Longman method
A method for the approximate calculation of a definite integral
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where has exactly
roots
inside the interval
,
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and satisfies the conditions stated below. Let
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then , where
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It is assumed that preserves its sign on the interval
, has different signs on adjacent intervals, and
,
. Such a function
is said to be oscillatory. The calculation of
by means of a quadrature formula for large
is difficult, since a good approximation of an oscillatory function on the whole interval
is impossible in practice. The use of the equality
leads to the need to calculate all integrals
, which is also inadvisable in the case of large
.
The approximate calculation of in Longman's method is based on the equality (
)
![]() | (1) |
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In (1) the finite differences of as functions of the discrete argument
occur:
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If is such that on the right-hand side of (1) one can neglect terms containing finite differences of order
, then the approximate equality
![]() | (2) |
can be used to calculate . To calculate the right-hand side of (2) it is sufficient to know the first
values
, that is, the values
, and the last
values
. Longman's method consists in the use of (2) for an approximate calculation of the sum
.
If in the integral the upper limit of integration
and
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then instead of (1) one must use the equality
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(the Euler transform) and replace the series on the right-hand side by a partial sum.
The method was proposed by I.M. Longman [1].
References
[1] | I.M. Longman, "A method for the numerical evaluation of finite integrals of oscillatory functions" Math. Comput. , 14 : 69 (1960) pp. 53–59 |
[2] | P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984) |
Longman method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Longman_method&oldid=16230