---
title: 'Voigt Function: Definition and Computation'
url: https://www.emergentmind.com/topics/voigt-function
type: topic
---

# Voigt Function: Definition and Computation

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 $w(z) = e^{-z^2} \operatorname{erfc}(-i z)$. The normalization convention is $\int_{-\infty}^{\infty} H(a, u) \, du = 1$ [1806.10338].


## 2. Integral Forms, Fourier and Scale-Mixture Representations

Alternative integral representations reveal different structural and computational aspects:
- Probability-integral:  
  $K(x, y) = \frac{y}{\pi} \int_{-\infty}^{\infty} \frac{e^{-t^2}}{(x-t)^2 + y^2} dt$
- Laplace/Fourier domain:  
  $K(x, y) = \frac{1}{\sqrt{\pi}} \int_0^{\infty} e^{-t^2/4 - yt} \cos(xt) dt$, $y > 0$
- Scale-mixture (probabilistic) form:  
  The Voigt profile is a scale mixture of Gaussians, with the mixing distribution a Lévy law. Explicitly,
  \[
  V(x; \sigma, \gamma) = \int_{\sigma^2}^{\infty} \frac{1}{\sqrt{2\pi v}} \exp\left(-\frac{x^2}{2 v}\right) f_L(v; \sigma, \gamma) dv,
  \]
  where $f_L$ is a Lévy density [2508.13252].

The characteristic function is $\varphi_X(t) = \exp\left(-\frac{1}{2}\sigma^2 t^2 - \gamma |t|\right)$.

## 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$n$"):  
  Approximates $w(z)$ as a sum of simple poles, with $n$ even,
  \[
  w(z) = \sum_{k=1}^{n/2} \left[ \frac{\alpha_k + i\beta_k}{z - x_k + i\delta} - \frac{\alpha_k - i\beta_k}{z + x_k + i\delta} \right]
  \]
  where $x_k$, $\omega_k$ are Gauss-Hermite nodes/weights and $\delta$ is a shift parameter [1806.11560]. For $n=16$ ($\delta=1.3118$), uniform relative errors are $\lesssim 8\times10^{-5}$; with $n=20$ and $\delta=1.55$, errors drop below $10^{-6}$ even in the wings [1806.11560].

- **Weideman's Rational Approximation**:  
  A single rational function
  \[
  w(z) \approx \frac{P_N(z)}{Q_N(z)}
  \]
  with $N$ adjustable; $N = 32$ provides $<10^{-6}$ uniform error for $0 \le x \le 20$, $10^{-6} \le y \le 10^2$ [1806.10338, 2010.09804].

- **Hybrid ("H-W") Two-Region Methods**:  
  For $|x| + y > R_{\text{th}}$ (typically $R_{\text{th}} \approx 15$), switch to Humlíček's asymptotic expansion:
  \[
  w(z) \sim \frac{i}{\sqrt{\pi}z}\left[1 + \frac{1}{2z^2} + \frac{3}{4z^4} + \cdots\right]
  \]
  Elsewhere, deploy the main rational expansion. This achieves high-throughput, $\sim 10^8$ calls/s, with spectroscopically meaningful $10^{-5}$ accuracy [1806.10338].

- **Residue Calculus Rational Approximations**:  
  Approximations requiring only elementary operations, e.g. the 16-term Abrarov & Quine expansion achieves relative error $\sim10^{-14}$, outperforming earlier schemes in both accuracy and speed by avoiding poles for $y \ge 0$ [1504.00322].

- **Fourier Expansion and Incomplete Cosine Expansion**:  
  Truncated Fourier or incomplete cosine series expansions, e.g. the $N=23$ exponential series of [1205.1768] allow double-precision accuracy ($10^{-15}$) at speeds surpassing Weideman or Humlíček for ultra-large-scale applications [1407.0533, 1205.1768].

