You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: |
    We present a rational approximation for the Dawsons integral of real argument and show how it can be implemented for accurate and rapid computation of the Voigt function at small $y <  < 1$. The algorithm based on this approach enables computation with accuracy exceeding ${10^{ - 10}}$ within the domain $0 \le x \le 15$ and $0 \le y \le {10^{ - 6}}$. Due to rapid performance the proposed rational approximation runs the algorithm without deceleration.\
    [**Keywords:**]{} Dawsons integral, Voigt function, Faddeeva function, complex error function, rational approximation, spectral line broadening
author:
- '**S. M. Abrarov[^1] and B. M. Quine$^{*}$[^2]**'
date: 'May, 7 2015'
title: '**A rational approximation for the Dawsons integral of real argument**'
---

Introduction
============

The Dawsons integral is defined by [@Abramowitz1972; @Cody1970; @McCabe1974; @Rybicki1989] $$\label{eq_1}
F\left( z \right) = {e^{ - {z^2}}}\int\limits_0^z {{e^{{t^2}}}dt},$$ where $z = x + iy$ is a complex argument. This integral is closely related to the complex error function, also known as the Faddeeva function [@Abramowitz1972; @Faddeyeva1961; @Schreier1992] $$\label{eq_2}
w\left( z \right) = {e^{ - {z^2}}}\left( {1 + \frac{{2i}}{{\sqrt \pi  }}\int\limits_0^z {{e^{{t^2}}}dt} } \right).$$ In particular, comparing equations and we can easily find a relation between the Dawsons integral and the complex error function $$\label{eq_3}
F\left( z \right) = \frac{{i\sqrt \pi  }}{2}\left[ {{e^{ - {z^2}}} - w\left( z \right)} \right].$$ As we can see from this identity, the Dawsons integral is simply a reformulation of the complex error function.

The Dawsons integral cannot be taken analytically in terms of elementary functions in a closed form and, therefore, it must be solved numerically. Several useful approximations for the Dawsons integral have been reported [@Cody1970; @McCabe1974; @Rybicki1989]. In this paper we show a new approach that can be successfully implemented for efficient computation.

The complex error function can be alternatively represented as (see Appendix A) $$\label{eq_4}
w\left( {x,y} \right) = \frac{1}{{\sqrt \pi  }}\int\limits_0^\infty  {\exp \left( { - {t^2}/4} \right)\exp \left( { - yt} \right)\exp \left( {ixt} \right)dt}.$$ and expressed as a sum [@Schreier1992] $$w\left( {x,y} \right) = K\left( {x,y} \right) + iL\left( {x,y} \right), \qquad  y \ge 0,$$ where its real and imaginary parts are (see Appendix B) $$\label{eq_5}
K\left( {x,y} \right) = \frac{y}{\pi }\int\limits_{ - \infty }^\infty  {\frac{{{e^{ - {t^2}}}}}{{{y^2} + {{\left( {x - t} \right)}^2}}}dt},$$ and $$\label{eq_6}
L\left( {x,y} \right) = \frac{1}{\pi }\int\limits_{ - \infty }^\infty  {\frac{{\left( {x - t} \right){e^{ - {t^2}}}}}{{{y^2} + {{\left( {x - t} \right)}^2}}}dt},$$ respectively. The equation is known as the Voigt function [@Schreier1992; @Armstrong1967; @Armstrong1972; @Letchworth2007; @Abrarov2015a], widely used in many branches of Applied Mathematics [@Srivastava1987; @Abrarov2011; @Pagnini2010], Physics [@Edwards1992; @Quine2002; @Christensen2012; @Berk2013; @Sonnenschein2012; @Quine2013] and Astronomy [@Emerson1996]. Since the function has no a specific name, it can be regarded as the $L$-function.

Due to symmetric properties of the Voigt function $$K\left( {x, - \left| y \right|} \right) = K\left( { - x, - \left| y \right|} \right) =  - K\left( {x,\left| y \right|} \right),$$ it is sufficient to consider only the ${{\rm{I}}^{{\rm{st}}}}$ and ${\rm{I}}{{\rm{I}}^{{\rm{nd}}}}$ quadrants in order to cover the entire complex plane. Therefore, we will imply further that $y \ge 0$.

