naginterfaces.library.roots.lambertw_​complex

naginterfaces.library.roots.lambertw_complex(branch, offset, z)

lambertw_complex computes the values of Lambert’s $W$ function $W\left(z\right)$.

For full information please refer to the NAG Library document for c05bb

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/c05/c05bbf.html

Parameters

branchint

The branch required.

offsetbool

Controls whether or not $z$ is being specified as an offset from $-exp(-1)$.

zcomplex

If $offset=True$, $z$ is the offset $\Delta z$ from $-exp(-1)$ of the intended argument to $W$; that is, $W\left(\beta \right)$ is computed, where $\beta =-exp(-1)+\Delta z$.

If $offset=False$, $z$ is the argument $z$ of the function; that is, $W\left(\beta \right)$ is computed, where $\beta =z$.

Returns

wcomplex

The value $W\left(\beta \right)$: see also the description of $z$.

residfloat

The residual $|W\left(\beta \right)exp\left(W\left(\beta \right)\right)-\beta |$: see also the description of $z$.

Warns

NagAlgorithmicWarning

(errno $1$)

For the given offset $z$, $W$ is negligibly different from $-1$: $Re\left(z\right)=⟨value⟩$ and $Im\left(z\right)=⟨value⟩$.

(errno $1$)

$z$ is close to $-exp(-1)$. Enter $z$ as an offset to $-exp(-1)$ for greater accuracy: $Re\left(z\right)=⟨value⟩$ and $Im\left(z\right)=⟨value⟩$.

(errno $2$)

The iterative procedure used internally did not converge in $⟨value⟩$ iterations. Check the value of $resid$ for the accuracy of $w$.

Notes

lambertw_complex calculates an approximate value for Lambert’s $W$ function (sometimes known as the ‘product log’ or ‘Omega’ function), which is the inverse function of

$$f\left(w\right)=w{e}^w\text{for}w\in C\text{.}$$

The function $f$ is many-to-one, and so, except at $0$, $W$ is multivalued. lambertw_complex allows you to specify the branch of $W$ on which you would like the results to lie by using the argument $branch$. Our choice of branch cuts is as in Corless et al. (1996), and the ranges of the branches of $W$ are summarised in Figure [label omitted].

[figure omitted]

For more information about the closure of each branch, which is not displayed in Figure [label omitted], see Corless et al. (1996). The dotted lines in the Figure denote the asymptotic boundaries of the branches, at multiples of $\pi$.

The precise method used to approximate $W$ is as described in Corless et al. (1996). For $z$ close to $-exp(-1)$ greater accuracy comes from evaluating $W(-exp(-1)+\Delta z)$ rather than $W\left(z\right)$: by setting $offset=True$ on entry you inform lambertw_complex that you are providing $\Delta z$, not $z$, in $z$.

References

Corless, R M, Gonnet, G H, Hare, D E G, Jeffrey, D J and Knuth, D E, 1996, On the Lambert $W$ function, Advances in Comp. Math. (3), 329–359