
certain values of u because of the conventional choice of branch cuts
that Mathematica makes. This requires some fixes.

The following nomenclature has been adopted by many.
(See, for example, Sato's book on Levy processes).
If you write f(u) = exp{lambda(u)}, then the unique
single-valued function lambda(u) is called the "distinguished logarithm".
The function exp{alpha lambda} is called the "distinguished alpha power of 
f".

My problem is that I have an analytic expression that has the form

f(u) = C(u) BesselI[nu, z(u)],

where both C(u) and z(u) are continuous (complex-valued) functions of u.

Since BesselI[nu, z] = z^{nu} times a function of even powers,
then every time z(u) crosses the negative axis, I get an unwanted
(and wrong, for my problem) discontinuity in f(u).

I correct this by multiplying Mathematica's Bessel function
return by exp(-2 n Pi I nu), where n = n(u) is an appropriate integer.

If I use the above notation, I guess this means I have
defined the "distinguished Bessel function".
However, a Google search does not turn up this term and
I'm not sure I want to adopt it. This leads to my questions:

1. Has anyone heard of a better or more conventional name in the
    particular case of the Bessel function?

2. Does anyone know or want to suggest any general naming conventions,
    perhaps less awkward, for the case of general multi-valued functions?

Thanks!
alan


 



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