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

This work focuses on estimating the number of terms of a Fourier-Bessel series of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p{\prime}$</tex> required to get within a certain error of a Bessel function of a fixed order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p$</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p\neq p{\prime}$</tex>. Our approach consists of two steps: one, constructing an invariant over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n^{\mathrm{t}\mathrm{h}}$</tex> order Hankel transform; and two, observing the effect of expanding a suitably scaled Bessel function of a fixed order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p$</tex> in its Fourier-Bessel series of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p{\prime}$</tex>. 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.