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Math Reading Group

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(no subject) [Sep. 1st, 2006|09:35 pm]
Math Reading Group

synchcola
Argh. I intend to put off reading chapter two of Special Functions until sometime later when I have lots of time. Check back in 1996.

Thanks for reading! :)
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Week 14 (August 13 to August 19) [Aug. 15th, 2006|11:15 am]
Math Reading Group

synchcola
Section 2.6 - Dilogarithms

- The dilogarithm, ∑n=1 xn / n2, and friends.
- Several identities for the dilogarithm.

A nice plot of the dilogarithm etc.Collapse )
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Lazy Week 2 (August 6 to August 12) [Aug. 6th, 2006|05:52 am]
Math Reading Group

synchcola
I've been busier than usual lately (which still isn't very busy). This is another extension. u_u;;

Have Non-Cooperative Games. ^_^;
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Week 12 (July 30 - August 5) [Jul. 30th, 2006|01:27 am]
Math Reading Group

synchcola
Section 2.5 - Contiguous Relations

- Relations between three contiguous 2F1-s.
- Jacobi polynomials!
- The Rodriguez formula for the Jacobi polynomials
- Hints at more stuff (contiguous relations for 3F2-s, the theory of moments)

Random interesting thing TBA. I didn't really get around to reading Section 2.4 yet >_>
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Week 12 (July 23 - July 29) [Jul. 23rd, 2006|12:47 am]
Math Reading Group

synchcola
Section 2.4 - The Barnes Integral for the Hypergeometric Function

- The Mellin transform (but why does it work?)
- Proof of Barnes's integral using residues and contour-integral estimates.
- Some sideways integrals of Γ(s)-products, and some hypergeometric identities.

I think these identities are proved in this way because nice proofs appear, not because contour integration is the natural way to deal with these particular identities.

Prof. Doron Zeilberger discovered a wonderful algorithm to produce proofs of most of these identities (it is in the book too) but the proofs are so unintuitive and boring to check that there is still a big market for "elegant" memorable proofs of things. Besides, I haven't even memorized his algorithm yet! not for lack of trying.

If this interests you you can read Petkovšek, Wilf, and Zeilberger, "A=B", which is on the Internet. You can also read Concrete Mathematics, which has a lucid explanation. Check out Zeilberger's Maple code, on his site.
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Lazy week (July 16 - July 22) [Jul. 19th, 2006|12:25 am]
Math Reading Group

synchcola
I've been distracted cause it's my birthday this week. Also, Section 2.3 turned out to be pretty hard. So, this week is still for reading Section 2.3.

I could have just posted the blurb for Section 2.4 and skipped the difficult part. Aren't you proud of me? :P

The analyticity of the Mellin transform.Collapse )
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Week 9 - July 9 to July 15 [Jul. 9th, 2006|06:33 am]
Math Reading Group

synchcola
Section 2.3 - The Hypergeometric Equation

- The P symbol, and the way the exponents in the P symbol change if the function be modified.
- The singular differential equation for the hypergeometric function, and its two solutions.
- 2F1 in terms of F(1-x).

The P symbol in AMSTeXCollapse )
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Week 8 (July 2 to July 8) [Jul. 2nd, 2006|03:01 am]
Math Reading Group

synchcola
Section 2.2 - Euler's Integral Representation

- Euler's integral for 2F1(a, b; c; x) and how to use it to increase the order of a pFq.
- The value of 2F1(a, b; c; 1) due to Gauss.
- Pfaff's transformation; the Pfaff-Saalchütz sum for a balanced 3F2.
- Dougall's identity (note typo: 1 + d + e should be 1 + a + e.)

This week's random interesting thing is going to be an elaboration of Remark 2.2.3, eventually. >_>
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Week 7 (June 25 to July 1) [Jun. 25th, 2006|01:02 am]
Math Reading Group

synchcola
Section 2.1 - The Hypergeometric Series

- Definition of the hypergeometric series.
- Its convergence away from |x| = 1. Its convergence for |x| = 1. Its behaviour near 1.

Theorem 2.1.2 makes a statement about conditional convergence, but the proof is not explicit.

Proof of part of Theorem 2.1.2Collapse )

Appendix F - Series Solutions of Differential Equations

- Differential equations near a regular singular point.
- The indicial equation.

The concept of a differential equation with only regular singular points seems to be complicated. (gah, the big R symbol...)
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Week 6 (June 18 to June 24) [Jun. 18th, 2006|02:37 am]
Math Reading Group

synchcola
Whoops! Section 1.10 is this week.

Section 1.10 - Gauss and Jacobi Sums

- The gamma and beta functions in a finite field.
- Characters on Z(p) and L-series. The exercises are still waiting. :P

Section 1.12 - The p-adic Gamma Function

- The gamma function on p-adic numbers (actually it's more like the factorial, since there is no integral involved.)
- A Mellin transform for the p-adic numbers?
- Next week is Chapter 2!
- Hooray.

Random interesting thing TBA. :)
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