In our recent publication [@Abrarov2015a] we have shown that a new sampling methodology based on incomplete expansion of the sinc function [@Abrarov2015b] leads to a rational approximation of the Voigt function for rapid and accurate computation (see also the Matlab source code in [@Abrarov2015a]) $$\label{eq_7}
\begin{aligned}
\kappa \left( {x,y} \right) &\buildrel \Delta \over = \sum\limits_{m = 1}^{{m_{\max }}} {\frac{{{\alpha _m}\left( {{\beta _m} + {y^2} - {x^2}} \right) + {\gamma _m}y\left( {{\beta _m} + {x^2} + {y^2}} \right)}}{{\beta _m^2 + 2{\beta _m}\left( {{y^2} - {x^2}} \right) + {{\left( {{x^2} + {y^2}} \right)}^2}}}} \\
&\Rightarrow K\left( {x,y} \right) \approx \kappa \left( {x,y + \varsigma /2} \right),
\end{aligned}$$ where $${\alpha _m} = \frac{{\sqrt \pi  \left( {m - 1/2} \right)}}{{2m_{\max }^2h}}\sum\limits_{n =  - N}^N {{e^{{\varsigma ^2}/4 - {n^2}{h^2}}}\sin \left( {\frac{{\pi \left( {m - 1/2} \right)\left( {nh + \varsigma /2} \right)}}{{{m_{\max }}h}}} \right)},$$ $${\beta _m} = {\left( {\frac{{\pi \left( {m - 1/2} \right)}}{{2{m_{\max }}h}}} \right)^2},$$ $${\gamma _m} = \frac{1}{{{m_{\max }}\sqrt \pi  }}\sum\limits_{n =  - N}^N {{e^{{\varsigma ^2}/4 - {n^2}{h^2}}}\cos \left( {\frac{{\pi \left( {m - 1/2} \right)\left( {nh + \varsigma /2} \right)}}{{{m_{\max }}h}}} \right)},$$ $h = 0.293$, ${m_{\max }} = 12$, $\varsigma  = 2.75$ and $N = 23$ (for higher accuracy the coefficients $h$ and ${m_{\max }}$ can be taken as $0.25$ and $16$, respectively). This rational approximation is faster than the Weidemans rational approximation [@Weideman1994] by factor greater than two (see [@Abrarov2015a] for details). Furthermore, with the same number of summation terms it is by several orders of the magnitude more accurate than the Weidemans rational approximation in the domain of practical interest $0 < x < 40,000$ and ${10^{ - 4}} < y < {10^{ - 2}}$ [@Quine2002; @Wells1999] required for applications using the HITRAN molecular spectroscopic database [@Rothman2013]. In general, the series approximation provides accurate results while $y \ge {10^{ - 6}}$. However, its accuracy deteriorates with decreasing parameter $y$ and, consequently, it cannot cover a narrow band region $y < {10^{ - 6}}$ along $x$-axis. It should be noted that the accuracy deterioration of the Voigt function at small $y <  < 1$ is a very common problem in most known approximations (see for example [@Armstrong1967; @Amamou2013; @Abrarov2015c]).

Although the computation of the Voigt function at $y < {10^{ - 6}}$ is required relatively rare in practice, it, nevertheless, has to be taken into account in algorithmic implementation. In this work we propose a rational approximation for the Dawsons integral of real argument and show how its implementation can resolve effectively such a problem in computation of the Voigt function that occurs at small $y <  < 1$.

