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One way is to think of Bessel functions as some sort of non-abelian analogues of the exponential functionsfunction. In fact, you can form a complete orthogonal system of $L^2 (\mathbb{R^+}, x^{-1}d x)$ using the Bessel functions. See Chapter 16 of Iwaniec-Kowalski (page 411).

For manipulations, the formulae given in Appendix B of Iwaniec's Spectral Methods book usually suffice.

Yes, Dan Brown is more appropriate than Gradhsteyn and Rizhik for airport reading.

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One way is to think of Bessel functions as some sort of non-abelian analogues of exponential functions. In fact, you can form a complete orthogonal system of $L^2 (\mathbb{R^+}, x^{-1}d x)$ using the Bessel functions. See Chapter 16 of Iwaniec-Kowalski (page 411).

For manipulations, the formulae given in Appendix B of Iwaniec's Spectral Methods book usually suffice.

Yes, Dan Brown is more appropriate than Gradhsteyn and Rizhik for airport reading.