- **Chebyshev Polynomial Subinterval Expansion**:  
  The domain is subdivided in $(u, a)$; in each, a bivariate Chebyshev polynomial provides an efficient interpolant. Combined with continued-fraction and Taylor/Dawson expansions for special regions, a uniform $10^{-6}$ accuracy with a $2$–$20\times$ speed advantage over classical codes has been achieved in line transfer applications [2411.00917].

| Algorithm              | Max Rel. Error | Representative Domain | Notes                              |
|------------------------|----------------|----------------------|-------------------------------------|
| Humlíček $n=16$        | $8\times10^{-5}$ | Whole, $y>10^{-6}$   | Two-branch, fastest                |
| Weideman $N=32$        | $<10^{-6}$      | $0\le x \le 20$, $y\geq10^{-6}$ | Single rational, slower            |
| Residue rational (A&Q) | $10^{-14}$      | $0<x<4\times10^4$, $10^{-4}<y<10^2$ | No poles, fastest in HITRAN regime |
| Fourier exp. ($N=23$)  | $10^{-15}$      | All $z$              | Highly vectorizable, no special funcs |
| Chebyshev-subinterval  | $10^{-6}$       | $|u|\le200$, $0\leq a\leq50$ | Optimized for F90, line-by-line    |


## 4. Special Regimes: Small $y$ and Dawson's Integral

For $y \ll 1$, 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:
\[
K(x, y) \approx e^{-x^2} - \frac{2y}{\sqrt{\pi}} \bigl[1 - 2xF(x)\bigr]
\]
where $F(x)$ is the Dawson function, itself accurately approximated by a short rational expansion [1505.04683, 2112.02078]. Series-based or Taylor-Euler approaches yield $<10^{-10}$ relative errors uniformly for $0 \le x \le 15$, $0 \le y \le 10^{-6}$, 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:
\[
V_{\rm pv}(x) = N \left[ (1-\eta) G(x; H) + \eta L(x; H) \right] + C(x)
\]
with $H(\sigma, \gamma)$, $\eta(\sigma, \gamma)$, and empirical correction $C(x)$. Analytical forms include Whiting (1968), Matveev (1972), Kielkopf (1973), Thompson–Mihalas–Davis (1987), and Liu–Jeffries (2001) [1901.08366]. The best, Kielkopf’s, remains within $10^{-4}$ 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 $C$)  | General, $y\ll1$       |
| 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, $\alpha_V$) is a frequent requirement for convolution windows and empirical database line lists. Wang et al. (2020) provide a piecewise polynomial expansion achieving uniform errors $<10^{-17}$, using a 31-term series in the Gaussian or Lorentzian regime and a rational expansion in between [2111.15628]. This super-accurate formula vastly exceeds classical fits like Olivero–Longbothum (1977) ($\sim 10^{-4}$ error) with no special-function calls:
\[
\begin{cases}
\alpha_V \approx \alpha_G \Gamma_1(r)              & r \leq 0.6993   \\
\alpha_V \approx (\alpha_L+\alpha_G)\sum u_k D^k   & 0.6993 < r < 8.25 \\
\alpha_V \approx \alpha_L \Gamma_2(\eta)           & r \geq 8.25
\end{cases}
\] 

## 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) [2508.13252].
- **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 [1711.09304].

## 8. Computational Implementations and Best Practices

Efficient implementations balance accuracy and throughput:
- For $10^{-4}$–$10^{-6}$ relative error, combine Humlíček–Weideman or Chebyshev subinterval routines;
- For $<10^{-13}$, use Fourier/incomplete-cosine expansion or residue rational routines with precomputed coefficients;
- For extreme $y \ll 10^{-6}$, deploy tailored Dawson-integral expansions;
- Always precompute and store expansion coefficients, vectorize inner loops, and validate against machine-precision references for large $|x|$ or small $y$ domains [1806.10338, 2411.00917].

## References

- [1806.10338] Comprehensive review of state-of-the-art Voigt function implementations, spectroscopic accuracy requirements, and recommended algorithms.
- [1806.11560] Humlíček’s generalized rational approximation for rapid, 5–6 digit accurate Voigt/Faddeeva evaluation.
- [1504.00322] Residue-based rational approximations achieving $10^{-14}$-level accuracy.
- [1407.0533], [1205.1768] Incomplete cosine and Fourier expansion methods for high-precision evaluation.
- [2411.00917] Two-variable Chebyshev polynomial expansion, combining continued-fraction/Taylor expansions for core and asymptotic domains.
- [2112.02078], [1505.04683] Small-$y$ asymptotics via Taylor/Dawson and Hermite polynomial expansions.
- [2111.15628] HWHM formula achieving $<10^{-17}$ relative error.
- [1901.08366] Critical survey of closed-form pseudo-Voigt approximations.
- [2508.13252] Probabilistic structure and duality theory for Voigt and related distributions.
- [2010.09804] 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.

Source: https://www.emergentmind.com/topics/voigt-function