Results and discussion
======================

Derivation
----------

As we have shown recently, the exponential function can be expanded as a series [@Abrarov2015a] $$\label{eq_8}
\exp \left( { - {x^2}} \right) \approx \frac{1}{2}\sum\limits_{m = 1}^{{m_{\max }}} {\left[ {\frac{{{A_m} + \left( {x + i\varsigma /2} \right){B_m}}}{{C_m^2 - {{\left( {x + i\varsigma /2} \right)}^2}}} + \frac{{{A_m} + \left( { - x + i\varsigma /2} \right){B_m}}}{{C_m^2 - {{\left( { - x + i\varsigma /2} \right)}^2}}}} \right]},$$ where $${A_m} = \alpha_m = \frac{{\sqrt \pi  \left( {m - 1/2} \right)}}{{2m_{\max }^2h}}\sum\limits_{n =  - N}^N {{e^{{\varsigma ^2}/4 - {n^2}{h^2}}}\sin \left( {\frac{{\pi \left( {m - 1/2} \right)\left( {nh + \varsigma /2} \right)}}{{{m_{\max }}h}}} \right)},$$ $${B_m} = - i \gamma_m = - \frac{i}{{{m_{\max }}\sqrt \pi  }}\sum\limits_{n =  - N}^N {{e^{{\varsigma ^2}/4 - {n^2}{h^2}}}\cos \left( {\frac{{\pi \left( {m - 1/2} \right)\left( {nh + \varsigma /2} \right)}}{{{m_{\max }}h}}} \right)}$$ and $${C_m^2} = \beta_m = \left( \frac{{\pi \left( {m - 1/2} \right)}}{{2{m_{\max }}h}} \right)^2.$$ This equation has been used to obtain the rational approximation [@Abrarov2015a]. We may also utilize it for the $L$-function by substituting the exponential function approximation into equation . This leads to the following integral $$\label{eq_9}
\begin{aligned}
L\left( {x,y}\right) &\approx \\
&\frac{1}{{2\pi }}\int\limits_{ - \infty }^\infty  {\frac{{x - t}}{{{y^2} + {{\left( {x - t} \right)}^2}}}\sum\limits_{m = 1}^{{m_{\max }}} {\left[ {\frac{{{A_m} + \left( {t + i\varsigma /2} \right){B_m}}}{{C_m^2 - {{\left( {t + i\varsigma /2} \right)}^2}}} + \frac{{{A_m} + \left( { - t + i\varsigma /2} \right){B_m}}}{{C_m^2 - {{\left( { - t + i\varsigma /2} \right)}^2}}}} \right]} } \,dt.
\end{aligned}$$ The integrand of this integral is analytic everywhere over the entire complex plane except $2 + 4{m_{\max }}$ isolated points $$\begin{aligned}
\left\{ x - iy,x + iy, - {C_m} - i\varsigma /2,{C_m} - i\varsigma /2, - {C_m} + i\varsigma /2,{C_m} \right. &+ \left. i\varsigma /2 \right\}, \\
m &\in \left\{ {1,2,3,\,\, \ldots \,\,m_{\text{max}}} \right\}
\end{aligned}$$ where we can observe the singularities. In order to integrate equation (9), we may choose conveniently a contour ${C_{ccw}}$ in counterclockwise (CCW) direction on the upper-half complex plane as a semicircle centered at the origin with infinite radius. Consequently, since the domain enclosed by contour ${C_{ccw}}$ covers only the half of complex plane, the number of the isolated points is reduced twice $${t_r} = \left\{ {x + iy, - {C_m} + i\varsigma /2,{C_m} + i\varsigma /2} \right\}, \quad m \in \left\{ {1,2,3,\,\, \ldots \,\,{m_{\max }}} \right\}.$$ Using the Residue Theorems formula now $$\frac{1}{{2\pi i}}\oint\limits_{{C_{ccw}}} {f\left( t \right)} \,dt = \sum\limits_{r = 1}^{1 + 2{m_{\max }}} {{\rm{Res}}\left[ {f\left( t \right),{t_r}} \right]},$$ where $f\left( t \right)$ is the integrand of integral , we obtain a rational approximation for the $L$-function as follows $$\label{eq_10}
\begin{aligned}
\lambda \left( {x,y} \right) &\buildrel \Delta \over = \sum\limits_{m = 1}^{{m_{\max }}} {\frac{{x\left[ {2{\alpha _m}y + {\gamma _m}\left( {{x^2} + {y^2} - {\beta _m}} \right)} \right]}}{{\beta _m^2 + 2{\beta _m}\left( {{y^2} - {x^2}} \right) + {{\left( {{x^2} + {y^2}} \right)}^2}}}} \\
 &\Rightarrow L\left( {x,y} \right) \approx \lambda \left( {x,y + \varsigma /2} \right).
\end{aligned}$$ Consequently, the complex error function can be calculated as $$\label{eq_11}
w\left( {x,y} \right) \approx \kappa \left( {x,y + \varsigma /2} \right) + i\lambda \left( {x,y + \varsigma /2} \right).$$

The computational testing we performed with equation shows that its imaginary part $\lambda \left( {x,y + \varsigma /2} \right)$ remains always accurate regardless how small the parameter $y$ is taken. Therefore the computation of the Voigt function at small $y <  < 1$ is our main objective. We may attempt to resolve this problem by using the following approximation that has been reported in our recent work [@Abrarov2015c] $$w\left( {x,y <  < 1} \right) \approx {e^{{{\left( {ix - y} \right)}^2}}}\left[ {1 + \frac{{i{e^{{x^2}}}}}{{\sqrt \pi  }}\left( {2F\left( x \right) - \frac{{1 - {e^{2ixy}}}}{x}} \right)} \right],$$ where the Dawsons integral of real argument is denoted as $F\left( x \right) \equiv F\left( {x,y = 0} \right)$. Taking the real part of the equation above yields the approximation of the Voigt function $$\label{eq_12}
K\left( {x,y <  < 1} \right) \approx {e^{{y^2} - {x^2}}}\cos \left( {2xy} \right) - \frac{{2{e^{{y^2}}}}}{{\sqrt \pi  }}\left[ {y\,{\rm{sinc}}\left( {2xy} \right) - F\left( x \right)\sin \left( {2xy} \right)} \right],$$ where we imply that the sinc function satisfies $$\left\{ {{\rm{sinc}}\left( {2xy \ne 0} \right) = \sin \left( {2xy} \right)/\left( {2xy} \right),\,\,{\rm{sinc}}\left( {2xy = 0} \right) = 1} \right\}.$$

