<p> H. T. Davis Gamma(x) </p>

0f61b683-46a0-4cfd-a65a-67c8f89ce7b6 29 4 5 decimal

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Hermite polynomials => n x H 1 - (k^n! (* (k! (n 2 k * - (! (* / 2 x * (n 2 k * -^* (k 0 n 2 / Floor sum :

y'' 2 x * (y' * - λ 1 - (y * + 0 ≡

Normalised wave function n x Ψ 1 π / (14 /^12 n^n! (* sqrt / * n x H * x 2^- 2 / exp * :

01012 -^Ψ (2^0.56419

n 4 :

x V 12 / x 2^* : x Ψ.0 0 x Ψ (2^12 / + : x Ψ.1 1 x Ψ (2^32 / + : x Ψ.2 2 x Ψ (2^52 / + : x Ψ.3 3 x Ψ (2^72 / + : 51 line

n x Ψ 2 ^

plot x V x Ψ.0 x Ψ.1 x Ψ.2 x Ψ.3 51 sys :

plot

Wave function for the 
harmonic oscillator n x Ψ n x H x 2^- 2 / exp * :

Probanility densities
of the eigenstates n x PDψ n x Ψ (2^:

The expectation function n x Es n x Ψ (2^x * :

The expectation value n x Es x 05 int 648

0 n x PDψ 2 1 sys 0 n x Es 2 1 sys <p> H. T. Davis Gamma(x) </p>

Abramowitz & Stegun, p. 256 c 1 0.5772156649015329 0.6558780715202538 - 0.0420026350340952 - 0.1665386113822915 0.0421977345555443 - 0.0096219715278770 - 0.0072189432466630 0.0011651675918591 - 0.0002152416741149 - 0.0001280502823882 0.0000201348547807 - 0.0000012504934821 - 0.0000011330272320 0.0000002056338417 - 0.0000000061160950 0.0000000050020075 0.0000000011812746 - 0.0000000001043427 0.0000000000077823 0.0000000000036968 - 0.0000000000005100 0.0000000000000206 - 0.0000000000000054 - 0.0000000000000014 0.0000000000000001 261 mat :

1 z Γ / c k el z k^* k 1 ∞ sum ≡

z Γpoly c k el z k^* k 126 sum :

ε 1021 -^:

z Γa 1 z Γpoly / :

x c 10 x ≤ (x 1 ≤ (& cases :

x Γpoly x c *

Gamma(x)

x Γ x Γa 0 x ≤ (x 1 ≤ (& x 1 - (x 1 - Γa * 1 x ≤ (x 2 ≤ (& x 2 - (x 1 - (* x 2 - Γa * 2 x ≤ (x 3 ≤ (& x 3 - (x 2 - (* x 1 - (* x 3 - Γa * 3 x ≤ (x 4 ≤ (& x 4 - (x 3 - (* x 2 - (* x 1 - (* x 4 - Γa * 4 x ≤ (x 5 ≤ (& x 5 - (x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 5 - Γa * 5 x ≤ (x 6 ≤ (& x 6 - (x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 6 - Γa * 6 x ≤ (x 7 ≤ (& x 7 - (x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 7 - Γa * 7 x ≤ (x 8 ≤ (& x 8 - (x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 8 - Γa * 8 x ≤ (x 9 ≤ (& x 9 - (x 8 - (* x 7 - (* x 6 - (* x 5 - (* x 4 - (* x 3 - (* x 2 - (* x 1 - (* x 9 - Γa * 9 x ≤ (x 10 ≤ (& hello ! cases :

15 D Γ(x) 0.5 Γ 1.5 Γ 2.5 Γ 3.5 Γ 4.5 Γ 5.5 Γ 6.5 Γ 7.5 Γ 8.5 Γ 10.5 Γ 101 mat 1.77245385090552 0.886226925452759 1.32934038817914 3.32335097044785 11.6317283965675 52.3427777845536 287.885277815045 1871.25430579779 14034.4072934834 hello ! 101 mat

15 D H.T. Davis Γ(x) x Γ x Γ :

x c 10 x ≤ (x 5 ≤ (& cases :

min Γ(x 1.461632144968362 0.885603194410889 12 mat

Γ(x) 15 Decimals ... sanity checked vs Mathcad 11 Abramowitz & Stegun Table 6.1 is only 10 D

x Γ x c * 1 1 o 5 red 1 5 mat 2 1 o 5 red 1 5 mat 3 2 o 5 red 1 5 mat 4 6 o 5 red 1 5 mat 5 24 o 5 red 1 5 mat 1.4 0.885 . 14 black 1 5 mat 7 1 sys

π Γ 2.28803779534003

x Γ x Gamma :

π Γ 2.28804133935697