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Year 2022, Volume: 5 Issue: 4, 266 - 272, 01.12.2022
https://doi.org/10.33401/fujma.1166846

Abstract

References

  • [1] F. Olver, D. Lozier, R. Boisvert, C. Clark (Eds.), Handbook of Mathematical Functions, Cambridge University Press, 2010.
  • [2] G. Matviyenko, On the evaluation of Bessel functions,Appl. Comput. Harmon. Anal., 1 (1993), 116–135. https://doi.org/10.1006/acha.1993.1009
  • [3] C. Schwartz, Numerical calculation of Bessel functions, Int. J. Mod. Phys. C, 23(12) (2012), 1250084. https://doi.org/10.1142/S0129183112500842
  • [4] J. Bremer, An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments, (2017), arXiv https://arxiv.org/abs/1705.07820
  • [5] E. A. Karatsuba, Calculation of Bessel functions via the summation of series, Numer. Analys. Appl., 12 (2019), 372–387. https://doi.org/10.1134/S1995423919040050
  • [6] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover Publ. New York, 1965.
  • [7] F. B. Gross, New approximations to J0 and J1 Bessel functions, IEEE transactions on antennas and propagation, 43(8) (1995), 904–907. https://doi.org/10.1109/8.402217
  • [8] L. Li, F. Li, F. B. Gross, A new polynomial approximation for Jn Bessel functions,Appl. Math. Comput., 183 (2006), 1220–1225. https://doi.org/10.1016/j.amc.2006.06.047
  • [9] V. Barsan, S. Cojocaru, Physical applications of a simple approximation of Bessel functions of integer order, Eur. J. Phys., 28(5) (2007), 983. https://doi.org/10.1088/0143-0807/28/5/021
  • [10] H. E. Fettis, Numerical calculation of certain definite integrals by Poisson’s summation formula, Math. Tables Aids. Comp., 9 (1955), 85–92. https://doi.org/10.2307/2002063
  • [11] W. Rehwald, Representation of Bessel functions through trigonometric functions or power series and asymtotic formulas with complex arguments (in German), Frequenz, 21 (1967), 321–328. https://doi.org/10.1515/FREQ.1967.21.10.320
  • [12] R. A. Waldron, Formulas for computation of approximate values of some Bessel functions, Proc. of the IEEE 69(12) (1981) 1586–1588. https://doi.org/10.1109/PROC.1981.12211
  • [13] N. M. Blachman, S. H. Mousavinezhad, Trigonometric Approximations for Bessel Functions, IEEE Transactions on Aerospace and Electronic Systems AES-22(1) (1986). https://doi.org/10.1109/TAES.1986.310686
  • [14] M. T. Abuelma’atti, Trigonometric approximations for some Bessel functions, Active and Passive Elec. Comp., 22 (1999), 75–85. https://downloads.hindawi.com/journals/apec/1999/015810.pdf
  • [15] A. Al-Jarrah, K. M. Dempsey, M. L. Glasser, Generalized series of Bessel functions, J. Comp. Appl. Math., 143 (2002), 1–8. https://doi.org/10.1016/S0377-0427(01)00505-2
  • [16] A. H. Stroud, J. P. Kohli, Computation of Jn by numerical integration, Commun. of the ACM, 12(4) (1969), 236–238. https://doi.org/10.1145/362912.362942
  • [17] P. Baratella, M. Garetto, G. Vinardi, Approximation of the Bessel function Jn (x) by numerical integration, J. Comp. Appl. Math., 7(2) (1981), 87–91. https://www.sciencedirect.com/science/article/pii/0771050X81900401
  • [18] L. N. Trefethen, J. A. C. Weideman, The exponentially convergent trapezoidal rule, SIAM Review, 56(3) (2014), 385–458. https://doi.org/10.1137/130932132
  • [19] L. G. Molinari, Some Neumann-Bessel series and the Laplacian on polygons, J. Math. Phys., 62(5) (2021), 053502, 7 pages. https://doi.org/10.1063/5.0037872
  • [20] H. Jelitto, Generalized trigonometric power sums covering the full circle, J. Appl. Math. Phys., 10 (2022), 405–414. https://doi.org/10.4236/jamp.2022.102031
  • [21] K. S. Williams and Z. Nan-Yue, Evaluation of two trigonometric sums, Math. Slovaca, 44(5) (1994) 575–583. http://dml.cz/dmlcz/136630.
  • [22] B. C. Berndt and B. P. Yeap, Explicit evaluations and reciprocity theorems for finite trigonometric sums, Adv. Appl. Math. 29 (2002) 358–385. https://www.sciencedirect.com/science/article/pii/S0196885802000209
  • [23] A. Gervois and M. L. Mehta, Some trigonometric identities encountered by McCoy and Orrick, J. Math. Phys. 36 (1995) 5098–5109. https://doi.org/10.1063/1.531345
  • [24] B. C. Berndt, S. Kim and A. Zaharescu, Exact evaluations and reciprocity theorems for finite trigonometric sums, arXiv:2210.00180[math.NT] (2022). https://doi.org/10.48550/arXiv.2210.00180
  • [25] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, Vol. 1: Elementary functions, Gordon and Breach, 1986.
  • [26] C. M. da Fonseca, M. L. Glasser, V. Kowalenko, Basic trigonometric power sums with applications, The Ramanujan Journal, 42(2) (2017), 401–428. https://link.springer.com/article/10.1007/s11139-016-9778-0
  • [27] M. Merca, A note on cosine power sums, Journal of Integer Sequences, 15 (2012), Article ID 12.5.3. https://cs.uwaterloo.ca/journals/JIS/VOL15/Merca1/merca6.pdf

