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Use DocumenterCitations.jl to manage dependencies (#486)
* Add DocumenterCitations as doc dependency * Add citations to docstrings * Minor docstring fixes * Fix typo citet -> cite
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‎docs/Project.toml

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[deps]
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Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
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DocumenterCitations = "daee34ce-89f3-4625-b898-19384cb65244"
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SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
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[sources]
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SpecialFunctions = {path = ".."}
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[compat]
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Documenter = "~0.27"
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Documenter = "1"

‎docs/make.jl

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using SpecialFunctions, Documenter
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using DocumenterCitations
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# `using SpecialFunctions` for all doctests
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DocMeta.setdocmeta!(SpecialFunctions, :DocTestSetup, :(using SpecialFunctions); recursive=true)
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bib = CitationBibliography(
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joinpath(@__DIR__, "src", "refs.bib");
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style = :authoryear,
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)
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makedocs(modules=[SpecialFunctions],
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sitename="SpecialFunctions.jl",
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authors="Jeff Bezanson, Stefan Karpinski, Viral B. Shah, et al.",
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format = Documenter.HTML(; assets = String[]),
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pages=["Home" => "index.md",
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"Overview" => "functions_overview.md",
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"Reference" => "functions_list.md"],
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#warnonly=[:missing_docs],
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plugins=[bib],
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checkdocs=:exports,
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)
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deploydocs(repo="github.com/JuliaMath/SpecialFunctions.jl.git")

‎docs/src/functions_list.md

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eta
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zeta
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```
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## Citations
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```@bibliography
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```

‎docs/src/functions_overview.md

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| [`digamma(x)`](@ref SpecialFunctions.digamma) | [digamma function](https://en.wikipedia.org/wiki/Digamma_function) (i.e. the derivative of `loggamma` at `x`) |
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| [`invdigamma(x)`](@ref SpecialFunctions.invdigamma) | [invdigamma function](http://bariskurt.com/calculating-the-inverse-of-digamma-function/) (i.e. inverse of `digamma` function at `x` using fixed-point iteration algorithm) |
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| [`trigamma(x)`](@ref SpecialFunctions.trigamma) | [trigamma function](https://en.wikipedia.org/wiki/Trigamma_function) (i.e the logarithmic second derivative of `gamma` at `x`) |
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| [`polygamma(m,x)`](@ref SpecialFunctions.polygamma) | [polygamma function](https://en.wikipedia.org/wiki/Polygamma_function) (i.e the (m+1)-th derivative of the `loggamma` function at `x`) |
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| [`polygamma(m,x)`](@ref SpecialFunctions.polygamma) | [polygamma function](https://en.wikipedia.org/wiki/Polygamma_function) (i.e the ``(m+1)``-th derivative of the `loggamma` function at `x`) |
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| [`gamma(a,z)`](@ref SpecialFunctions.gamma(::Number,::Number)) | [upper incomplete gamma function ``\Gamma(a,z)``](https://en.wikipedia.org/wiki/Incomplete_gamma_function) |
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| [`loggamma(a,z)`](@ref SpecialFunctions.loggamma(::Number,::Number)) | accurate `log(gamma(a,x))` for large arguments |
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| [`gamma_inc(a,x,IND)`](@ref SpecialFunctions.gamma_inc) | [incomplete gamma function ratio P(a,x) and Q(a,x)](https://en.wikipedia.org/wiki/Incomplete_gamma_function) (i.e evaluates P(a,x) and Q(a,x) for accuracy specified by IND and returns tuple (p,q)) |
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| [`gamma_inc_inv(a,p,q)`](@ref SpecialFunctions.gamma_inc_inv) | [inverse of incomplete gamma function ratio P(a,x) and Q(a,x)](https://en.wikipedia.org/wiki/Incomplete_gamma_function) (i.e evaluates x given P(a,x)=p and Q(a,x)=q) |
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| [`gamma_inc(a,x,IND)`](@ref SpecialFunctions.gamma_inc) | [incomplete gamma function ratio ``P(a,x)`` and ``Q(a,x)``](https://en.wikipedia.org/wiki/Incomplete_gamma_function) (i.e evaluates ``P(a,x)`` and ``Q(a,x)`` for accuracy specified by IND and returns tuple (p,q)) |
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| [`gamma_inc_inv(a,p,q)`](@ref SpecialFunctions.gamma_inc_inv) | [inverse of incomplete gamma function ratio ``P(a,x)`` and ``Q(a,x)``](https://en.wikipedia.org/wiki/Incomplete_gamma_function) (i.e evaluates x given ``P(a,x)=p`` and ``Q(a,x)=q``) |
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| [`beta(x,y)`](@ref SpecialFunctions.beta) | [beta function](https://en.wikipedia.org/wiki/Beta_function) at `x,y` |
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| [`logbeta(x,y)`](@ref SpecialFunctions.logbeta) | accurate `log(beta(x,y))` for large `x` or `y` |
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| [`logabsbeta(x,y)`](@ref SpecialFunctions.logabsbeta) | accurate `log(abs(beta(x,y)))` for large `x` or `y` |
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| [`logabsbinomial(x,y)`](@ref SpecialFunctions.logabsbinomial) | accurate `log(abs(binomial(n,k)))` for large `n` and `k` near `n/2` |
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| [`beta_inc(a,b,x,y)`](@ref SpecialFunctions.beta_inc) | [incomplete beta function ratio Ix(a,b) and Iy(a,b)](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function) (i.e evaluates Ix(a,b) and Iy(a,b) and returns tuple (p,q)) |
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| [`beta_inc_inv(a,b,p,q)`](@ref SpecialFunctions.beta_inc_inv) | Inverse of the incomplete beta function (i.e evaluates x given ``I_{x}(a, b) = p``) |
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| [`beta_inc(a,b,x,y)`](@ref SpecialFunctions.beta_inc) | [incomplete beta function ratio ``I_x(a,b)`` and ``I_y(a,b)``](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function) (i.e evaluates ``I_x(a,b)`` and ``I_y(a,b)`` and returns tuple (p,q)) |
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| [`beta_inc_inv(a,b,p,q)`](@ref SpecialFunctions.beta_inc_inv) | Inverse of the incomplete beta function (i.e evaluates ``x`` given ``I_{x}(a, b) = p``) |
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## Exponential and Trigonometric Integrals

