dan z email 2.2.11
PDF · 2 pages · 186.5 KB
Open PDF file
Phil writes to Dan about two errors in GR7 item 3.197.211 on page 317: the c parameter of F(a,b;c;z) should be λ-ν, and the first condition should be |arg(β/u)|<π. He gives his derivation from the basic hypergeometric integral representation plus a Kummer relation, and compares it with ET II 201(8). A postscript lists Mehler-transform integrals with Legendre functions P_{-1/2+iτ} that he does not find in GR7.
AI-written summary; may contain errors.
Extracted text (machine-read; may contain errors)
Hi Dan,
I would like to report two e rrata for GR7 (my list grows very slowly ).
integral in qu estion: GR7 page 3 17 item 3.19 7.211
Here is what GR7 shows:
There are two errors in this form ula.
• the c parameter of F(a,b;c;z) should b e λ-ν and not λ-µ
• the first condition sh ould be | arg(β/u) | < π and not | arg(u/β) | < π
The ET II 201 (8) integral quoted appears on page 146 of m y Russian edition as follows:
and one can make the substitutions α→β, y→u to make the com parison. Also needed is the first Ku mmer
relation say from HT I page 105, which d oes 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 t hing in ET II, I derived it from scratch using t he fundam ental integral representation
for F(a,b;c;z) and som e variable changes . Here is the sequence of events starting with HT I p 59 (10)
1
∫0 1 dt tb-1 (1-tz)-a (1-t)c-b-1 = B(b,c-b) F(a,b;c;z) // do x = 1/t :
∫1 ∞ dx 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:
∫u ∞ dy 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:
∫u ∞ dy 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 :
∫u ∞ dx 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
∫u ∞ dx 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 31 7
I see fro m the superscript 3.197.211 that this integral has had some "histor y" in GR.
Best regards,
Phil Lucht
Salt Lake City
31 Januar y 2011
P.S. I do n't see this in your errata PDF dated April 11, 2008 located at http://www.mathtable.co m/gr .
P.P.S. The ab ove integral arose when I was doing some Mehler transform integrals. It seems odd to m e
that these rel atively simple (to state, not alway s to do) integrals do not appear in GR7, though som e of
this form appear on page 788. For example, these do not appear in GR7 :
∫0 ∞ dτ P-1/2+iτ(y) cos(aτ)/cos h(πτ) = (1/ 2 ) 1/y + cha
∫0 ∞ dτ P-1/2+iτ(y) cos(aτ) /cosh2(πτ) = ( 2 /π ) (1/ y-cha ) tan-1[ y-cha /1+cha ]
∫0 ∞ dτ P-1/2+iτ(y) ch(bτ) /cos h2(πτ) = ( 2 /π ) (1/ y-cosb ) cot-1[1+cosb / y-cosb ] |b| < 2 π
These and many others a ppear in PB M 2003 Spe cial Functions vol 3 a nd/or in Boeing Report 246 of
Oberhettinger and Higgins. The P-1/2+iτ(y) appear in both sp herical and toroida l harm onics.
2