According to equation the Dawsons integral of complex argument can be written now as $$F\left( {x,y} \right) \approx \frac{{i\sqrt \pi  }}{2}\left\{ {{e^{ - {{\left( {x + iy} \right)}^2}}} - \left[ {\kappa \left( {x,y + \varsigma /2} \right) + i\lambda \left( {x,y + \varsigma /2} \right)} \right]} \right\}$$ and since at $K\left( {x,y = 0} \right) = \exp \left( { - {t^2}} \right)$ (see Appendix B), from equation above we get an approximation for the Dawsons integral of real argument $$\label{eq_13}
F\left( {x,y = 0} \right) \approx \frac{{\sqrt \pi  }}{2}\lambda \left( {x,\varsigma /2} \right).$$ Lastly, using the series approximation the equation can be expressed as given by $$\label{eq_14}
F\left( x \right) \approx \frac{{\sqrt \pi  }}{2}\sum\limits_{m = 1}^{{m_{\max }}} {\frac{{x\left[ {{\alpha _m}\varsigma  + {\gamma _m}\left( {{x^2} + {\varsigma ^2}/4 - {\beta _m}} \right)} \right]}}{{\beta _m^2 + 2{\beta _m}\left( {{\varsigma ^2}/4 - {x^2}} \right) + {{\left( {{x^2} + {\varsigma ^2}/4} \right)}^2}}}}.$$

Figure 1 shows the difference $\varepsilon \left( t \right)$ between the original Dawsons integral of real argument and its rational approximation . As we can see from this figure, despite only $12$ summation terms involved in the series approximation , the difference $\varepsilon \left( t \right)$ remains within a narrow range $ \pm 7 \times {10^{ - 9}}$. This confirms a rapid convergence of equation that makes it suitable for practical applications. Therefore, we can use it effectively as a supplement for computation of the Voigt function at small $y <  < 1$.

![image](fig1.pdf){width="24pc"}

Since at $y <  < 1$ the following functions can be approximated as ${e^{{y^2}}} \approx 1$, $\cos \left( {2xy} \right) \approx 1$, ${\rm{sinc}}\left( {2xy} \right) \approx 1$ and $\sin \left( {2xy} \right) \approx 2xy,$ the equation is significantly simplified as follows $$K\left( {x,y <  < 1} \right) \approx {e^{ - {x^2}}} - \frac{{2y}}{{\sqrt \pi  }}\left[ {1 - 2xF\left( x \right)} \right]$$ and according to equation it can also be represented in form $$\label{eq_15}
K\left( {x,y <  < 1} \right) \approx {e^{ - {x^2}}} - \frac{{2y}}{{\sqrt \pi  }}\left[ {1 - \sqrt \pi  x\lambda \left( {x,\varsigma /2} \right)} \right].$$

The computational time of the approximation is mostly taken by $\lambda $-function. Since the approximations and are about equally rapid, the approximation is also as fast as the approximation . Thus, we can see that implementation of the Dawsons function of the real argument does not decelerate the computation of the Voigt function at small $y <  < 1$.

Implementation
--------------

When the input parameters $x$ and $y$ are large enough, say if the condition $\left| {x + iy} \right| > 15$ is satisfied, many rational approximations become effective for computation. For example, the Gauss–Hermite quadrature, the Taylor expansion series [@Letchworth2007] or the Laplace continued fraction [@Gautschi1970; @Jones1988; @Poppe1990a] can be used. Therefore, despite that the rational approximations and can cover accurately the entire domain of practical interest $0 < x < 40,000$ and ${10^{ - 4}} < y < {10^2}$ [@Quine2002; @Wells1999] required for applications using the HITRAN molecular spectral database [@Rothman2013], we may restrict them only within the domain $\left| {x + iy} \right| \le 15$ that is considered the most difficult for rapid and accurate computation. Thus, the computation of the Voigt function inside the domain $\left| {x + iy} \right| \le 15$ can be implemented in accordance with the following scheme $$K\left( {x,y} \right) \approx \left\{ \begin{aligned}
&\kappa \left( {x,y + \varsigma /2} \right), \hspace{90pt} \left| {x + iy} \right| \le 15 \cap {10^{ - 6}} \le y \le 15\\
&{e^{ - {x^2}}} - \frac{{2y}}{{\sqrt \pi  }}\left[ {1 - \sqrt \pi  x\lambda \left( {x,\varsigma /2} \right)} \right], \hspace{5pt} \left| {x + iy} \right| \le 15 \cap 0 \le y < {10^{ - 6}}.
\end{aligned} \right.$$

