ags@pucc-i (Seaman) (02/16/84)
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."
ags%pucc-i@jett.UUCP (Seaman) (02/16/84)
Relay-Version: version B 2.10.1 6/24/83; site akgua.UUCP Posting-Version: version B 2.10.1 6/24/83; site pucc-i Path: akgua!clyde!burl!ulysses!mhuxl!ihnp4!inuxc!pur-ee!CS-Mordred!Pucc-H:Pucc-I:ags Message-ID: <205@pucc-i> 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."
akt@mcnc.UUCP (Amit Thakur) (02/19/84)
all this talk of i^i has got me wondering overtime: i^2=(-1) a real number ln(i)=ln|i|+i(pi/2 + 2*n*pi)= 0 + i(pi/2 + 2n*pi) ln(a^b) = b*ln(a) then does it follow that ln(-1)=ln(i^2)= 2i(pi/2 + 2n*pi)= i*pi(1+4n)?? in general, ln(x), x real and x < 0, then ln(x)= i*pi(1+4n) + ln(|x|) this would mean that we could take logarithms of negative numbers. has anyone ever thought of this before? if this is valid, then the only number for which ln(x) is not defined is x=0. akt at ...decvax!mcnc!akt
akt@mcnc.UUCP (Amit Thakur) (02/19/84)
i been doing some more wondering overnight:
ln(-1) = i*pi(1+2n*pi) because exp{ i*pi(1+2n*pi)} = -1.
i think this is a better solution.
also: exp{ +-i*2n*pi } = 1. 0 = ln(1) = +-i*2n*pi
but 0 <> +-i*2n*pi for n <> 0.
note that +-i*2n*pi = +-i*2n*pi{exp[i*arctan(+-2n*pi/0)]} =
+-i*2n*pi{exp[i*arctan(+-infinity)]} = +-i*2n*pi{exp[i*+-pi/2]}
applying euler's formula gives us +-i*2n*pi again, as it should.
thus, if we let n=0, then obviously, 0 = +-i*2n*pi, and we
can then say that ln(0) = 0. or, if we choose any other n,
and force 0 = +-i*2n*pi, we can say that ln(0) = +-i*2n*pi.
there: ln() is now defined over the entire complex plane!
question: i know i divided by zero to arrive at the above
result, but i don't see why this isn't valid, especially since
the exp() function is defined as an infinite series.
please: no flames to the effect that i should go back and study
my high school algebra. tell me why this is not valid in *this*
particular case.
of course, using ln(x)/ln(b), one could easily find log (x).
b
akt at ...decvax!mcnc!akt
p.s. this chain of thoughts was started by the idea that pure
negative imaginary numbers had logs, but not negative reals.
and why should zero be so lonesome all by itself?csc@watmath.UUCP (Computer Sci Club) (02/21/84)
The complex log fuction can be defined in a number of ways.
Perhaps the most intuitive is as the inverse of the exponential
function (which can be defined as the sum of an infinite series).
Hence we define log(z) to be a complex number such that exp(log(z))=z.
Now any non zero complex number z can be written as exp(r + it),
(r and t real numbers). r is uniquely determined but t isn't.
If t is valid so is t+2n(pi) with n an integer. Therefore there
are an infinitely many complex numbers g such that exp(g)=z.
(if z is non zero). Hence log is multi valued. There is no
complex number g such that exp(g)=0, therefore log(0) is undefined.
Hence log is a multi valued function defined on the complex plane
minus zero.
It does not make sense to define log(0)=0 as then
exp(log(0))=1. The article which argued that log(0)=0 contained
a division by zero which implied i*2n(pi)=i*2n(pi)exp(i*(pi)/2)
or as exp(i*pi/2)=i this implies i*2n(pi)=-2n(pi). A contradiction.
It is because dividing by zero leads to such contradictions that
such division is not defined.
One can define a single valued log function by choosing one
value of t for each z, usually done by restricting t to some half
open interval of length 2(pi). However one cannnot do this in such
a way as to have the resulting function continuous on the complex
plane (minus zero). Also such equations as log(ab)=log(a) + log(b)
and log(exp(z))=z cannot hold for all a,b,z. The usual practice is
to use whichever "branch" (ie. choice of an interval for t) that
is most convenient for the task at hand.
William Hughesakt@mcnc.UUCP (Amit Thakur) (02/22/84)
i sent out a retraction of my claim that ln(0)=0, but apparently it got lost somewhere. basically, i said: perhaps i need to go back and study my high school algebra. i was taking real value of ln(1) (=0) and setting it equal to the imaginary value of ln(1) (=+-2n*pi), from the logic that if a=b and b=c, then a=c. but then -1.414... = 2**(0.5) = 1.414, but -1.414 <> 1.414. thus, 0 <> 2n*pi, for n <> 0. thus, my argument for ln(0) was not valid. what happened was that i stayed up till 4:30am reading news on saturday night (my saturday nights are really fun, lemme tell ya!:-)) and then got up early sunday morning and posted the value of ln(0). i hereby retract any and all silliness in my previous article. akt at ...decvax!mcnc!akt