[Rafael-SOWNet]

Part of #2195, item 3: polygamma, digamma and harmonicnumber compile, in FastExpression and in the LINQ compiler.

Numerics/PolygammaFunction is a double-precision kernel that follows the interpreter's algorithm:

• The reflection (DLMF 5.15.6), left of Re z = 1/2 near the axis.
• The recurrence (DLMF 5.15.5), to Re w >= 16 + n.
• The asymptotic series (DLMF 5.11.2, 5.15.8), with the Bernoulli numbers taken exactly from the tangent numbers, as the interpreter does, then rounded.
• NaN at the poles and at an order that is not whole.

Two things the decimal algorithm can afford and double precision cannot:

• Off the axis there is no reflection. There cot(pi z) is near ±i, where 1 + cot^2 vanishes, and Q_n's terms cancelled to seven digits at polygamma(7, -2 - 3i). The recurrence alone passes no pole off the axis.
• cot is taken with its argument reduced by the nearest integer, and near a half as tan(pi (1/2 - r)). At a half-whole z, cos(pi z) is a rounding away from 0, and pi^(n + 1) and Q_n's coefficients magnify it: polygamma(10, -9/2) was off by 2e-3 relative. Complex.Tan is taken only of a small argument; near pi/2 it computes its imaginary part as 0/0.

The harmonic numbers are the sum for a whole n up to 10000. For a negative whole n they have no value, except for a whole order <= 0. Elsewhere, for a whole order from 1 to 64, they are the polygamma form, as #2200 has them. All three functions are new since 2.6.0, so their entries say they compile and there is no 2.6.0 row.

Checked against mpmath (dps 30), compiled and evaluated alike:

• 7,170 polygamma points: orders 0 to 10 on the real axis from -5 to 5 off the poles, and 6,000 random points, a third of them near the negative axis.
• 2,997 harmonic points of orders 1, 2 and 3, real and complex.

All agree to 1e-9, and so does evaluation.

Tests:

• PolygammaCompilationTest compares both compilers with evaluation on orders 0, 1, 2, 3, 7 and 1/2, at 15 arguments: poles, half-integers, left of 1/2, far out, and off the axis. The same grid is run for harmonicnumber, and digamma is checked on its own.
• UserInvalidExceptions pins gamma(2, x) in place of polygamma(1, x) as a node with no compiled form.
• Full suite 17943/0, interactive trio, AOT smoke 70/0, all frameworks build.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

[codecov]

Codecov Report

❌ Patch coverage is 88.54962% with 15 lines in your changes missing coverage. Please review.
✅ Project coverage is 86.62%. Comparing base (33b0bd8) to head (d1186d2).

Files with missing lines	Patch %	Lines
Sources/AngouriMath/Numerics/PolygammaFunction.cs	87.61%	6 Missing and 7 partials ⚠️
...gouriMath/Functions/Compilation/IntoFE/Compiler.cs	80.00%	2 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2215      +/-   ##
==========================================
+ Coverage   86.60%   86.62%   +0.01%     
==========================================
  Files         304      305       +1     
  Lines       67207    67338     +131     
  Branches    19509    19544      +35     
==========================================
+ Hits        58206    58330     +124     
- Misses       4894     4897       +3     
- Partials     4107     4111       +4     

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.17% <88.54%> (+0.01%)	⬆️
fsharp	32.97% <ø> (ø)
interactive	40.72% <ø> (ø)

Flags with carried forward coverage won't be shown. Click here to find out more.

Files with missing lines	Coverage Δ
...ath/Functions/Compilation/IntoFE/FastExpression.cs	96.31% <100.00%> (+0.09%)	⬆️
...ctions/Compilation/IntoLinq/CompilationProtocol.cs	90.88% <100.00%> (+0.10%)	⬆️
...h/Functions/Compilation/IntoLinq/MathAllMethods.cs	71.11% <100.00%> (+0.65%)	⬆️
...gouriMath/Functions/Compilation/IntoFE/Compiler.cs	86.54% <80.00%> (-0.21%)	⬇️
Sources/AngouriMath/Numerics/PolygammaFunction.cs	87.61% <87.61%> (ø)

... and 11 files with indirect coverage changes

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