[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
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. It may be of some use to note that 1/gamma(z) is an entire function. -- Gene
________________________________ 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
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Hmm, the binomial coefficient on complex numbers could be thought of as b(z,w) := gamma(z+w)/(gamma(z)gamma(w)), 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 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. It may be of some use to note that 1/gamma(z) is an entire function.
-- Gene
________________________________ 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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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
On 3/30/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
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 ...
Around the negative real axis, I think the continuous arg() surface resembles a multi-storey carpark, with spiral staircases of ramps in alternating senses around the poles. Two cars setting off from the same point and navigating up neighbouring ramps in opposite senses will collide again on the next floor/sheet --- have I got that right?
It may be of some use to note that 1/gamma(z) is an entire function.
It took a while for this penny to drop. But yes --- if the discussion of binomial coefficients were recast instead in terms of 1/Gamma(x+1) say, a lot of singularities could be dispatched.
-- 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
On 3/30/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Around the negative real axis, I think the continuous arg() surface resembles a multi-storey carpark, with spiral staircases of ramps in alternating senses around the poles. Two cars setting off from the same point and navigating up neighbouring ramps in opposite senses will collide again on the next floor/sheet --- have I got that right?
Sorry, that was not quite correct: rather u = arg(gamma(z)) spirals left-handedly around every pole, clockwise looking down along the u-axis. For some reason this simpler situation seemed harder to visualise ... WFL
participants (3)
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Dan Asimov -
Eugene Salamin -
Fred Lunnon