On Sat, Mar 29, 2014 at 5:40 PM, Eugene Salamin <gene_salamin@yahoo.com>wrote:
You can do abs(gamma(z)) as a contour plot, and you and do exp(i arg(gamma(z))) in color and not have to worry about the discontinuity in arg(gamma(z)). You can even do a color contour plot showing both together. But arg(gamma(z)) is multiple valued and depends on which way you go around the poles.
You're right --- so in my scheme, arg() should have an infinite winding point at each pole. How are these connected, I wonder ...
It may be of some use to note that 1/gamma(z) is an entire function.
-- Gene
Ah, that must be why I keep coming across those mysterious colour plots! Well, it's a neat solution to the problem I suppose; but I hanker after a more traditional presentation, which is so much more attractive, besides easier to interpret! On 3/30/14, Dan Asimov <asimov@msri.org> wrote:
Hmm, the binomial coefficient on complex numbers could be thought of as
b(z,w) := gamma(z+w)/(gamma(z)gamma(w)),
Not quite --- there's a linear factor missing there.
the reciprocal of the beta function beta(z,w). Since the poles of gamma are at {0, -1, -2, ...}, if z and w are integers with z+w being a pole of gamma, then at least one of z or w must also be a pole. Since all poles are simple, this means that a pole in the numerator is canceled out by a pole in the denominator. (IF z and w are integers!)
--Dan
This topic is discussed in detail at https://www.dropbox.com/s/anykne0pd55ehjg/binomial.pdf WFL
________________________________ From: Fred Lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Saturday, March 29, 2014 5:19 PM Subject: [math-fun] Continuous argument
Preparing a seminar on binomial coefficient extension --- avoiding the nuclear option? --- for a computer science audience, I prepared plots of re, im, abs, arg parts of the gamma function qua function of a complex variable. It struck me that the arg plot would be much improved if one could avoid it snapping back to the principal value when passing thru' angles (+/1)pi --- proceeding instead continuously in whichever direction happened to be current at that point.
How can arg() be hacked to achieve such an effect?
[ Note that arg(gamma(z)) really is discontinuous around the poles at non-positive integer z --- but that is not an issue here, since special measures must be taken in that neighbourhood anyway. Rather appropriate subject line for the list, what? ]
Fred Lunnon