A note on trigonometric approximations of Bessel functions of the first kind, and trigonometric power sums

Year 2022, Volume: 5 Issue: 4, 266 - 272, 01.12.2022
https://doi.org/10.33401/fujma.1166846

Abstract

I reconsider the approximation of Bessel functions with finite sums of trigonometric functions, in the light of recent evaluations of Neumann-Bessel series with trigonometric coefficients. A proper choice of angle allows for an efficient choice of the trigonometric sum. Based on these series, I also obtain straightforward non-standard evaluations of new parametric sums with powers of cosine and sine functions.

References

  • [1] F. Olver, D. Lozier, R. Boisvert, C. Clark (Eds.), Handbook of Mathematical Functions, Cambridge University Press, 2010.
  • [2] G. Matviyenko, On the evaluation of Bessel functions,Appl. Comput. Harmon. Anal., 1 (1993), 116–135. https://doi.org/10.1006/acha.1993.1009
  • [3] C. Schwartz, Numerical calculation of Bessel functions, Int. J. Mod. Phys. C, 23(12) (2012), 1250084. https://doi.org/10.1142/S0129183112500842
  • [4] J. Bremer, An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments, (2017), arXiv https://arxiv.org/abs/1705.07820
  • [5] E. A. Karatsuba, Calculation of Bessel functions via the summation of series, Numer. Analys. Appl., 12 (2019), 372–387. https://doi.org/10.1134/S1995423919040050
  • [6] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover Publ. New York, 1965.
  • [7] F. B. Gross, New approximations to J0 and J1 Bessel functions, IEEE transactions on antennas and propagation, 43(8) (1995), 904–907. https://doi.org/10.1109/8.402217
  • [8] L. Li, F. Li, F. B. Gross, A new polynomial approximation for Jn Bessel functions,Appl. Math. Comput., 183 (2006), 1220–1225. https://doi.org/10.1016/j.amc.2006.06.047
  • [9] V. Barsan, S. Cojocaru, Physical applications of a simple approximation of Bessel functions of integer order, Eur. J. Phys., 28(5) (2007), 983. https://doi.org/10.1088/0143-0807/28/5/021
  • [10] H. E. Fettis, Numerical calculation of certain definite integrals by Poisson’s summation formula, Math. Tables Aids. Comp., 9 (1955), 85–92. https://doi.org/10.2307/2002063
  • [11] W. Rehwald, Representation of Bessel functions through trigonometric functions or power series and asymtotic formulas with complex arguments (in German), Frequenz, 21 (1967), 321–328. https://doi.org/10.1515/FREQ.1967.21.10.320
  • [12] R. A. Waldron, Formulas for computation of approximate values of some Bessel functions, Proc. of the IEEE 69(12) (1981) 1586–1588. https://doi.org/10.1109/PROC.1981.12211
  • [13] N. M. Blachman, S. H. Mousavinezhad, Trigonometric Approximations for Bessel Functions, IEEE Transactions on Aerospace and Electronic Systems AES-22(1) (1986). https://doi.org/10.1109/TAES.1986.310686
  • [14] M. T. Abuelma’atti, Trigonometric approximations for some Bessel functions, Active and Passive Elec. Comp., 22 (1999), 75–85. https://downloads.hindawi.com/journals/apec/1999/015810.pdf
  • [15] A. Al-Jarrah, K. M. Dempsey, M. L. Glasser, Generalized series of Bessel functions, J. Comp. Appl. Math., 143 (2002), 1–8. https://doi.org/10.1016/S0377-0427(01)00505-2
  • [16] A. H. Stroud, J. P. Kohli, Computation of Jn by numerical integration, Commun. of the ACM, 12(4) (1969), 236–238. https://doi.org/10.1145/362912.362942
  • [17] P. Baratella, M. Garetto, G. Vinardi, Approximation of the Bessel function Jn (x) by numerical integration, J. Comp. Appl. Math., 7(2) (1981), 87–91. https://www.sciencedirect.com/science/article/pii/0771050X81900401
  • [18] L. N. Trefethen, J. A. C. Weideman, The exponentially convergent trapezoidal rule, SIAM Review, 56(3) (2014), 385–458. https://doi.org/10.1137/130932132
  • [19] L. G. Molinari, Some Neumann-Bessel series and the Laplacian on polygons, J. Math. Phys., 62(5) (2021), 053502, 7 pages. https://doi.org/10.1063/5.0037872
  • [20] H. Jelitto, Generalized trigonometric power sums covering the full circle, J. Appl. Math. Phys., 10 (2022), 405–414. https://doi.org/10.4236/jamp.2022.102031
  • [21] K. S. Williams and Z. Nan-Yue, Evaluation of two trigonometric sums, Math. Slovaca, 44(5) (1994) 575–583. http://dml.cz/dmlcz/136630.
  • [22] B. C. Berndt and B. P. Yeap, Explicit evaluations and reciprocity theorems for finite trigonometric sums, Adv. Appl. Math. 29 (2002) 358–385. https://www.sciencedirect.com/science/article/pii/S0196885802000209
  • [23] A. Gervois and M. L. Mehta, Some trigonometric identities encountered by McCoy and Orrick, J. Math. Phys. 36 (1995) 5098–5109. https://doi.org/10.1063/1.531345
  • [24] B. C. Berndt, S. Kim and A. Zaharescu, Exact evaluations and reciprocity theorems for finite trigonometric sums, arXiv:2210.00180[math.NT] (2022). https://doi.org/10.48550/arXiv.2210.00180
  • [25] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, Vol. 1: Elementary functions, Gordon and Breach, 1986.
  • [26] C. M. da Fonseca, M. L. Glasser, V. Kowalenko, Basic trigonometric power sums with applications, The Ramanujan Journal, 42(2) (2017), 401–428. https://link.springer.com/article/10.1007/s11139-016-9778-0
  • [27] M. Merca, A note on cosine power sums, Journal of Integer Sequences, 15 (2012), Article ID 12.5.3. https://cs.uwaterloo.ca/journals/JIS/VOL15/Merca1/merca6.pdf
There are 27 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Luca Guido Molinari 0000-0002-5023-787X

