15.10 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

\[z = W(z)e^{W(z)} \]

Categories: Special functions ·

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).

Categories: Special functions ·

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.

Categories: Special functions ·

Function: nzeta (z) ¶

The Plasma Dispersion Function

\[{\rm nzeta}(z) = i\sqrt{\pi}e^{-z^2}(1-{\rm erf}(-iz)) \]

Categories: Special functions ·

Function: nzetar (z) ¶

Returns realpart(nzeta(z)).

Categories: Special functions ·

Function: nzetai (z) ¶

Returns imagpart(nzeta(z)).

Categories: Special functions ·