implement Lambert W function

This copies the implementation of the Lambert W function and
related functions and data from jlapeyre/LambertW.jl to
SpecialFunctions.jl

Author: alyst

README.md

@@ -1,7 +1,7 @@
 # SpecialFunctions.jl
 
 Special mathematical functions in Julia, including Bessel, Hankel, Airy, error, Dawson, exponential (or sine and cosine) integrals,
-eta, zeta, digamma, inverse digamma, trigamma, and polygamma functions.
+eta, zeta, Lambert's W, digamma, inverse digamma, trigamma, and polygamma functions.
 Most of these functions were formerly part of Base in early versions of Julia.
 
 CI (Linux, macOS, FreeBSD, Windows):

docs/src/functions_list.md

@@ -69,4 +69,7 @@ SpecialFunctions.beta
 SpecialFunctions.logbeta
 SpecialFunctions.logabsbeta
 SpecialFunctions.logabsbinomial
+SpecialFunctions.lambertw
+SpecialFunctions.lambertwbp
+SpecialFunctions.omega
 ```

docs/src/functions_overview.md

@@ -95,3 +95,9 @@ Here the *Special Functions* are listed according to the structure of [NIST Digi
 |:-------- |:----------- |
 | [`eta(x)`](@ref SpecialFunctions.eta)                         | [Dirichlet eta function](https://en.wikipedia.org/wiki/Dirichlet_eta_function) at `x`                                                                           |
 | [`zeta(x)`](@ref SpecialFunctions.zeta)                       | [Riemann zeta function](https://en.wikipedia.org/wiki/Riemann_zeta_function) at `x`                                                                             |
+
+## [Lambert's W Function](https://dlmf.nist.gov/4.13)
+| Function | Description |
+|:-------- |:----------- |
+| [`lambertw(x, [k=0])`](@ref SpecialFunctions.lambertw)        | [Lambert's W function](https://en.wikipedia.org/wiki/Lambert_W_function) at `x` for `k`-th branch   |
+| [`lambertwbp(x, [k=0])`](@ref SpecialFunctions.lambertwbp)    | Accurate value of ``1 + W_k(-\frac{1}{\mathrm{e}} + x)`` for small `x`                              |

src/SpecialFunctions.jl

@@ -78,7 +78,9 @@ export
     expintx,
     sinint,
     cosint,
-    lbinomial
+    lbinomial,
+    lambertw,
+    lambertwbp
 
 include("bessel.jl")
 include("erf.jl")
@@ -89,6 +91,7 @@ include("gamma.jl")
 include("gamma_inc.jl")
 include("betanc.jl")
 include("beta_inc.jl")
+include("lambertw.jl")
 include("chainrules.jl")
 include("deprecated.jl")