Note on the principle of stationary phase

✍ Scribed by R.N. Bhattacharya; Iva Basu

• Publisher: Elsevier Science
• Year: 1979
• Tongue: English
• Weight: 306 KB
• Volume: 16
• Category: Article
• ISSN: 0010-4655
• DOI: 10.1016/0010-4655(79)90085-7

✦ Synopsis

The principle of stationary phase as enunciated by Kelvin [11in 1887 concerns the approximate evaluation of an integral of a rapidly fluctuating function. The well-known formula is derived on the tacit assumption of the amplitude function ~(x) being non-zero at the isolated stationary point o of the phase function f(x). The present note points out that the vanishing of ~(x) at this point alters the order of approximation. The significant contribution to the integral comes in this case from the first non-vanishing even order derivative of ~(x) at x = o and the corresponding formula for the approximate value has been worked out. The formula has been further generalized for the case when the first non-vanishing derivatives of~andf are /~(cs)andfm( 0~) respectively, m, n being both positive integers.