‎docs/src/refs.bib

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@article{berry_1990,
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title = {Algorithm AS R83: A Remark on Algorithm AS 109: Inverse of the Incomplete Beta Function Ratio},
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ISSN = {00359254, 14679876},
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URL = {http://www.jstor.org/stable/2347779},
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author = {K. J. Berry and P. W. Mielke and G. W. Cran},
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journal = {Journal of the Royal Statistical Society. Series C (Applied Statistics)},
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number = {2},
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pages = {309--310},
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publisher = {Royal Statistical Society, Oxford University Press},
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volume = {39},
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year = {1990}
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}
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@article{blair_1976,
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title = {Rational Chebyshev Approximations for the Inverse of the Error Function},
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author = {Blair, J. M. and Edwards, C. A. and Johnson, J. H.},
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date = {1976},
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year = {1976},
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journal = {Mathematics of Computation},
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shortjournal = {Math. Comp.},
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volume = {30},
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number = {136},
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pages = {827--830},
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issn = {0025-5718, 1088-6842},
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doi = {10.1090/S0025-5718-1976-0421040-7},
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url = {https://www.ams.org/mcom/1976-30-136/S0025-5718-1976-0421040-7/},
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langid = {english}
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}
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@article{brent_1978,
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title = {A Fortran Multiple-Precision Arithmetic Package},
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author = {Brent, Richard P.},
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date = {1978-03},
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year = {1978},
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journal = {ACM Transactions on Mathematical Software},
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shortjournal = {ACM Trans. Math. Softw.},
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volume = {4},
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number = {1},
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pages = {57--70},
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issn = {0098-3500, 1557-7295},
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doi = {10.1145/355769.355775},
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url = {https://dl.acm.org/doi/10.1145/355769.355775},
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langid = {english}
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}
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@article{cran_1977,
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title = {Remark AS R19 and Algorithm AS 109: A Remark on Algorithms: AS 63: The Incomplete Beta Integral AS 64: Inverse of the Incomplete Beta Function Ratio},
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author = {Cran, G. W. and Martin, K. J. and Thomas, G. E.},
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journal = {Journal of the Royal Statistical Society. Series C (Applied Statistics)},
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number = {1},
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issn = {00359254, 14679876},
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url = {http://www.jstor.org/stable/2346887},
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pages = {111--114},
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publisher = {Royal Statistical Society, Oxford University Press},
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volume = {26},
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year = {1977}
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}
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@article{chattamvelli_1997,
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title = {Computing the Non-Central Beta Distribution Function},
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author = {Chattamvelli, R. and Shanmugam, F.},
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date = {1997-03-01},
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year = {1997},
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journal = {Journal of the Royal Statistical Society Series C: Applied Statistics},
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volume = {46},
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number = {1},
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pages = {146--156},
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issn = {0035-9254, 1467-9876},
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doi = {10.1111/1467-9876.00055},
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url = {https://academic.oup.com/jrsssc/article/46/1/146/6990591},
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langid = {english}
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}
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@article{didonato_1986,
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title = {Computation of the incomplete gamma function ratios and their inverse},
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author = {DiDonato, Armido R and Morris, Alfred H},
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volume = {12},
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issn = {0098-3500, 1557-7295},
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url = {https://dl.acm.org/doi/10.1145/22721.23109},
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doi = {10.1145/22721.23109},
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abstract = {An algorithm is given for computing the incomplete gamma function ratios $P(a, x)$ and $Q(a, x)$ for $a \geq 0$, $x \geq 0$, $a + x \neq 0$. Temme's uniform asymptotic expansions are used. The algorithm is robust; results accurate to 14 significant digits can be obtained. An' extensive set of coefficients for the Temme expansions is included. An algorithm, employing third-order Schröder iteration supported by Newton-Raphson iteration, is provided for computing $x$ when $a$, $P(a, x)$, and $Q(a, x)$ are given. Three iterations at most are required to obtain 10 significant digit accuracy for $x$.},
