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Re: Distinguished logarithm, branch cuts, etc.

• To: mathgroup at smc.vnet.net
• Subject: [mg64904] Re: Distinguished logarithm, branch cuts, etc.
• From: "sashap" <pavlyk at gmail.com>
• Date: Tue, 7 Mar 2006 06:11:55 -0500 (EST)
• References: <duh2tv$5r3$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

You have to be careful here.

An analytic function like BesselI[ nu, z ] *is* a continuous map from
complex plane
to a multi-sheeted Riemann surface. It is when you project from this
Riemann surface,
branch-cuts appear.

It seems that your problem would benefit from the branch-cut convention
different from
the one adopted in Mathematica. Provided z[u] can never take on purely
imaginary
values possible modification could be

   (z[u]^2)^(nu/2)*BesselI[nu, z[u]]/(z[u])^nu

or equivalently

   BesselI[nu, Sqrt[ z[u]^2] ]

Oleksandr Pavlyk
Wolfram Research