Incomplete Faddeeva Function

• The incomplete Faddeeva function is a generalized plasma dispersion function defined by restricting its integration domain, capturing the effects of depleted Maxwellian distributions near boundaries.
• It employs analytic continuation, asymptotic expansions, and rational approximations to manage singularities and branch cuts in plasma kinetic calculations.
• Its application in modeling linear dielectric response and wave dispersion in bounded plasmas provides a robust tool for researchers addressing non-Maxwellian phenomena.

The incomplete Faddeeva function, also known as the incomplete plasma dispersion function (IPDF), is a generalization of the standard plasma dispersion function in which the defining integral is restricted to a semi-infinite domain. This function arises naturally in the context of kinetic plasma theory whenever particle velocity distributions are Maxwellian only on finite or semi-infinite intervals, such as in the presence of boundary sheaths or double layers where the electron distribution is depleted below a cutoff velocity. The IPDF encapsulates essential analytic, asymptotic, and numerical properties distinct from its complete counterpart and is instrumental in modeling linear dielectric response and wave dispersion in bounded or non-Maxwellian plasmas (Baalrud, 2013).

1. Definition and Mathematical Formulation

For $\Im\{w\} > 0$, the incomplete plasma dispersion function $Z(\nu,w)$ is defined as

$$Z(\nu,w) = \frac{1}{\sqrt{\pi}} \int_{\nu}^{\infty} \frac{e^{-t^2}}{t-w}\,dt,$$

where $\nu \in \mathbb{R}$ is the cut-off parameter. Analytic continuation to $\Im\{w\} \le 0$ follows the Landau prescription. The standard plasma dispersion function is recovered in the limit $\nu\to-\infty$: $$Z(w) = Z(-\infty, w) = \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} \frac{e^{-t^2}}{t-w}\,dt.$$ The function $Z(w)$ is related to the Faddeeva function $w(z)$ by

$$w(z) = e^{-z^2}\,\operatorname{erfc}(-i z), \qquad Z(w) = i\sqrt{\pi}\, w(w),$$

valid for $Z(\nu,w)$0. Thus, the IPDF generalizes both $Z(\nu,w)$1 and $Z(\nu,w)$2 to semi-infinite integration domains (Baalrud, 2013, Zaghloul, 2017).

2. Analytic Properties and Relation to Complete Functions

The IPDF can be expressed as the difference between the complete plasma dispersion function and a one-sided correction integral: $Z(\nu,w)$3 For real $Z(\nu,w)$4 with $Z(\nu,w)$5, the singularity at $Z(\nu,w)$6 is located within the semi-infinite integration domain, necessitating the Plemelj formula to extract the discontinuity: $Z(\nu,w)$7 where $Z(\nu,w)$8 is the Heaviside function. There is a branch cut on the real axis for $Z(\nu,w)$9 with a logarithmic singularity at $$Z(\nu,w) = \frac{1}{\sqrt{\pi}} \int_{\nu}^{\infty} \frac{e^{-t^2}}{t-w}\,dt,$$0 in the real part. For arbitrary cutoffs, the relation

$$Z(\nu,w) = \frac{1}{\sqrt{\pi}} \int_{\nu}^{\infty} \frac{e^{-t^2}}{t-w}\,dt,$$1

enables construction of models for piecewise Maxwellian distributions (Baalrud, 2013).

3. Asymptotic Expansions and Series

The asymptotic behavior of $$Z(\nu,w) = \frac{1}{\sqrt{\pi}} \int_{\nu}^{\infty} \frac{e^{-t^2}}{t-w}\,dt,$$2 splits into two principal regimes:

• Small $$Z(\nu,w) = \frac{1}{\sqrt{\pi}} \int_{\nu}^{\infty} \frac{e^{-t^2}}{t-w}\,dt,$$3:

$$Z(\nu,w) = \frac{1}{\sqrt{\pi}} \int_{\nu}^{\infty} \frac{e^{-t^2}}{t-w}\,dt,$$4

with

$$Z(\nu,w) = \frac{1}{\sqrt{\pi}} \int_{\nu}^{\infty} \frac{e^{-t^2}}{t-w}\,dt,$$5

and $$Z(\nu,w) = \frac{1}{\sqrt{\pi}} \int_{\nu}^{\infty} \frac{e^{-t^2}}{t-w}\,dt,$$6, $$Z(\nu,w) = \frac{1}{\sqrt{\pi}} \int_{\nu}^{\infty} \frac{e^{-t^2}}{t-w}\,dt,$$7 denoting incomplete gamma functions; $$Z(\nu,w) = \frac{1}{\sqrt{\pi}} \int_{\nu}^{\infty} \frac{e^{-t^2}}{t-w}\,dt,$$8 is as defined in (Baalrud, 2013).