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language = {en},
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number = {4},
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urldate = {2025-03-08},
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journal = {ACM Transactions on Mathematical Software},
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month = dec,
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year = {1986},
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pages = {377--393},
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}
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@article{didonato_1992,
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title = {Algorithm 708: Significant Digit Computation of the Incomplete Beta Function Ratios},
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shorttitle = {Algorithm 708},
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author = {Didonato, Armido R. and Morris, Alfred H.},
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date = {1992-09},
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year = {1992},
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journal = {ACM Transactions on Mathematical Software},
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shortjournal = {ACM Trans. Math. Softw.},
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volume = {18},
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number = {3},
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pages = {360--373},
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issn = {0098-3500, 1557-7295},
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doi = {10.1145/131766.131776},
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url = {https://dl.acm.org/doi/10.1145/131766.131776},
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langid = {english}
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}
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@article{fukushima_2009,
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title = {Fast Computation of Complete Elliptic Integrals and Jacobian Elliptic Functions},
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author = {Fukushima, Toshio},
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date = {2009-12},
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year = {2009},
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journal = {Celestial Mechanics and Dynamical Astronomy},
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shortjournal = {Celest. Mech. Dyn. Astr.},
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volume = {105},
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number = {4},
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pages = {305--328},
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issn = {0923-2958, 1572-9478},
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doi = {10.1007/s10569-009-9228-z},
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url = {http://link.springer.com/10.1007/s10569-009-9228-z},
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langid = {english}
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}
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@article{fukushima_2015,
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title = {Precise and Fast Computation of Complete Elliptic Integrals by Piecewise Minimax Rational Function Approximation},
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author = {Fukushima, Toshio},
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date = {2015-07},
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year = {2015},
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journal = {Journal of Computational and Applied Mathematics},
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volume = {282},
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pages = {71--76},
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issn = {03770427},
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doi = {10.1016/j.cam.2014.12.038},
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url = {https://www.sciencedirect.com/science/article/pii/S0377042715000023},
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langid = {english}
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}
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@article{lenth_1987,
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title = {Algorithm AS 226: Computing Noncentral Beta Probabilities},
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shorttitle = {Algorithm AS 226},
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author = {Lenth, Russell V.},
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date = {1987},
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year = {1987},
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journal = {Applied Statistics},
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volume = {36},
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number = {2},
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eprint = {10.2307/2347558},
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eprinttype = {jstor},
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pages = {241},
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issn = {00359254},
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doi = {10.2307/2347558},
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url = {https://www.jstor.org/stable/10.2307/2347558?origin=crossref},
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urldate = {2025-02-19}
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}
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@article{macleod_1996,
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title = {Rational Approximations, Software and Test Methods for Sine and Cosine Integrals},
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author = {MacLeod, Allan J.},
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date = {1996-09},
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year = {1996},
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journal = {Numerical Algorithms},
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shortjournal = {Numer. Algor.},
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volume = {12},
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number = {2},
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pages = {259--272},
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issn = {1017-1398, 1572-9265},
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doi = {10.1007/BF02142806},
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url = {http://link.springer.com/10.1007/BF02142806},
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langid = {english}
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}

