(41) Γ(x+1) [HP-41C]
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09-10-2020, 10:56 PM
(This post was last modified: 09-11-2020 01:28 PM by Albert Chan.)
Post: #10
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RE: Γ(x+1) [HP-41C]
I tried turning Stirling's formula's correction to Gamma, in continued fraction form.
Amazingly, every divide gives me back 2 terms. XCas> c3(x) := 1 + 1/(12x-1/2+1/(720/293*x+1/(7211316/4406147*x))) \(x \rightarrow 1+\frac{1}{\Large 12x- \frac{1}{2}+\frac{1}{\frac{720}{293}x+\frac{1}{\frac{7211316}{4406147}x}}}\) XCas> series(c3(x), x=inf, polynom) \(1+\frac{1}{12}\left({1\over x}\right) +\frac{1}{288} \left({1\over x}\right)^2 -\frac{139}{51840} \left({1\over x}\right)^3 -\frac{571}{2488320} \left({1\over x}\right)^4 +\frac{163879}{209018880} \left({1\over x}\right)^5 +\frac{5246819}{75246796800} \left({1\over x}\right)^6 \) All terms matches correctly to Series of Gamma(x) / (sqrt(2*pi/x) * (x/e)^x), x=inf Below, we define 3 functions, FNS(x)=sinc(pi*x) , FNF(n)=n! , FNG(x)=Γ(x) Code: 10 DEF FNS(X)=SIN(ACOS(-1)*MOD(X,2))/(PI*X) ! = sinc(pi*x) >RUN >FNF(5), FNF(10), FNF(15) 120 3628800 1.307674368E12 >X=1.1 ! check reflection formula (FNG does not do reflection) >FNF(X)*FNF(-X)*FNS(X), FNG(1+X)*FNG(1-X)*FNS(X) 1 1 >FOR X=1 TO 2 STEP .1 @ G=FNG(X) @ X,G,GAMMA(X)-G @ NEXT X 1 1 0 1.1 .951350769865 .000000000002 1.2 .918168742399 .000000000001 1.3 .897470696308 -.000000000002 1.4 .887263817504 -.000000000001 1.5 .886226925454 -.000000000001 1.6 .893515349285 .000000000003 1.7 .908638732849 .000000000004 1.8 .931383770985 -.000000000005 1.9 .961765831906 .000000000001 2 1 0 |
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Messages In This Thread |
(41) Γ(x+1) [HP-41C] - Gerson W. Barbosa - 04-29-2020, 09:45 PM
RE: Γ(x+1) [HP-41C] - Gerson W. Barbosa - 04-30-2020, 08:35 PM
RE: Γ(x+1) [HP-41C] - Albert Chan - 05-01-2020, 11:59 PM
RE: Γ(x+1) [HP-41C] - Gerson W. Barbosa - 05-02-2020, 11:04 AM
RE: Γ(x+1) [HP-41C] - pinkman - 04-30-2020, 09:58 PM
RE: Γ(x+1) [HP-41C] - Gerson W. Barbosa - 05-01-2020, 08:46 PM
RE: Γ(x+1) [HP-41C] - Gerson W. Barbosa - 05-01-2020, 05:59 PM
RE: Γ(x+1) [HP-41C] - Gerson W. Barbosa - 05-03-2020, 05:29 PM
RE: Γ(x+1) [HP-41C] - Gerson W. Barbosa - 05-09-2020, 02:42 PM
RE: Γ(x+1) [HP-41C] - Albert Chan - 09-10-2020 10:56 PM
RE: Γ(x+1) [HP-41C] - Albert Chan - 09-13-2020, 12:49 PM
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