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Math Reading Group

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

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

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

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

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

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

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

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

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

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

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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