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Sep 29, 2025
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Remove trailing whitespace, and fix wrapping #24

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34 changes: 15 additions & 19 deletions README.md
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Original file line number Diff line number Diff line change
Expand Up @@ -6,18 +6,18 @@
[![Aqua QA](https://raw.githubusercontent.com/JuliaTesting/Aqua.jl/master/badge.svg)](https://github.com/JuliaTesting/Aqua.jl)
[![JET QA](https://img.shields.io/badge/JET.jl-%E2%9C%88%EF%B8%8F-%23aa4444)](https://github.com/aviatesk/JET.jl)

This package provides functions to compute the zeros of the J and Y functions,
and the zeros of their derivatives, where J and Y are Bessel functions of the first and second kind, respectively.

For all functions described below, the order `nu::Real` is a finite number and `n::Integer` is a positive integer.
When `nu isa AbstractFloat`, the returned value has the same type as `nu`. When `nu isa Integer`, the usual
promotion rules apply, so that for most builtin integer types the output type will be `Float64`. However,
when `nu isa BigInt` the output type will be `BigFloat`.

When the output type is `Float64`, the exported functions (`besselj_zero`,
`bessely_zero`, `besselj_deriv_zero`, and `bessely_deriv_zero`) will use lookup tables to rapidly
return function zeros if the order `nu` is one of the first few values of `0, 1, ...` and the enumerator
`n` is one of the first values of `1, 2, 3, ...`. See the individual function docstrings for the actual
This package provides functions to compute the zeros of the J and Y functions, and the zeros of
their derivatives, where J and Y are Bessel functions of the first and second kind, respectively.

For all functions described below, the order `nu::Real` is a finite number and `n::Integer` is a
positive integer. When `nu isa AbstractFloat`, the returned value has the same type as `nu`. When
`nu isa Integer`, the usual promotion rules apply, so that for most builtin integer types the output
type will be `Float64`. However, when `nu isa BigInt` the output type will be `BigFloat`.

When the output type is `Float64`, the exported functions (`besselj_zero`, `bessely_zero`,
`besselj_deriv_zero`, and `bessely_deriv_zero`) will use lookup tables to rapidly return function
zeros if the order `nu` is one of the first few values of `0, 1, ...` and the enumerator `n` is one
of the first values of `1, 2, 3, ...`. See the individual function docstrings for the actual
extents of the lookup tables.

### Exported Functions
Expand All @@ -28,7 +28,7 @@ extents of the lookup tables.
besselj_zero(nu, n)
```

Return the `n`th zero of the Bessel J function of order `nu`.
Return the `n`th zero of the Bessel J function of order `nu`.

#### bessely_zero(nu, n)

Expand All @@ -44,8 +44,7 @@ Return the `n`th zero of the Bessel Y function of order `nu`.
besselj_deriv_zero(nu, n)
```

Return the `n`th nonvanishing zero of the derivative of the Bessel J
function of order `nu`.
Return the `n`th nonvanishing zero of the derivative of the Bessel J function of order `nu`.

#### bessely_deriv_zero(nu, n)

Expand Down Expand Up @@ -81,8 +80,7 @@ Asymptotic formula for the `n`th zero for the Bessel Y function of order `nu`.
FunctionZeros.besselj_deriv_zero_asymptotic(nu, n)
```

Asymptotic formula for the `n`th nonvanishing zero of the derivative of the
Bessel J function of order `nu`.
Asymptotic formula for the `n`th nonvanishing zero of the derivative of the Bessel J function of order `nu`.


#### FunctionZeros.bessely_deriv_zero_asymptotic(nu, n)
Expand All @@ -92,5 +90,3 @@ FunctionZeros.bessely_deriv_zero_asymptotic(nu, n)
```

Asymptotic formula for the `n`th zero of the derivative of the Bessel Y function of order `nu`.


