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### 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 ·

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