Abstract
Consider the double integral
I(λ, α) = ∫∫Dg(x, y, α)eiλƒ(x, y, α) dxdy,
where λ is a large positive variable and a is an auxiliary parameter. We consider the case in which the phase function ƒ(x, y, α) has two simple stationary points (x+(α), y+(α)) and (x_(α), y_(α)) in D, which coalesce at a point (x0, y0) as a approaches a critical value α0. The point (x0, y0) can either be an interior point of D or a boundary point of D. Asymptotic expansions are derived in both cases, which hold uniformly in a neighbourhood of α0. Our derivation is mathematically rigorous.
© 2000 The Royal Society.
I(λ, α) = ∫∫Dg(x, y, α)eiλƒ(x, y, α) dxdy,
where λ is a large positive variable and a is an auxiliary parameter. We consider the case in which the phase function ƒ(x, y, α) has two simple stationary points (x+(α), y+(α)) and (x_(α), y_(α)) in D, which coalesce at a point (x0, y0) as a approaches a critical value α0. The point (x0, y0) can either be an interior point of D or a boundary point of D. Asymptotic expansions are derived in both cases, which hold uniformly in a neighbourhood of α0. Our derivation is mathematically rigorous.
© 2000 The Royal Society.
| Original language | English |
|---|---|
| Pages (from-to) | 407-431 |
| Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |
| Volume | 456 |
| Issue number | 1994 |
| DOIs | |
| Publication status | Published - 8 Feb 2000 |
Research Keywords
- Double integral
- Incomplete airy function
- Stationary point
- Uniform asymptotic expansion
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