View source for Longman method

A method for the approximate calculation of a definite integral

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l0608202.png" /> has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l0608203.png" /> roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l0608204.png" /> inside the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l0608205.png" />,

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and satisfies the conditions stated below. Let

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then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l0608208.png" />, where

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It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082010.png" /> preserves its sign on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082011.png" />, has different signs on adjacent intervals, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082013.png" />. Such a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082014.png" /> is said to be oscillatory. The calculation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082015.png" /> by means of a quadrature formula for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082016.png" /> is difficult, since a good approximation of an oscillatory function on the whole interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082017.png" /> is impossible in practice. The use of the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082018.png" /> leads to the need to calculate all integrals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082019.png" />, which is also inadvisable in the case of large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082020.png" />.

The approximate calculation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082021.png" /> in Longman's method is based on the equality (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082022.png" />)

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In (1) the finite differences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082025.png" /> as functions of the discrete argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082026.png" /> occur:

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082030.png" /> is such that on the right-hand side of (1) one can neglect terms containing finite differences of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082031.png" />, then the approximate equality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

can be used to calculate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082033.png" />. To calculate the right-hand side of (2) it is sufficient to know the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082034.png" /> values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082035.png" />, that is, the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082036.png" />, and the last <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082037.png" /> values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082038.png" />. Longman's method consists in the use of (2) for an approximate calculation of the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082039.png" />.

If in the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082040.png" /> the upper limit of integration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060820/l06082041.png" /> 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 [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Longman,   "A method for the numerical evaluation of finite integrals of oscillatory functions"  ''Math. Comput.'' , '''14''' :  69  (1960)  pp. 53–59</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.J. Davis,   P. Rabinowitz,   "Methods of numerical integration" , Acad. Press  (1984)</TD></TR></table>