 Fresnel tail expansion

ID: fresnel-tail-expansion

For

A = π /8

, define real tail functions by

Q + i P = e^{- i λ^{2}} \int_{λ}^{\infty} e^{i t^{2}} d t

. Then

C = A + P sin λ^{2} - Q cos λ^{2}

and

S = A - P cos λ^{2} - Q sin λ^{2}

P \sim \sum_{m \geq 0} \frac{( - 1 ) ^{m} ( 4 m - 1 )!!}{2 ^{2 m + 1} λ ^{4 m + 1}}, Q \sim \sum_{m \geq 0} \frac{( - 1 ) ^{m} ( 4 m + 1 )!!}{2 ^{2 m + 2} λ ^{4 m + 3}},

where

(- 1)!! = 1

. These are asymptotic expansions, rather than convergent infinite sums at a fixed argument.

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