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

$$\mathrm{n}\mathrm{z}\mathrm{e}\mathrm{t}\mathrm{a}(z)=i\sqrt{\pi }{e}^{-z^2}(1-\mathrm{e}\mathrm{r}\mathrm{f}(-iz))$$

Categories: Special functions ·

Function: nzetar (z) ¶

Returns realpart(nzeta(z)).

Categories: Special functions ·

Function: nzetai (z) ¶

Returns imagpart(nzeta(z)).

Categories: Special functions ·