Error analysis
--------------

Let us define the relative error as $$\Delta  = \left| {\frac{{K\left( {x,y} \right) - {K_{ref.}}\left( {x,y} \right)}}{{{K_{ref.}}\left( {x,y} \right)}}} \right|,$$ where ${K_{ref.}}\left( {x,y} \right)$ is the reference, to quantify the accuracy of the approximation . The highly accurate reference values can be generated, for example, by using the Algorithm 680 [@Poppe1990b], the Algorithm 916 [@Zaghloul2011] or a new algorithm descried in the recent work [@Karbach2014].

![image](fig2.pdf){width="22pc"}

Figure 2 show the logarithm of the relative error ${\log _{10}}\Delta $ for the Voigt function inside the domain $0 \le x \le 15$ and $0 \le y \le {10^{ - 6}}$. As we can see from this figure, despite that only $12$ summation terms involved in computation, the accuracy of approximation in this domain is better than ${10^{ - 10}}$.

Thus we can see that the approach based on implementation of the rational approximation for the Dawsons integral of real argument is efficient and practical.

Conclusion
==========

A rational approximation for the Dawsons integral of real argument is presented. It can be utilized as an efficient supplement for accurate computation of the Voigt function at small $y <  < 1$. In particular, this approach enables computation with accuracy exceeding ${10^{ - 10}}$ inside the domain $0 \le x \le 15$ and $0 \le y \le {10^{ - 6}}$. Due to rapid performance the algorithmic implementation of the rational approximation does not decelerate the computation of the Voigt function.

Acknowledgments {#acknowledgments .unnumbered}
===============

This work is supported by National Research Council Canada, Thoth Technology Inc. and York University. The authors wish to thank to Prof. Ian McDade and Dr. Brian Solheim for discussions and constructive suggestions.

Appendix A {#appendix-a .unnumbered}
==========

There is a simple proof of equation . The equation can be rearranged in form $$w\left( z \right) = {e^{ - {z^2}}} + \frac{{2i}}{{\sqrt \pi  }}F\left( z \right)$$ and since $${e^{ - {z^2}}} = \frac{1}{{\sqrt \pi  }}\int\limits_0^\infty  {\exp \left( { - {t^2}/4} \right)\cos \left( {zt} \right)dt},$$ $$F\left( z \right) = \frac{1}{2}\int\limits_0^\infty  {\exp \left( { - {t^2}/4} \right)\sin \left( {zt} \right)dt},$$ we can write $$\begin{aligned}
w\left( z \right) = \frac{1}{{\sqrt \pi  }}\int\limits_0^\infty  {\exp \left( { - {t^2}/4} \right)\left( {\cos \left( {zt} \right) + i\sin \left( {zt} \right)} \right)dt} \\
 = \frac{1}{{\sqrt \pi  }}\int\limits_0^\infty  {\exp \left( { - {t^2}/4} \right) {\exp \left( {izt} \right)} dt} 
\end{aligned}$$ or $$\begin{aligned}
w\left( {x,y} \right) = \frac{1}{{\sqrt \pi  }}\int\limits_0^\infty  {\exp \left( { - {t^2}/4} \right)\exp \left( {i\left( {x + iy} \right)t} \right)dt} \\
 = \frac{1}{{\sqrt \pi  }}\int\limits_0^\infty  {\exp \left( { - {t^2}/4} \right)\exp \left( { - yt} \right)\exp \left( {ixt} \right)dt} .
\end{aligned}$$ This completes the proof.

Appendix B {#appendix-b .unnumbered}
==========

Due to discontinuity at $t = x$ the functions and cannot be defined rigorously at $y = 0$. However, since the following limits exist (see for example [@Armstrong1967; @Letchworth2007; @Zaghloul2011] for details) $$\mathop {\lim }\limits_{y \to 0} K\left( {x,y} \right) = {e^{ - {x^2}}},$$ and $$\mathop {\lim }\limits_{y \to 0} L\left( {x,y} \right) = \frac{2}{{\sqrt \pi  }}F\left( {x,y = 0} \right)$$ we may imply that $K\left( {x,y = 0} \right)$ and $L\left( {x,y = 0} \right)$ are defined and equal to ${e^{ - {x^2}}}$ and $\frac{2}{{\sqrt \pi  }}F\left( {x,y = 0} \right)$, respectively.

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