[Maxima-bugs] [maxima:bugs] #5716 diff() of gamma_incomplete() and similar functions is wrong for z < 0 with domain : real

[David S. <tom...@us...>]

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**[bugs:#5716] diff\(\) of gamma\_incomplete\(\) and similar functions is wrong for z < 0 with domain : real**

**Status:** open
**Group:** None
**Labels:** diff gamma\_incomplete domain real 
**Created:** Sun Oct 25, 2026 03:05 PM UTC by David Scherfgen
**Last Updated:** Sun Oct 25, 2026 03:05 PM UTC
**Owner:** nobody


By Claude:

With `domain : real`, the default, `(-8)^(1/3)` is the real root `-2`. `gamma_incomplete(a,z)` is on the principal branch, but its derivative `-%e^-z*z^(a-1)` is simplified with the real root, so it is wrong for `z < 0`:

```
(%i1) d : diff(gamma_incomplete(1/3, z), z);
(%o1) -%e^-z/z^(2/3)
(%i2) float(rectform(subst(z = -8, d)));
(%o2) -745.2394967604321
```

The right value is `%e^8*(1+sqrt(3)*%i)/8`, about `372.62+645.40*%i`, which a numerical derivative of `gamma_incomplete(1/3, z)` at `z = -8` confirms. The same happens for an argument that is negative for `x > 0`:

```
(%i3) diff(gamma_incomplete(1/3, -x^3), x);
(%o3) 3*%e^x^3
```

For `x > 0` this should be `3*%e^x^3*(-1/2-sqrt(3)*%i/2)`. `gamma_incomplete_lower`, `gamma_incomplete_regularized`, `gamma_incomplete_generalized`, `beta_incomplete`, `beta_incomplete_regularized`, `beta_incomplete_generalized`, `struve_h` and `struve_l` have the same problem in their derivatives.


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