## Maxima – 104 – Funzioni speciali – 10 Continuo da qui, copio dal Reference Manual, PDF scaricabile da qui, sono a p.311.

Funzioni di Struve

The Struve functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 12, o la Wiki.

`struve_h (v, z)`
The Struve Function `H` of order `v` and argument `z`.

`struve_l (v, z)`
The Modified Struve Function `L` of order `v` and argument `z`.

Funzioni ipergeometriche

The Hypergeometric Functions are defined in Abramowitz and Stegun, Handbook of Matematical Functions, Chapters 13 and 15, o la Wiki.

Maxima has very limited knowledge of these functions. They can be returned from function `hgfred`.

`%m [k,u] (z)`
Whittaker `M` function `M[k,u](z) = exp(-z/2)*z^(1/2+u)*M(1/2+u-k,1+2*u,z)`.

`%w [k,u] (z)`
Whittaker `W` function.

`%f [p,q] ([a],[b],z)`
The `pFq(a1,a2,..ap;b1,b2,..bq;z)` `hypergeometric` function, where `a` a list of length `p` and `b` a list of length `q`.

`hypergeometric ([a1, ..., ap],[b1, ... ,bq], x)`
The hypergeometric function. Unlike Maxima’s `%f` hypergeometric function, the function `hypergeometric` is a simplifying function; also, `hypergeometric` supports complex double and big floating point evaluation. For the Gauss hypergeometric function, that is `p = 2` and `q = 1`, floating point evaluation outside the unit circle is supported, but in general, it is not supported.

When the option variable `expand_hypergeometric` is `true` (default is `false`) and one of the arguments `a1` through `ap` is a negative integer (a polynomial case), `hypergeometric` returns an expanded polynomial.

``````(%i1) hypergeometric([],[],x);
x
(%o1)                                 %e``````

Polynomial cases automatically expand when `expand_hypergeometric` is `true`:

``````(%i2) hypergeometric([-3],,x);
(%o2)                    hypergeometric([- 3], , x)
(%i3) hypergeometric([-3],,x), expand_hypergeometric : true;
3        2
x      3 x    3 x
(%o3)                      (- ---) + ---- - --- + 1
504     56     7``````

Both double float and big float evaluation is supported:

``````(%i4) hypergeometric([5.1],[7.1 + %i],0.42);
(%o4)              1.346250786375333 - 0.0559061414208204 %i
(%i5) hypergeometric([5,6],, 5.7 - %i);
(%o5)           0.007375824009774945 - 0.001049813688578673 %i
(%i6) hypergeometric([5,6],, 5.7b0 - %i), fpprec : 30;
(%o6) 7.37582400977494674506442010824b-3
- 1.04981368857867315858055393376b-3 %i``````

Funzioni paraboliche del cilindro

The Parabolic Cylinder Functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 19, o la Wiki.

Maxima has very limited knowledge of these functions. They can be returned from function `hgfred`.

`parabolic_cylinder_d (v, z)`
The parabolic cylinder function.

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