Re: [math-fun] Re: product integrals

12 Nov 2002

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Subj:	[math-fun] Re: product integrals

Date: Tue, 12 Nov 2002 01:40:01 -0600
From: mcintosh@servidor.unam.mx
Subject: [math-fun] Re: product integrals
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...
[editorial comment:  Gosper calls these Prodigals.  I don't think
...
they have any non-obvious proerties, but it is strange there's no
...
"official" name. --Rich]
On the contrary, they can be seen in all places and at all times (at least 
since we've had calculus). The reason that you don't hear more about them is 
that, after the initial excitement of discovery, they don't do much for you. 
In solving differential equations, they are equivalent to using Euler's method. 

There, their one great advantage (or two) is that by Gerschgorin's criterion, 
the solution matrix is invertible and solutions from initial conditions are 
thereby unique. #2 is that Huyghen's principle is valid, the solution after a 
while is a valid initial condition for continuing. 

Any attempt to multiply out the (1 + matrix epsilons) to get an integral 
formula makes the matrizant, which also arises from Picard's method. The 
problem is that those epsilons may not commute, in fact most assuredly will 
not, so terms cannot be just gathered up to form an exponential. Nevertheless 
the solution matrix for systems od linear differential equations has all the 
essential properties of an exponential, such as convergence, growth, and 
so on. 

Most of yesterday's postings cover this ground pretty well. One thing which 
I remember particularly well was "Feynman's Magnificent New Operator Calculus" 
wherein he had just doscovered time ordering. This was Cornell, 1949; one can 
look up the paper in Physical Review. The same ideas in field theory were 
called "Wick Ordering." 

I am kind of busy discussing all this right now in my complex variable class, 
under the rubrik "complex functions which satisfy differential equations." 
One of the nice techniques is the sum law for exponents, which changes form 
in the noncommutative case. Pursuing this in a certain direction leads to the 
Campbell-Hausdorff series. This is usually a bit complicated to derive, so 
I was interested in the remark that such a thing is guaranteed in matrix groups 
when the two matrices do not have a negative pair of eigenvalues. 

Can a reference to this be supplied? 

A good reference on the topic is an article by Wilhelm Magnus in Communications 
in Mathematical Physics, the Courant Institute house organ, late 50's. 

The formula for compounding rotations is a magnificent application of these 
ideas; what is not so well known is that if one makes pseudo-quaternions out 
of the real 2x2 matrices you get a similar formula with a lorentz metric. That 
is what you should use for the arc length of a rectangular hyperbola, and avoid 
elliptic integrals. But only if it is rectangular. 

- hvm 

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Re: [math-fun] Re: product integrals

MCKAY＠vax2.concordia.ca