‎src/bessel.jl

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"""
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sphericalbesselj(nu, x)
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Spherical bessel function of the first kind at order `nu`, ``j_ν(x)``. This is the non-singular
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Spherical Bessel function of the first kind at order `nu`, ``j_ν(x)``. This is the non-singular
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solution to the radial part of the Helmholz equation in spherical coordinates.
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"""
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sphericalbessely(nu, x)
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Spherical bessel function of the second kind at order `nu`, ``y_ν(x)``. This is
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Spherical Bessel function of the second kind at order `nu`, ``y_ν(x)``. This is
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the singular solution to the radial part of the Helmholz equation in spherical
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coordinates. Sometimes known as a spherical Neumann function.
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‎src/beta_inc.jl

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See also: [`beta_inc`](@ref)
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# Implementation
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`BFRAC(A,B,X,Y,LAMBDA,EPS)` from Didonato and Morris (1982)
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`BFRAC(A,B,X,Y,LAMBDA,EPS)` from [Didonato and Morris (1992)](@cite didonato_1992)
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"""
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function beta_inc_cont_fraction(a::Float64, b::Float64, x::Float64, y::Float64, lambda::Float64, epps::Float64)
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@assert a > 1.0
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See also: [`beta_inc`](@ref)
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# Implementation
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`BASYM(A,B,LAMBDA,EPS)` from Didonato and Morris (1982)
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`BASYM(A,B,LAMBDA,EPS)` from [Didonato and Morris (1992)](@cite didonato_1992)
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"""
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function beta_inc_asymptotic_symmetric(a::Float64, b::Float64, lambda::Float64, epps::Float64)
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@assert a >= 15.0
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See also: [`beta_inc`](@ref)
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# Implementation
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`FPSER(A,B,X,EPS)` from Didonato and Morris (1982)
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`FPSER(A,B,X,EPS)` from [Didonato and Morris (1992)](@cite didonato_1992)
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"""
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# Implementation
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`APSER(A,B,X,EPS)` from [Didonato and Morris (1992)](@cite didonato_1992)
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function beta_inc_power_series1(a::Float64, b::Float64, x::Float64, epps::Float64)
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# Implementation
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`BPSER(A,B,X,EPS)` from Didonato and Morris (1982)
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`BPSER(A,B,X,EPS)` from [Didonato and Morris (1992)](@cite didonato_1992)
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‎src/betanc.jl