• Large $$Z(\nu,w) = \frac{1}{\sqrt{\pi}} \int_{\nu}^{\infty} \frac{e^{-t^2}}{t-w}\,dt,$$9:

$\nu \in \mathbb{R}$0

with

$\nu \in \mathbb{R}$1

These coefficients enable systematic construction of Padé approximants matching the function’s asymptotic behavior (Baalrud, 2013).

4. Differential, Recurrence, and Moment Relations

The IPDF satisfies an inhomogeneous first-order ODE: $\nu \in \mathbb{R}$2 Higher-order derivatives obey the recurrence

$\nu \in \mathbb{R}$3

Generalized moments,

$\nu \in \mathbb{R}$4

can be expressed via generating functions involving Hermite polynomials and the derivatives of $\nu \in \mathbb{R}$5 (Baalrud, 2013).

5. Numerical Evaluation Strategies

Evaluation approaches depend on the position of $\nu \in \mathbb{R}$6 relative to the integration domain:

• Direct quadrature is effective when $\nu \in \mathbb{R}$7 and the singularity lies outside.
• Subtraction of the incomplete contribution: For $\nu \in \mathbb{R}$8,

$\nu \in \mathbb{R}$9

leverages existing numerical routines for $\Im\{w\} \le 0$0 or $\Im\{w\} \le 0$1.

• Continued-fraction (McCabe–Murphy) expansions furnish rapidly convergent approximations, merging small- and large-$\Im\{w\} \le 0$2 behaviors.
• Padé rational approximations of the form

$\Im\{w\} \le 0$3

provide practical accuracy away from the singularity at $\Im\{w\} \le 0$4.

For the complete Faddeeva function, highly accurate and efficient approximations combine continued fractions, Hui’s $\Im\{w\} \le 0$5, and Humlíček’s $\Im\{w\} \le 0$6 rational forms via region-based domain partitioning in the $\Im\{w\} \le 0$7 plane, achieving max-relative errors $\Im\{w\} \le 0$8 across practical input ranges (Zaghloul, 2017).

6. Applications in Plasma Kinetics and Wave Dispersion

The IPDF is of central importance when the underlying distribution functions are Maxwellian only on semi-infinite intervals, such as the “depleted Maxwellian’’ near wall sheaths. The linear dielectric function,

$\Im\{w\} \le 0$9

acquires incomplete $\nu\to-\infty$0 functions for each domain-limited interval. For a depleted electron population,

$\nu\to-\infty$1

the dielectric is

$\nu\to-\infty$2

where $\nu\to-\infty$3. Asymptotics of each term yield ion-acoustic and Langmuir wave relations. The IPDF encodes the absence of resonant particles below the cutoff, permitting undamped or weakly damped modes at high $\nu\to-\infty$4, as manifested in the discontinuous imaginary part of $\nu\to-\infty$5 (Baalrud, 2013).

7. Practical Considerations and Algorithmic Implementation

Numerical evaluation of the related Faddeeva function, and hence $\nu\to-\infty$6, leverages region-dependent rational or continued fraction approximations. The algorithm partitions the $\nu\to-\infty$7 plane and selects among the following forms:

• Continued-fraction truncations for $\nu\to-\infty$8 beyond specific thresholds
• Hui’s $\nu\to-\infty$9 rational approximation for small/moderate $$Z(w) = Z(-\infty, w) = \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} \frac{e^{-t^2}}{t-w}\,dt.$$0
• Humlíček’s $$Z(w) = Z(-\infty, w) = \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} \frac{e^{-t^2}}{t-w}\,dt.$$1 approximation near the real axis

A single evaluation therefore typically demands only polynomial evaluation and a modest number of floating-point operations, achieving efficiency superior to “full-range” methods for large batches. Derivatives of $$Z(w) = Z(-\infty, w) = \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} \frac{e^{-t^2}}{t-w}\,dt.$$2 are computed directly via

$$Z(w) = Z(-\infty, w) = \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} \frac{e^{-t^2}}{t-w}\,dt.$$3

and Cauchy–Riemann relations yield all real derivatives necessary for applications in spectral line broadening and dielectric response computations (Zaghloul, 2017).

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References:

(Baalrud, 2013): "The incomplete plasma dispersion function: properties and application to waves in bounded plasmas" (Zaghloul, 2017): "Algorithm 985: Simple, efficient, and relatively accurate approximation for the evaluation of the Faddeyeva function"