Voigt Function: Definition and Computation

• Voigt function is defined as the convolution of a Gaussian and Lorentzian distribution, forming a fundamental lineshape model in spectroscopy.
• Advanced numerical algorithms such as Humlíček’s and Weideman’s rational approximations enable high-precision and efficient evaluation over diverse parameter ranges.
• Its practical applications span quantitative spectroscopy, astrophysics, and atmospheric science, addressing challenges in special regimes like small y-values with refined methods.

The Voigt function, denoted conventionally as $K(x, y)$ or $V(x; \sigma, \gamma)$, is the real part of the complex error function or Faddeeva function and arises as the normalized convolution of a Lorentzian (Cauchy) and a Gaussian distribution. This function is central to quantitative spectroscopy, astrophysics, atmospheric science, and radiative transfer, providing the line shape when both Doppler (Gaussian) and collisional/pressure (Lorentzian) broadening are present. Despite its ubiquity, the Voigt function lacks elementary closed-form expressions, necessitating sophisticated high-precision numerical algorithms for its evaluation in large-scale and high-throughput computational environments.

1. Mathematical Definition and Representations

The standard Voigt profile $V(x; \sigma, \gamma)$ is defined as the convolution

$$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx',$$

where $$G(x; \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma^2}\right)$$ is a Gaussian and $$L(x; \gamma) = \frac{1}{\pi} \frac{\gamma}{x^2 + \gamma^2}$$ is a Lorentzian (Cauchy) with half-width at half-maximum $\gamma$. Setting $u = x/(\sqrt{2}\sigma)$ and $a = \gamma/(\sqrt{2}\sigma)$ yields the normalized form

$$H(a, u) = \frac{a}{\pi} \int_{-\infty}^{+\infty} \frac{e^{-t^2}}{(u - t)^2 + a^2} \, dt = \Re w(u + i a)$$

with $V(x; \sigma, \gamma)$0. The normalization convention is $V(x; \sigma, \gamma)$1 (Schreier, 2018).

2. Integral Forms, Fourier and Scale-Mixture Representations

Alternative integral representations reveal different structural and computational aspects:

• Probability-integral:

$V(x; \sigma, \gamma)$2

• Laplace/Fourier domain:

$V(x; \sigma, \gamma)$3, $V(x; \sigma, \gamma)$4

• Scale-mixture (probabilistic) form: The Voigt profile is a scale mixture of Gaussians, with the mixing distribution a Lévy law. Explicitly,

$V(x; \sigma, \gamma)$5

where $V(x; \sigma, \gamma)$6 is a Lévy density (Cannas, 18 Aug 2025).

The characteristic function is $V(x; \sigma, \gamma)$7.

3. Numerical Evaluation: Rational Approximations and Hybrid Methods

Direct numeric quadrature of the Voigt integral is inefficient and inaccurate in parameter regimes relevant to line-by-line spectroscopic modeling, motivating the development of rational approximations and hybrid algorithms.

• Humlíček's Rational Approximation ("cpf$V(x; \sigma, \gamma)$8"): Approximates $V(x; \sigma, \gamma)$9 as a sum of simple poles, with $V(x; \sigma, \gamma)$0 even,

$V(x; \sigma, \gamma)$1

where $V(x; \sigma, \gamma)$2, $V(x; \sigma, \gamma)$3 are Gauss-Hermite nodes/weights and $V(x; \sigma, \gamma)$4 is a shift parameter (Schreier, 2018). For $V(x; \sigma, \gamma)$5 ($V(x; \sigma, \gamma)$6), uniform relative errors are $V(x; \sigma, \gamma)$7; with $V(x; \sigma, \gamma)$8 and $V(x; \sigma, \gamma)$9, errors drop below $$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx',$$0 even in the wings (Schreier, 2018).

• Weideman's Rational Approximation:

A single rational function

$$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx',$$1

with $$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx',$$2 adjustable; $$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx',$$3 provides $$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx',$$4 uniform error for $$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx',$$5, $$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx',$$6 (Schreier, 2018, Schreier et al., 2020).

• Hybrid ("H-W") Two-Region Methods:

For $$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx',$$7 (typically $$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx',$$8), switch to Humlíček's asymptotic expansion:

$$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx',$$9

Elsewhere, deploy the main rational expansion. This achieves high-throughput, $$G(x; \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma^2}\right)$$0 calls/s, with spectroscopically meaningful $$G(x; \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma^2}\right)$$1 accuracy (Schreier, 2018).

