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### D.1.1 Bessel function calling conventions

The virtual code interface simplifies the gsl C language API by excluding the facilities for error estimates, omitting certain array valued functions, and subsuming sets of related functions within common ones where possible.

The functions with names in the following group take an argument of the form `(n,x)`, where `n` identifies the member of the function family, and `x` is the argument to the function.

- `J` regular cylindrical Bessel functions
- `Y` irregular cylindrical Bessel functions
- `I` regular modified cylindrical Bessel functions
- `K` irregular modified cylindrical Bessel functions

For these functions, `n` can be either a natural number encoded as in Representation of Numeric and Textual Data, or a floating point number encoded as in `math`. The latter case specifies functions of a fractional order. The relevant gsl function is called based on the value and type of the parameter.

Two further related families of functions follow the same calling convention.

- `Isc` scaled regular modified cylindrical Bessel functions
- `Ksc` scaled irregular modified cylindrical Bessel functions

The foregoing functions are related to those above by an exponential scale factor as documented in the gsl reference manual.

Functions with names in the following group also take an argument of the form `(n,x)`, but are not defined for fractional orders and so require a natural number for `n`.

- `j` regular spherical Bessel functions
- `y` irregular spherical Bessel functions
- `isc` regular modified spherical Bessel functions
- `ksc` irregular modified spherical Bessel functions

The functions in the remaining group follow idiosyncratic calling conventions.

- `zJ0`, `zJ1` These take a natural number `n` and return the `n`th root of the regular cylindrical Bessel functions of order 0 or 1, respectively.
- `zJnu` This takes a pair `(nu,n)` where `nu` is the (fractional) order of a regular cylindrical Bessel function, `n` is a natural number. It returns the `n`th zero of the function.
- `lnKnu` This takes a pair of floating point numbers `(nu,x)` where `nu` is the (fractional) order of an irregular modified cylindrical Bessel and `x` is the argument to the function, and it returns the natural log of the function.

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