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```math
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I_{x}(a,b; \lambda) = \sum_{j=0}^{\infty} q(\lambda/2,j) I_{x}(a+j,b;0)
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```
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For ``\lambda < 54`` : algorithm suggested by Lenth(1987) in `ncbeta_tail(a,b,lambda,x)`.
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Else for ``\lambda \geq 54``: modification in Chattamvelli(1997) in
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For ``\lambda < 54``: algorithm suggested by [Lenth (1987)](@cite lenth_1987) in `ncbeta_tail(a,b,lambda,x)`.
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Else for ``\lambda \geq 54``: modification in [Chattamvelli (1997)](@cite chattamvelli_1997) in
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`ncbeta_poisson(a,b,lambda,x)` by using both forward and backward recurrences.
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"""
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‎src/ellip.jl

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``\alpha`` by ``k = \sin \alpha``.
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# Implementation
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Using piecewise approximation polynomial as given in
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> 'Fast Computation of Complete Elliptic Integrals and Jacobian Elliptic Functions',
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> Fukushima, Toshio. (2014). F09-FastEI. Celest Mech Dyn Astr,
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> DOI 10.1007/s10569-009-9228-z,
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> <https://pdfs.semanticscholar.org/8112/c1f56e833476b61fc54d41e194c962fbe647.pdf>
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For ``m<0``, followed by
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> Fukushima, Toshio. (2014).
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> 'Precise, compact, and fast computation of complete elliptic integrals by piecewise
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> minimax rational function approximation'.
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> Journal of Computational and Applied Mathematics. 282.
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> DOI 10.13140/2.1.1946.6245.,
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> <https://www.researchgate.net/publication/267330394>
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Using piecewise approximation polynomial as given in [fukushima_2009](@citet).
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For ``m < 0``, followed by [fukushima_2015](@citet).
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``\alpha`` by ``k=\sin \alpha``.
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# Implementation
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Using piecewise approximation polynomial as given in
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> 'Fast Computation of Complete Elliptic Integrals and Jacobian Elliptic Functions',
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> Fukushima, Toshio. (2014). F09-FastEI. Celest Mech Dyn Astr,
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> DOI 10.1007/s10569-009-9228-z,
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> <https://pdfs.semanticscholar.org/8112/c1f56e833476b61fc54d41e194c962fbe647.pdf>
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For ``m<0``, followed by
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> Fukushima, Toshio. (2014).
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> 'Precise, compact, and fast computation of complete elliptic integrals by piecewise
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> minimax rational function approximation'.
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> Journal of Computational and Applied Mathematics. 282.
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> DOI 10.13140/2.1.1946.6245.,
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> <https://www.researchgate.net/publication/267330394>
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Using piecewise approximation polynomial as given in [fukushima_2015](@citet)
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For ``m<0``, followed by [fukushima_2015](@citet).
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As suggested in this paper, the domain is restricted to ``(-\infty,1]``.
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"""
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ellipe(m::Real) = _ellipe(float(m))

‎src/erf.jl

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See also: [`erf(x)`](@ref erf).
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# Implementation
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Using the rational approximants tabulated in:
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> J. M. Blair, C. A. Edwards, and J. H. Johnson,
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> "Rational Chebyshev approximations for the inverse of the error function",
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> Math. Comp. 30, pp. 827--830 (1976).
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> <https://doi.org/10.1090/S0025-5718-1976-0421040-7>,
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> <http://www.jstor.org/stable/2005402>
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combined with Newton iterations for `BigFloat`.
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Using the rational approximants tabulated in [Blair (1976)](@cite blair_1976) combined with
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Newton iterations for `BigFloat`.
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"""
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erfinv(x::Real) = _erfinv(float(x))
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See also: [`erfc(x)`](@ref erfc).
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# Implementation
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Using the rational approximants tabulated in:
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> J. M. Blair, C. A. Edwards, and J. H. Johnson,
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> "Rational Chebyshev approximations for the inverse of the error function",
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> Math. Comp. 30, pp. 827--830 (1976).
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> <https://doi.org/10.1090/S0025-5718-1976-0421040-7>,
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> <http://www.jstor.org/stable/2005402>
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Using the rational approximants tabulated in [Blair (1976)](@cite blair_1976)
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combined with Newton iterations for `BigFloat`.
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"""
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erfcinv(x::Real) = _erfcinv(float(x))

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