Gamma function, SinhIntegral, CoshIntegral

[Albert Chan]

(11-06-2022 06:27 PM)robmio Wrote:  HP PRIME --> Gamma (4/5, -6) --> 294.845140024
MATHEMATICA --> Gamma [4/5, -6] --> 238.757-172.621*i.

Mathematica Gamma, converted to HP Prime Gamma
Note: it is not Abs[Gamma[4/5,-6]] ≈ 294.624

Gamma[4/5] + Abs[Gamma[4/5] - Gamma[4/5,-6]]
= Gamma[4/5] + (Gamma[4/5] - Gamma[4/5,-6]) / (-1)^(4/5)
= 294.845140024 ...

HP Prime Gamma, back to Mathematica Gamma:

CAS> gamma(a,x) := when(x<0, [Gamma(a),Gamma(a,x)] * [1+(-1)^a,-(-1)^a], Gamma(a,x))

CAS> gamma(4/5,-6.)      → 238.757077078-172.62130796*i

Quote:I hadn't really noticed: why does HP PRIME return the absolute value?

It is just a guess, but some integral result is more elegant.

CAS> int(e^x^3)      → 1/3*(Gamma(1/3,-x^3) - Gamma(1/3))
CAS> Ans(x=6.)        → 5.96393809188e91

Mathematica:

∫(e^x^3) = -(x Γ(1/3, -x^3))/(3 (-x^3)^(1/3))

∫(e^x^3, x=0..6) ≈ (5.964E91 + 0.7733*i) - (-0.4465 + 0.7733*i) ≈ 5.964E91