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In the Bateman Manuscript Project's Table of Integral Transforms, there are several identities for Y-transforming (or similarly but more familiarly Hankel-transforming) certain special functions with exponential prefactors. Because the combination we (collaborative we, or should it be conspiratorial we?) wish to transform does not appear, we are searching for a reference in which one of these results quoted in the Table of Integral Transforms is derived so that perhaps we could adapt to our situation. If anyone knows of such a source, we would be very grateful.

Specifically, this is the transform we want to compute:

$$\int_0^\infty Y_\nu (wy) \left( wy \right)^{1/2} H_\nu \left( zy \right) e^{-i a y^2} \, dy$$

Here are some similar results from the Table of Integral Transforms.

(These results are for Hankel transform, and are on p.51 of Table of Integral Transforms - functions on the right are the Hankel transforms of those at left. The second pair of formulas of course assembles via Euler's formula into a version of the first formula where the coefficient in the exponential is imaginary):

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(For Y-transforms there is a slew of vaguely similar results e.g. this one from p.113, although none have references and most are problematic in that the arguments of the special function and the exponential are related in ways ours are not):

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(We can of course regard our transform as an H-transform instead of a Y-transform, and there the situation is similar in that related results appear without reference in the Table of Integral Transforms.)

If anyone can point us toward where such results originate and what methods are used for their derivation, that would be wonderful.

ETA: The Table of Integral Transforms can be found online here thanks to Caltech: https://authors.library.caltech.edu/43489/

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