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From: ags%pucc-i@jett.UUCP (Seaman)
Newsgroups: net.math
Subject: References on i ** i, "principal logs"
Message-ID: <205@pucc-i.UUCP>
Date: Thu, 16-Feb-84 10:19:48 EST
Article-I.D.: pucc-i.205
Posted: Thu Feb 16 10:19:48 1984
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Date: Thu, 16-Feb-84 10:19:48 EST
Date-Received: Fri, 17-Feb-84 09:18:49 EST
Organization: Purdue University Computing Center

A brief survey of the literature reveals that there is no universal agreement
as to what the "principal value of the logarithm" means.  I quote from Ahlfors,
"Complex Analysis" page 47:

	"By convention the logarithm of a positive number shall always mean
the real logarithm, unless the contrary is stated.  The symbol a ** b, where
a and b are arbitrary complex numbers except for the condition a <> 0, is
always interpreted as an equivalent of exp(b log a).  If a is restricted to
positive numbers, log a shall be real, and a ** b has a single value.
Otherwise log a is the complex logarithm, and a ** b has in general infinitely
many values which differ by factors exp(2*PI*i*n*b).  There will be a single
value if and only if b is an integer n, and then a ** b can be interpreted as
a power of a or a ** (-1).  If b is a rational number with the reduced form
p/q, then a ** b has exactly q values and can be represented as (a**p) ** 1/q."

Note that this specifically denies assigning any unique value to i ** i.
Ahlfors also points out that w = exp(i*y) has a unique solution with y in
[0,2*PI), but declines to identify this value as a "principle value".  Note
that the choice of interval here differs from (-PI,PI] which some people
on the net have claimed as the "universally recognized principal branch".

Other textbooks have a variety of definitions.  For instance, Knopp's
"Theory of Functions" defines the principle value of the logarithm of z as
a path integral, from 1 to z, of du/u.  The path is constrained to lie
in the complex plane cut by the negative real axis, thus avoiding winding
problems.  The integral is path-independent within this domain.  Note that
"the principal logarithm of -1" has no meaning according to this definition.
The same definition can be found in James and James, "Mathematics Dictionary".

Weinberger's "A First Course in Partial Differential Equations" also uses
the path integral definition, but does not specify any particular location
for the cut.  He merely points out that "If we make a cut extending from z=0
to infinity to prevent any winding, [the logarithm] is analytic in the
remainder of the z-plane."

I think the conclusion is clear.  There is no universal agreement on the
meaning of the "principal value" of the logarithm.  By contrast, try finding
a text which offers anything other than the conventional meaning for the
real square root function, i.e. the positive branch.
--

Dave Seaman
..!pur-ee!pucc-i:ags

"Against people who give vent to their loquacity
by extraneous bombastic circumlocution."