Bug when using 'Γ(x)' instead of 'Gamma(x)' during the search of a Root

[Wiljea]

CONTEXT: I am trying to calculate the Student factor corresponding to the confidence level ('Clvl') & the number of dof ('ν') by integrating the Student density function from the unknown value 'x' to infinity (represented here by 10, which provides enough coverage):

Clvl=0.99 @ for a confidence level of 99%
ν=19  @ for a sample of n=20 measurements (=> ν=n-1)
'Solve(∫(x;10;Γ((ν+1)/2)/(Γ(ν/2)*√(ν*Ⓒπ))*(1+t²/ν)↑(-(ν+1)/2);t)-(1-Clvl)/2=0;x)' Eval
@Expecting 2.861 => "∫ error: Invalid function"

where the Student density function has the correct form as it appears on the stack:

Therefore, I was wondering if DB48x (v0.9.18) is able to perform such inverse Student law calculation.

[Wiljea]

FYI, in the "Modern Physics" section of the Equation Library, the command "Solve" was able to compute with "∫" in many subsections. The only difference in the present case is that the variable 'x' to be solved for appears in the integration limit. I think that kind of function should also be treated by "Solve" whose acceptance criterion are now perhaps too restrictive.

[evgaster]

Clvl=0.99 @ for a confidence level of 99%
ν=19  @ for a sample of n=20 measurements (=> ν=n-1)
'Solve(∫(x;10;Γ((ν+1)/2)/(Γ(ν/2)*√(ν*Ⓒπ))*(1+t²/ν)↑(-(ν+1)/2);t)-(1-Clvl)/2=0;x)' Eval
@Expecting 2.861 => "∫ error: Invalid function"

Gives me (simulator 0.9.18):

It doesn't know "Solve".

Let's try:

'x' Purge
Clvl=0.99 @ for a confidence level of 99%
ν=19  @ for a sample of n=20 measurements (=> ν=n-1)
'∫(x;10;Gamma((ν+1)/2)/(Gamma(ν/2)*√(ν*Ⓒπ))*(1+t²/ν)↑(-(ν+1)/2);t)-(1-Clvl)/2=0' Eval
x
1
«Root» TimedEval

Gives me:

Let's spend some spaces (don't cost that much; ROI in readability is great) and move fixed parts before the integral and do the calculation as far as possible before root finding:

'x' Purge
Clvl=0.99 @ for a confidence level of 99%
ν=19  @ for a sample of n=20 measurements (=> ν=n-1)
'Gamma((ν + 1) / 2) / (Gamma(ν / 2) * √(ν * Ⓒπ)) * ∫(x; 10; (1 + t² / ν) ↑ (-(ν + 1) / 2); t) - (1 - Clvl) / 2 = 0'
ToDecimal
x
1
«Root» TimedEval

Gives me:

[Wiljea]

It works also in algebraic form (because that's the way it's done in the Equation Library) except that "Root" must be used instead of "Solve" and a trial value for 'x' must also be specified:

Clvl=0.99 @ for a confidence level of 99%
ν=19 @ for a sample of n=20 measurements (=> dof: ν=n-1)
'Root(Gamma((ν+1)/2)/(Gamma(ν/2)*√(ν*→Num(Ⓒπ)))*∫(x;10;(1+t²/ν)↑(-(ν+1)/2);t)=(1-Clvl)/2;x;2.0)' Eval
@Expecting x=2.86093 43661 6

which is the correct value.

However what is strange (and remains as a NEW BUG) is that if I use the symbol "Γ" for the Gamma function, only the first occurrence of Γ is correctly recognised:

Clvl=0.99 @ for a confidence level of 99%
ν=19 @ for a sample of n=20 measurements (=> dof: ν=n-1)
'Root(Γ((ν+1)/2)/(Γ(ν/2)*√(ν*→Num(Ⓒπ)))*∫(x;10;(1+t²/ν)↑(-(ν+1)/2);t)=(1-Clvl)/2;x;2.0)' Eval
@Failing x=2.86093 43661 6 => "Root error: Invalid function"

The latter point (2nd occurrence not computed) can be seen if you try:

ν=19
'Gamma((ν+1)/2)/(Gamma(ν/2)*√(ν*→Num(Ⓒπ)))' Eval @Expecting 0.39372 98072 93
'Γ((ν+1)/2)/(Γ(ν/2)*√(ν*→Num(Ⓒπ)))' Eval @Failing 0.39372 98072 93 => doesn't compute

showing:

To @c3d, I think it's a parsing error, if that rings a bell for you, it is analog to a similar error that we encountered for a quotient of factorial functions. If I remember well, it was in the Equation Library for the computation of the true period of a real pendulum (no small angle approximation) whose expression was given by an infinite series. Because of this new bug, I will maintain this ISSUE as open with a new title.

[c3d]

@Wiljea There is simply no Γ alias for Gamma. It's an oversight. It will be fixed in the next release.

I have a fix, but it may be a good idea for you to see if you can figure out the one-line change to fix it. Look into the file named ids.tbl.

[Wiljea]

@c3d I looked in the file ids.tbl and indeed, an alias is missing and would be necessary to enable the use of the 'Γ' character for the Gamma function, as well as for the LogGamma function. More precisely, the lines 390 & 391 shall be written as:
        NAMED(tgamma, "Gamma") ALIAS(tgamma, "Γ")
        NAMED(lgamma, "LogGamma") ALIAS(lgamma, "LogΓ")
But I don't know how to work within GitHub to advance or push these modifications.

I will close this ISSUE after the next release.

[c3d]

Don't worry, I'll deal with it. I'm glad you figured out the basis. There is a bit more to it to make it really good. It's pushed on dev now.