• Residue Calculus Rational Approximations:

Approximations requiring only elementary operations, e.g. the 16-term Abrarov & Quine expansion achieves relative error $$G(x; \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma^2}\right)$$2, outperforming earlier schemes in both accuracy and speed by avoiding poles for $$G(x; \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma^2}\right)$$3 (Abrarov et al., 2015).

• Fourier Expansion and Incomplete Cosine Expansion:

Truncated Fourier or incomplete cosine series expansions, e.g. the $$G(x; \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma^2}\right)$$4 exponential series of (Abrarov et al., 2012) allow double-precision accuracy ($$G(x; \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma^2}\right)$$5) at speeds surpassing Weideman or Humlíček for ultra-large-scale applications (Abrarov et al., 2014, Abrarov et al., 2012).

• Chebyshev Polynomial Subinterval Expansion:

The domain is subdivided in $$G(x; \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma^2}\right)$$6; in each, a bivariate Chebyshev polynomial provides an efficient interpolant. Combined with continued-fraction and Taylor/Dawson expansions for special regions, a uniform $$G(x; \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma^2}\right)$$7 accuracy with a $$G(x; \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma^2}\right)$$8–$$G(x; \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma^2}\right)$$9 speed advantage over classical codes has been achieved in line transfer applications (Zaghloul et al., 2024).

Algorithm	Max Rel. Error	Representative Domain	Notes
Humlíček $$L(x; \gamma) = \frac{1}{\pi} \frac{\gamma}{x^2 + \gamma^2}$$0	$$L(x; \gamma) = \frac{1}{\pi} \frac{\gamma}{x^2 + \gamma^2}$$1	Whole, $$L(x; \gamma) = \frac{1}{\pi} \frac{\gamma}{x^2 + \gamma^2}$$2	Two-branch, fastest
Weideman $$L(x; \gamma) = \frac{1}{\pi} \frac{\gamma}{x^2 + \gamma^2}$$3	$$L(x; \gamma) = \frac{1}{\pi} \frac{\gamma}{x^2 + \gamma^2}$$4	$$L(x; \gamma) = \frac{1}{\pi} \frac{\gamma}{x^2 + \gamma^2}$$5, $$L(x; \gamma) = \frac{1}{\pi} \frac{\gamma}{x^2 + \gamma^2}$$6	Single rational, slower
Residue rational (A&Q)	$$L(x; \gamma) = \frac{1}{\pi} \frac{\gamma}{x^2 + \gamma^2}$$7	$$L(x; \gamma) = \frac{1}{\pi} \frac{\gamma}{x^2 + \gamma^2}$$8, $$L(x; \gamma) = \frac{1}{\pi} \frac{\gamma}{x^2 + \gamma^2}$$9	No poles, fastest in HITRAN regime
Fourier exp. ($\gamma$0)	$\gamma$1	All $\gamma$2	Highly vectorizable, no special funcs
Chebyshev-subinterval	$\gamma$3	$\gamma$4, $\gamma$5	Optimized for F90, line-by-line

4. Special Regimes: Small $\gamma$6 and Dawson's Integral

For $\gamma$7, the Voigt function approaches a Gaussian in the core and shows Lorentzian wings, with rapid breakdown of generic rational approximations. In this regime, superconvergent schemes exploit expansions in Dawson’s integral: $\gamma$8 where $\gamma$9 is the Dawson function, itself accurately approximated by a short rational expansion (Abrarov et al., 2015, Wang, 2021). Series-based or Taylor-Euler approaches yield $u = x/(\sqrt{2}\sigma)$0 relative errors uniformly for $u = x/(\sqrt{2}\sigma)$1, $u = x/(\sqrt{2}\sigma)$2, matching quadruple-precision benchmarks.

5. Pseudo-Voigt Closed-form Approximations

Simple, empirical "pseudo-Voigt" formulas are based on direct linear combinations of Gaussian and Lorentzian: $u = x/(\sqrt{2}\sigma)$3 with $u = x/(\sqrt{2}\sigma)$4, $u = x/(\sqrt{2}\sigma)$5, and empirical correction $u = x/(\sqrt{2}\sigma)$6. Analytical forms include Whiting (1968), Matveev (1972), Kielkopf (1973), Thompson–Mihalas–Davis (1987), and Liu–Jeffries (2001) (Schreier, 2019). The best, Kielkopf’s, remains within $u = x/(\sqrt{2}\sigma)$7 relative error, but others can err by 1–5%. These are selected for rough fitting or when elementary functions only are permitted—accurate spectroscopic synthesis universally employs algorithmic (not closed-form) routines.

