Uniform asymptotic formula for the Aharonov Bohm wavefield

Abstract

A uniform asymptotic formula for the Aharonov–Bohm wavefield (that of a plane quantum wave scattered by a thin straight solenoid) far away from the solenoid is obtained in a direct way. Actually quite good accuracy is achieved even down to one wavelength away. The error is numerically of order radius^(−3/2) for all values of polar angle, including directly forwards. Several previous formulas, uniform and otherwise, for the far field limit exist in the literature. All contain the essential ingredient: the Fresnel integral (complex error function), but ordinarily the error in these formulas is of order radius^(−1/2) in the forwards direction where the Fresnel contribution is most important.