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A method for the approximate calculation of a definite integral

$$ 
I  =  \int\limits _ { a } ^ { b }  f ( x)  dx ,
$$

where  $  f $
has exactly  $  n $
roots  $  x _ {i} $
inside the interval  $  [ a, b] $,

$$ 
x _ {0}  =  a  <  x _ {1}  < \dots <  x _ {n}  <  b  =  x _ {n+1} ,
$$

and satisfies the conditions stated below. Let

$$ 
v _ {i}  =  ( - 1 )  ^ {i} \int\limits _ {x _ {i} } ^ { {x _ i+1} } f( x)  dx ,\ \ 
i = 0 \dots n ,
$$

then  $  I = S $, 
where

$$ 
S  =  \sum_{j=0}^ { n }  ( - 1 )  ^ {j} v _ {j} .
$$

It is assumed that  $  f $
preserves its sign on the interval  $  [ x _ {i} , x _ {i+1} ] $, 
has different signs on adjacent intervals, and  $  v _ {i} \neq 0 $, 
$  i = 0 \dots n $. 
Such a function  $  f $
is said to be oscillatory. The calculation of  $  I $
by means of a quadrature formula for large  $  n $
is difficult, since a good approximation of an oscillatory function on the whole interval  $  [ a , b ] $
is impossible in practice. The use of the equality  $  I = S $
leads to the need to calculate all integrals  $  v _ {j} $, 
which is also inadvisable in the case of large  $  n $.

The approximate calculation of  $  I $
in Longman's method is based on the equality ( $  n \geq  p $)

$$ \tag{1 }
S  =  \sum_{k=0}^ { p-1} ( - 1 )  ^ {k} 2 ^ {- k - 1 } \Delta  ^ {k} v _ {0} +
$$

$$ 
+ 
( - 1)  ^ {n} \sum_{k=0}^ { p-1} 2 ^ {- k - 1 } \Delta  ^ {k} v _ {n-k+2}  ^ {-p} ( - 1 )  ^ {p} \sum_{k=0}^{n-p} ( - 1 )  ^ {k} \Delta  ^ {p} v _ {k} .
$$

In (1) the finite differences of  $  v _ {j} $
as functions of the discrete argument  $  j $
occur:

$$ 
\Delta v _ {j}  =  v _ {j+1} - v _ {j} ,\  j= 0 \dots n- 1,
$$

$$ 
\Delta  ^ {r+1} v _ {j}  =  \Delta  ^ {r} v _ {j+1} - \Delta  ^ {r} v _ {j} ,
$$

$$ 
r  =  1 \dots p- 1; \  j  =  0 \dots n- r- 1 .
$$

If  $  v _ {j} $
is such that on the right-hand side of (1) one can neglect terms containing finite differences of order  $  p $, 
then the approximate equality

$$ \tag{2 }
S  \cong  \sum_{k=0}^{p-1} ( - 1 )  ^ {k} 2 ^ {- k - 1 } \Delta  ^ {k} v _ {0} + ( - 1 )  ^ {n} \sum _ {k=0}^ { p-1} 2 ^ {- k - 1 } \Delta  ^ {k} v _ {n-k}
$$

can be used to calculate  $  S $. 
To calculate the right-hand side of (2) it is sufficient to know the first  $  p $
values  $  v _ {j} $, 
that is, the values  $  v _ {0} \dots v _ {p-1} $, 
and the last  $  p $
values  $  v _ {n} \dots v _ {n-p+1} $. 
Longman's method consists in the use of (2) for an approximate calculation of the sum  $  S $.

If in the integral  $  I $
the upper limit of integration  $  b = + \infty $
and

$$ 
I  =  \sum_{i=0}^  \infty   ( - 1 )  ^ {i} v _ {i} ,
$$

then instead of (1) one must use the equality

$$ 
\sum_{i=0}^  \infty   ( - 1 )  ^ {i} v _ {i}  =  \sum_{i=0}^  \infty (-1)  ^ {i} 2^{-i-1} \Delta  ^ {i} v _ {0}
$$

(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>