Method	Max Rel. Error	Use Case
Whiting	5%	Rough/fast, teaching
Matveev	3% (with $u = x/(\sqrt{2}\sigma)$8)	General, $u = x/(\sqrt{2}\sigma)$9
Kielkopf	0.01%	High-precision, closed
Thompson	1.2%	Medium-accuracy
Liu	0.5%	Area-preserving

6. Super-Accuracy for Width (HWHM) and Asymptotic Behavior

The Voigt profile’s half-width-at-half-maximum (HWHM, $a = \gamma/(\sqrt{2}\sigma)$0) is a frequent requirement for convolution windows and empirical database line lists. Wang et al. (2020) provide a piecewise polynomial expansion achieving uniform errors $a = \gamma/(\sqrt{2}\sigma)$1, using a 31-term series in the Gaussian or Lorentzian regime and a rational expansion in between (Wang et al., 2021). This super-accurate formula vastly exceeds classical fits like Olivero–Longbothum (1977) ($a = \gamma/(\sqrt{2}\sigma)$2 error) with no special-function calls: $a = \gamma/(\sqrt{2}\sigma)$3

7. Extensions, Dualities, and Relativistic Generalizations

• Scale-mixture and Dual Profiles:

The Voigt profile is a normal scale mixture with a Lévy mixing law, providing a probabilistic structure for estimation and EM algorithms. The dual Voigt distribution, constructed via the characteristic function, is a truncated reflected normal, with finite moments (unlike the original Voigt) (Cannas, 18 Aug 2025).

• Relativistic Voigt:

In particle physics, convolution of the relativistic Breit–Wigner with a Gaussian leads to a relativistic Voigt profile, with quartic denominators and more intricate singularity structure. Its evaluation uses modifications of Faddeeva and erfc formulas to accommodate the new kernel (Kycia et al., 2017).

8. Computational Implementations and Best Practices

Efficient implementations balance accuracy and throughput:

• For $a = \gamma/(\sqrt{2}\sigma)$4–$a = \gamma/(\sqrt{2}\sigma)$5 relative error, combine Humlíček–Weideman or Chebyshev subinterval routines;
• For $a = \gamma/(\sqrt{2}\sigma)$6, use Fourier/incomplete-cosine expansion or residue rational routines with precomputed coefficients;
• For extreme $a = \gamma/(\sqrt{2}\sigma)$7, deploy tailored Dawson-integral expansions;
• Always precompute and store expansion coefficients, vectorize inner loops, and validate against machine-precision references for large $a = \gamma/(\sqrt{2}\sigma)$8 or small $a = \gamma/(\sqrt{2}\sigma)$9 domains (Schreier, 2018, Zaghloul et al., 2024).

• (Schreier, 2018) Comprehensive review of state-of-the-art Voigt function implementations, spectroscopic accuracy requirements, and recommended algorithms.
• (Schreier, 2018) Humlíček’s generalized rational approximation for rapid, 5–6 digit accurate Voigt/Faddeeva evaluation.
• (Abrarov et al., 2015) Residue-based rational approximations achieving $$H(a, u) = \frac{a}{\pi} \int_{-\infty}^{+\infty} \frac{e^{-t^2}}{(u - t)^2 + a^2} \, dt = \Re w(u + i a)$$0-level accuracy.
• (Abrarov et al., 2014, Abrarov et al., 2012) Incomplete cosine and Fourier expansion methods for high-precision evaluation.
• (Zaghloul et al., 2024) Two-variable Chebyshev polynomial expansion, combining continued-fraction/Taylor expansions for core and asymptotic domains.
• (Wang, 2021, Abrarov et al., 2015) Small-$$H(a, u) = \frac{a}{\pi} \int_{-\infty}^{+\infty} \frac{e^{-t^2}}{(u - t)^2 + a^2} \, dt = \Re w(u + i a)$$1 asymptotics via Taylor/Dawson and Hermite polynomial expansions.
• (Wang et al., 2021) HWHM formula achieving $$H(a, u) = \frac{a}{\pi} \int_{-\infty}^{+\infty} \frac{e^{-t^2}}{(u - t)^2 + a^2} \, dt = \Re w(u + i a)$$2 relative error.
• (Schreier, 2019) Critical survey of closed-form pseudo-Voigt approximations.
• (Cannas, 18 Aug 2025) Probabilistic structure and duality theory for Voigt and related distributions.
• (Schreier et al., 2020) Robust rational approximations in speed-dependent and Hartmann–Tran line shapes.

These developments enable robust, accurate, and high-throughput modeling of spectroscopic lineshapes essential for atmospheric, astrophysical, and laboratory-scale line-by-line computations.