14.10 Functions and Variables for Special Functions

Function: lambert_w (z)

The principal branch of Lambert’s W function W(z) (DLMF 4.13), the solution of $$ z = W(z)e^{W(z)} $$

Function: generalized_lambert_w (k, z)

The k-th branch of Lambert’s W function W(z) (DLMF 4.13), the solution of \(z=W(z)e^{W(z)}\).

The principal branch, denoted \(W_p(z)\) in DLMF, is lambert_w(z) = generalized_lambert_w(0,z).

The other branch with real values, denoted \(W_m(z)\) in DLMF, is generalized_lambert_w(-1,z).

Function: kbateman [v] (x)

The Bateman k function

$$ k_v(x) = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \cos(x \tan\theta-v\theta)d\theta $$

It is a special case of the confluent hypergeometric function. Maxima can calculate the Laplace transform of kbateman using laplace or specint, but has no other knowledge of this function.

Function: nzeta (z)

The Plasma Dispersion Function $$ {\rm nzeta}(z) = i\sqrt{\pi}e^{-z^2}(1-{\rm erf}(-iz)) $$

Function: nzetar (z)

Returns realpart(nzeta(z)).

Function: nzetai (z)

Returns imagpart(nzeta(z)).