20 changes: 10 additions & 10 deletions src/FunctionZeros.jl
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Expand Up @@ -2,7 +2,7 @@
"""
FunctionZeros
This module provides functions to compute the zeros of the J and Y functions,
and the zeros of their derivatives, where J and Y are Bessel functions of the first and second kind, respectively.
and the zeros of their derivatives, where J and Y are Bessel functions of the first and second kind, respectively.
"""
module FunctionZeros
import SpecialFunctions
Expand All @@ -15,7 +15,7 @@ const nupre_max = 1
const npre_max = 500

# Strings used in multiple function docstrings:
const speeddocstr = """For greater speed, table lookup is used for `Float64` outputs when
const speeddocstr = """For greater speed, table lookup is used for `Float64` outputs when
`nu ∈ 0:$nupre_max` and `n ∈ 1:$(npre_max)`."""
const argstr = """## Arguments
- `nu::Real`: The order of the Bessel function.
Expand Down Expand Up @@ -84,7 +84,7 @@ end
_besselj_zero(nu, n)

`n`th zero of the Bessel J function of order `nu`,
for `n` = `1,2,...`.
for `n` = `1,2,...`.

$argstr
"""
Expand All @@ -101,7 +101,7 @@ const jzero_pre = [_besselj_zero(nu, n) for nu in 0:nupre_max, n in 1:npre_max]
besselj_zero(nu, n)

Return the `n`th zero of the Bessel J function of order `nu`,
for `n` = `1,2,...`.
for `n` = `1,2,...`.

$argstr
$speeddocstr
Expand Down Expand Up @@ -146,7 +146,7 @@ const yzero_pre = [_bessely_zero(nu, n) for nu in 0:nupre_max, n in 1:npre_max]
bessely_zero(nu, n)

Return the `n`th zero of the Bessel Y function of order `nu`,
for `n` = `1,2,...`.
for `n` = `1,2,...`.

$argstr
$speeddocstr
Expand Down Expand Up @@ -185,7 +185,7 @@ bessely_deriv_zero_asymptotic(nu, n) = bessel_deriv_zero_asymptotic(nu, n, 2)
"""
bessel_deriv_zero_asymptotic(nu, n, kind=1)

Asymptotic formula for the `n`th zero of the the derivative of Bessel J (Y) function
Asymptotic formula for the `n`th zero of the the derivative of Bessel J (Y) function
of order `nu`. `kind == 1 (2)` for Bessel function of the first (second) kind, J (Y).
"""
function bessel_deriv_zero_asymptotic(nu_in::Real, n::Integer, kind=1)
Expand Down Expand Up @@ -218,11 +218,11 @@ end
_besselj_deriv_zero(nu, n)

Return the `n`th nonvanishing zero of the derivative of Bessel J function of order `nu`,
for `n` = `1,2,...`.
for `n` = `1,2,...`.

$argstr
"""
function _besselj_deriv_zero(nu::Real, n::Integer)
function _besselj_deriv_zero(nu::Real, n::Integer)
# Ref: https://dlmf.nist.gov/10.6.E1
iszero(nu) && (n += 1) # Skip the zero occuring at zero
return Roots.find_zero(bessel_deriv_zero_asymptotic(nu, n, 1)) do x
Expand All @@ -237,7 +237,7 @@ const jderivzero_pre = [_besselj_deriv_zero(nu, n) for nu in 0:nupre_max, n in 1
besselj_deriv_zero(nu, n)

Return the `n`th nonvanishing zero of the derivative of the Bessel J function of order `nu`,
for `n` = `1,2,...`.
for `n` = `1,2,...`.

$argstr
$speeddocstr
Expand Down Expand Up @@ -276,7 +276,7 @@ const yderivzero_pre = [_bessely_deriv_zero(nu, n) for nu in 0:nupre_max, n in 1
bessely_deriv_zero(nu, n)

Return the `n`th zero of the derivative of the Bessel Y function of order `nu`,
for `n` = `1,2,...`.
for `n` = `1,2,...`.

$argstr
$speeddocstr
Expand Down
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