In calculus, interchange of the order of integration is a methodology that transforms iterated integrals (or multiple integrals through the use of Fubini's theorem) of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. In some cases, the order of integration can be validly interchanged; in others it cannot.
The problem for examination is evaluation of an integral of the form
\iintD f(x,y) dxdy,
where D is some two-dimensional area in the xy–plane. For some functions f straightforward integration is feasible, but where that is not true, the integral can sometimes be reduced to simpler form by changing the order of integration. The difficulty with this interchange is determining the change in description of the domain D.
The method also is applicable to other multiple integrals.[1] [2]
Sometimes, even though a full evaluation is difficult, or perhaps requires a numerical integration, a double integral can be reduced to a single integration, as illustrated next. Reduction to a single integration makes a numerical evaluation much easier and more efficient.
Consider the iterated integral
| z | |
| \int | |
| a |
| x | |
| \int | |
| a |
h(y)dydx,
| z | |
| \int | |
| a |
| x | |
| \int | |
| a |
h(y) dy dx=
| z | |
| \int | |
| a |
h(y) dy
| z | |
| \int | |
| y |
dx=
| z | |
| \int | |
| a |
\left(z-y\right)h(y)dy.
This result can be seen to be an example of the formula for integration by parts, as stated below:[4]
an introduction to the general theory of infinite processes and of analytic functions, with an account of the principal transcendental functions
. Edmund Taylor Whittaker. E. T. Whittaker. George Neville Watson. G. N. Watson . §4.51, p. 75 . 0-521-58807-3 . 1927 . Cambridge University Press . 4th ed., repr .| z | |
| \int | |
| a |
f(x)g'(x)dx=\left[f(x)g(x)
| z | |
| \right] | |
| a |
-
| z | |
| \int | |
| a |
f'(x)g(x)dx
Substitute:
f(x)=
| x | |
| \int | |
| a |
h(y)dy~and~g'(x)=1.
Which gives the result.
For application to principal-value integrals, see Whittaker and Watson,[4] Gakhov,[5] Lu,[6] or Zwillinger.[7] See also the discussion of the Poincaré-Bertrand transformation in Obolashvili.[8] An example where the order of integration cannot be exchanged is given by Kanwal:[9]
| 1 | |
| (2\pii)2 |
| * | |
| \int | |
| L |
| d{\tau | |
| 1}{{\tau} |
1-
| ||||
| t} \int | ||||
| L |
=
| 1 | |
| 4 |
g(t) ,
while:
| 1 | |
| (2\pii)2 |
| * | |
| \int | |
| L |
g(\tau) d\tau\left(
| * | |
| \int | |
| L |
| d\tau1 | |
| \left(\tau1-t\right)\left(\tau-\tau1\right) |
\right)=0 .
The second form is evaluated using a partial fraction expansion and an evaluation using the Sokhotski - Plemelj formula:[10]
| ||||
| \int | ||||
| L |
=
| * | |
| \int | |
| L |
| d\tau1 | |
| \tau1-t |
=\pi i .
The notation
| * | |
| \int | |
| L |
A discussion of the basis for reversing the order of integration is found in the book Fourier Analysis by T.W. Körner.[11] He introduces his discussion with an example where interchange of integration leads to two different answers because the conditions of Theorem II below are not satisfied. Here is the example:
| infty | |
| \int | |
| 1 |
| x2-y2 | |
| \left(x2+y2\right)2 |
dy=\left[
| y | |
| x2+y2 |
| infty | |
| \right] | |
| 1 |
=-
| 1 | |
| 1+x2 |
\left[x\ge1\right] .
| infty | |
| \int | |
| 1 |
\left(
| infty | |
| \int | |
| 1 |
| x2-y2 | |
| \left(x2+y2\right)2 |
dy\right) dx=-
| \pi | |
| 4 |
.
| infty | |
| \int | |
| 1 |
\left(
| infty | |
| \int | |
| 1 |
| x2-y2 | |
| \left(x2+y2\right)2 |
dx\right) dy=
| \pi | |
| 4 |
.
Two basic theorems governing admissibility of the interchange are quoted below from Chaudhry and Zubair:[12]
The most important theorem for the applications is quoted from Protter and Morrey:[13]
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