The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving <sub>1</sub><i>F</i><sub>2</sub> Hypergeometric Functions That Arise from Them
The Bessel function of the first kind <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>J</mi><mi>N</mi></msub><mfenced separators="" open="(&q...
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MDPI AG
2024-02-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/13/2/134 |
| _version_ | 1797298923707039744 |
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| author | Jack C. Straton |
| author_facet | Jack C. Straton |
| author_sort | Jack C. Straton |
| collection | DOAJ |
| description | The Bessel function of the first kind <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>J</mi><mi>N</mi></msub><mfenced separators="" open="(" close=")"><mi>k</mi><mi>x</mi></mfenced></mrow></semantics></math></inline-formula> is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi><mi>N</mi></msub><mfenced separators="" open="(" close=")"><mi>k</mi><mi>x</mi></mfenced></mrow></semantics></math></inline-formula>. The purpose of these expansions in Legendre polynomials was not an attempt to rival established <i>numerical</i> methods for calculating Bessel functions but to provide a form for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>J</mi><mi>N</mi></msub><mfenced separators="" open="(" close=")"><mi>k</mi><mi>x</mi></mfenced></mrow></semantics></math></inline-formula> useful for <i>analytical</i> work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo> </mo><mspace width="-0.166667em"></mspace></mrow><mn>1</mn></msub><msub><mi>F</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mfenced separators="" open="(" close=")"><msup><mn>2</mn><mi>i</mi></msup><msup><mn>3</mn><mi>j</mi></msup><msup><mn>5</mn><mi>k</mi></msup><msup><mn>7</mn><mi>l</mi></msup><msup><mn>11</mn><mi>m</mi></msup><msup><mn>13</mn><mi>n</mi></msup><msup><mn>17</mn><mi>o</mi></msup><msup><mn>19</mn><mi>p</mi></msup></mfenced></mrow></semantics></math></inline-formula> multiplying powers of the coefficient <i>k</i>. |
| first_indexed | 2024-03-07T22:43:06Z |
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| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-07T22:43:06Z |
| publishDate | 2024-02-01 |
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| record_format | Article |
| series | Axioms |
| spelling | doaj.art-ec494ca71b45423399ef5cb18ba6cab92024-02-23T15:07:31ZengMDPI AGAxioms2075-16802024-02-0113213410.3390/axioms13020134The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving <sub>1</sub><i>F</i><sub>2</sub> Hypergeometric Functions That Arise from ThemJack C. Straton0Department of Physics, Portland State University, Portland, OR 97207-0751, USAThe Bessel function of the first kind <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>J</mi><mi>N</mi></msub><mfenced separators="" open="(" close=")"><mi>k</mi><mi>x</mi></mfenced></mrow></semantics></math></inline-formula> is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi><mi>N</mi></msub><mfenced separators="" open="(" close=")"><mi>k</mi><mi>x</mi></mfenced></mrow></semantics></math></inline-formula>. The purpose of these expansions in Legendre polynomials was not an attempt to rival established <i>numerical</i> methods for calculating Bessel functions but to provide a form for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>J</mi><mi>N</mi></msub><mfenced separators="" open="(" close=")"><mi>k</mi><mi>x</mi></mfenced></mrow></semantics></math></inline-formula> useful for <i>analytical</i> work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo> </mo><mspace width="-0.166667em"></mspace></mrow><mn>1</mn></msub><msub><mi>F</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mfenced separators="" open="(" close=")"><msup><mn>2</mn><mi>i</mi></msup><msup><mn>3</mn><mi>j</mi></msup><msup><mn>5</mn><mi>k</mi></msup><msup><mn>7</mn><mi>l</mi></msup><msup><mn>11</mn><mi>m</mi></msup><msup><mn>13</mn><mi>n</mi></msup><msup><mn>17</mn><mi>o</mi></msup><msup><mn>19</mn><mi>p</mi></msup></mfenced></mrow></semantics></math></inline-formula> multiplying powers of the coefficient <i>k</i>.https://www.mdpi.com/2075-1680/13/2/134Bessel functionsFourier–Legendre seriesLaplace seriesgeneralized hypergeometric functionspolynomial approximationscomputational methods |
| spellingShingle | Jack C. Straton The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving <sub>1</sub><i>F</i><sub>2</sub> Hypergeometric Functions That Arise from Them Axioms Bessel functions Fourier–Legendre series Laplace series generalized hypergeometric functions polynomial approximations computational methods |
| title | The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving <sub>1</sub><i>F</i><sub>2</sub> Hypergeometric Functions That Arise from Them |
| title_full | The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving <sub>1</sub><i>F</i><sub>2</sub> Hypergeometric Functions That Arise from Them |
| title_fullStr | The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving <sub>1</sub><i>F</i><sub>2</sub> Hypergeometric Functions That Arise from Them |
| title_full_unstemmed | The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving <sub>1</sub><i>F</i><sub>2</sub> Hypergeometric Functions That Arise from Them |
| title_short | The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving <sub>1</sub><i>F</i><sub>2</sub> Hypergeometric Functions That Arise from Them |
| title_sort | fourier legendre series of bessel functions of the first kind and the summed series involving sub 1 sub i f i sub 2 sub hypergeometric functions that arise from them |
| topic | Bessel functions Fourier–Legendre series Laplace series generalized hypergeometric functions polynomial approximations computational methods |
| url | https://www.mdpi.com/2075-1680/13/2/134 |
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