Efficient application of the Voigt functions in the Fourier transform
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912445719642112 |
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| author | Abrarov, Sanjar M. Siddiqui, Rehan Jagpal, Rajinder K. Quine, Brendan M. |
| author_facet | Abrarov, Sanjar M. Siddiqui, Rehan Jagpal, Rajinder K. Quine, Brendan M. |
| contents | In this work, we develop a method for rational approximation of the Fourier transform (FT) based on the real and imaginary parts of the complex error function \[ w(z) = e^{-z^2}(1 - {\rm{erf}}(-iz)) = K(x,y) + iL(x,y), \qquad z = x + iy, \] where $K(x,y)$ and $L(x,y)$ are known as the Voigt and imaginary Voigt functions, respectively. In contrast to our previous rational approximation of the FT, the expansion coefficients in this method are not dependent on the values of a sampled function. As the values of the Voigt functions remain the same, this approach can be used for rapid computation with help of look-up tables. Mathematica codes with some examples are presented. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_21003 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Efficient application of the Voigt functions in the Fourier transform Abrarov, Sanjar M. Siddiqui, Rehan Jagpal, Rajinder K. Quine, Brendan M. General Mathematics 42A38, 65D15 In this work, we develop a method for rational approximation of the Fourier transform (FT) based on the real and imaginary parts of the complex error function \[ w(z) = e^{-z^2}(1 - {\rm{erf}}(-iz)) = K(x,y) + iL(x,y), \qquad z = x + iy, \] where $K(x,y)$ and $L(x,y)$ are known as the Voigt and imaginary Voigt functions, respectively. In contrast to our previous rational approximation of the FT, the expansion coefficients in this method are not dependent on the values of a sampled function. As the values of the Voigt functions remain the same, this approach can be used for rapid computation with help of look-up tables. Mathematica codes with some examples are presented. |
| title | Efficient application of the Voigt functions in the Fourier transform |
| topic | General Mathematics 42A38, 65D15 |
| url | https://arxiv.org/abs/2504.21003 |