## `detools::arbFuns` -- number of arbitrary functions in the general solution of an involutive partial differential equation

#### Introduction

`detools::arbFuns``(q,alpha)` computes the number of arbitrary functions in the general solution of an involutive partial differential equation of order `q` and with Cartan characters `alpha`.

#### Call(s)

`detools::arbFuns(q, alpha)`

#### Parameters

`q` | - | the order of the equation: a positive integer.
`alpha` | - | the Cartan characters: a list of nonnegative integers.

#### Returns

a list of integers of the same length as `alpha`. The i-th entry gives the number of arbitrary functions with i arguments.

#### Related Functions

`detools::cartan`, `detools::hilbert`

#### Details

- `detools::arbFuns` performs a purely combinatorial calculation trying to express the Cartan characters of an involutive partial differential equations as the number of arbitrary functions in the general solution. For first order equations this make always sense; for higher order equations it is possible that negative values occur in the answer.

#### Example 1

How many arbitrary functions appear in the general solution of Maxwell's equations in electrodynamics? For three-dimensional space the four Cartan characters are 6, 6, 4 and 0. So we enter

    >> detools::arbFuns(1, [6, 6, 4, 0])

                                   [0, 2, 4, 0]

and obtain that (at least formally) the solution space can be parametrised by four functions of three variables and two functions of two variables.

#### Background

- This combinatorial counting of arbitrary functions is based on the so-called formal theory of differential equations and essentially analyses formal power series solutions. More details can be found in the article:
  - W.M. Seiler: On the arbitrariness of the general solution of an involutive partial differential equation, Journal of Mathematical Physics 35 (1994) 486-498

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