Publication Date December 1, 2022
Submission Date August 25, 2022
Acceptance Date November 2, 2022
Published in Issue Year 2022 Volume: 5 Issue: 4

Cite

APA Molinari, L. G. (2022). A note on trigonometric approximations of Bessel functions of the first kind, and trigonometric power sums. Fundamental Journal of Mathematics and Applications, 5(4), 266-272. https://doi.org/10.33401/fujma.1166846
AMA Molinari LG. A note on trigonometric approximations of Bessel functions of the first kind, and trigonometric power sums. Fundam. J. Math. Appl. December 2022;5(4):266-272. doi:10.33401/fujma.1166846
Chicago Molinari, Luca Guido. “A Note on Trigonometric Approximations of Bessel Functions of the First Kind, and Trigonometric Power Sums”. Fundamental Journal of Mathematics and Applications 5, no. 4 (December 2022): 266-72. https://doi.org/10.33401/fujma.1166846.
EndNote Molinari LG (December 1, 2022) A note on trigonometric approximations of Bessel functions of the first kind, and trigonometric power sums. Fundamental Journal of Mathematics and Applications 5 4 266–272.
IEEE L. G. Molinari, “A note on trigonometric approximations of Bessel functions of the first kind, and trigonometric power sums”, Fundam. J. Math. Appl., vol. 5, no. 4, pp. 266–272, 2022, doi: 10.33401/fujma.1166846.
ISNAD Molinari, Luca Guido. “A Note on Trigonometric Approximations of Bessel Functions of the First Kind, and Trigonometric Power Sums”. Fundamental Journal of Mathematics and Applications 5/4 (December 2022), 266-272. https://doi.org/10.33401/fujma.1166846.
JAMA Molinari LG. A note on trigonometric approximations of Bessel functions of the first kind, and trigonometric power sums. Fundam. J. Math. Appl. 2022;5:266–272.
MLA Molinari, Luca Guido. “A Note on Trigonometric Approximations of Bessel Functions of the First Kind, and Trigonometric Power Sums”. Fundamental Journal of Mathematics and Applications, vol. 5, no. 4, 2022, pp. 266-72, doi:10.33401/fujma.1166846.
Vancouver Molinari LG. A note on trigonometric approximations of Bessel functions of the first kind, and trigonometric power sums. Fundam. J. Math. Appl. 2022;5(4):266-72.

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