Investigation into the role of the Bessel function order in the Fourier-Bessel series and the Hankel Transform

Suketu P Patni; Vikram M Gadre

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arXiv · 2503.17722

Investigation into the role of the Bessel function order in the Fourier-Bessel series and the Hankel Transform

Suketu P Patni·Vikram M Gadre

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Abstract

This work focuses on estimating the number of terms of a Fourier-Bessel series of order $p'$ required to get within a certain error of a Bessel function of a fixed order $p$ where $p \neq p'$. Our approach consists of two steps: one, constructing an invariant over $n$ of the $n^{\text{th}}$ order Hankel transform; and two, observing the effect of expanding a suitably scaled Bessel function of a fixed order $p$ in its Fourier-Bessel series of order $p'$. We demonstrate a new error metric to simplify the error computations. Further, we generate an empirical model using numerical simulations and examine its capabilities in predicting the number of terms required.

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Suketu P Patni, Vikram M Gadre. 2025-03-22. Investigation into the role of the Bessel function order in the Fourier-Bessel series and the Hankel Transform. https://arxiv.org/abs/2503.17722

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