dan z email 2.2.11
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Email from Phil Lucht (Salt Lake City) to Dan, dated 31 January 2011, about integral 3.197.211 on page 317 of GR7. He says the c parameter of the hypergeometric function should be λ-ν rather than λ-μ, and that the first condition should be |arg(β/u)|<π. He checks the result against ET II and Kummer's relation and gives his own derivation from the basic integral representation. A postscript lists Mehler-transform integrals with P_{-1/2+iτ} missing from GR7.
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Hi Dan,
I would like to report two errata for GR7 (my list grows very slowly).
integral in question: GR7 page 317 item 3.197.211
Here is what GR7 shows:
There are two errors in this formula.
the c parameter of F(a,b;c;z) should be λ-ν and not λ-μ
the first condition should be | arg(β/u) | < π and not | arg(u/β) | < π
The ET II 201 (8) integral quoted appears on page 146 of my Russian edition as follows:
and one can make the substitutions α→β, y→u to make the comparison. Also needed is the first Kummer relation say from HT I page 105, which does not affect the c or z arguments of F(a,b;c;z),
=> F(-ν, λ-μ-ν; λ-ν; -α/y) = y-ν-μ (y+α)μ+ν F(λ,μ; λ-ν; -α/y)
Before I found this thing in ET II, I derived it from scratch using the fundamental integral representation for F(a,b;c;z) and some variable changes. Here is the sequence of events starting with HT I p 59 (10)
!Syntax Error, Idt tb-1 (1-tz)-a (1-t)c-b-1 = B(b,c-b) F(a,b;c;z) // do x = 1/t :
!Syntax Error, Idx xa-c (x-z)-a (x-1)c-b-1 = B(b,c-b) F(a,b;c; z) // agrees HT I p115 (6); do y = ux:
!Syntax Error, Idy ya-c (y-uz)-a (y-u)c-b-1 = u-b B(b,c-b) F(a,b;c;z) // let z = -β/u:
!Syntax Error, Idy ya-c (y+β)-a (y-u)c-b-1 = u-b B(b,c-b) F(a,b;c;-β/u) // a=-ν, b=λ-μ-ν, c=λ-ν :
!Syntax Error, Idx x-λ (x+β)ν (x-u)μ-1 = uλ+μ+ν B(λ-μ-ν,μ) F(-ν, λ-μ-ν; λ-ν; -β/u) // as in GR4 p 286
F(-ν, λ-μ-ν; λ-ν; -α/u) = u-ν-μ (y+α)μ+ν F(λ,μ; λ-ν; -α/u) // Kummer
!Syntax Error, Idx x-λ (x+β)ν (x-u)μ-1 = u-λ (β+u)μ+ν B(λ-μ-ν,μ) F(λ,μ; λ-ν; -β/u) // corrected GR7 p 317
I see from the superscript 3.197.211 that this integral has had some "history" in GR.
Best regards,
Phil Lucht
Salt Lake City
31 January 2011
P.S. I don't see this in your errata PDF dated April 11, 2008 located at http://www.mathtable.com/gr.
P.P.S. The above integral arose when I was doing some Mehler transform integrals. It seems odd to me that these relatively simple (to state, not always to do) integrals do not appear in GR7, though some of this form appear on page 788. For example, these do not appear in GR7 :
!Syntax Error, Idτ P-1/2+iτ(y) cos(aτ)/cosh(πτ) = (1/) 1/
!Syntax Error, Idτ P-1/2+iτ(y) cos(aτ) /cosh2(πτ) = (/π ) (1/) tan-1[ /]
!Syntax Error, Idτ P-1/2+iτ(y) ch(bτ) /cosh2(πτ) = (/π ) (1/) cot-1[/] |b| < 2π
These and many others appear in PBM 2003 Special Functions vol 3 and/or in Boeing Report 246 of Oberhettinger and Higgins. The P-1/2+iτ(y) appear in both spherical and toroidal harmonics.