Order of integration (calculus)

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For the summary statistic in time series, see Order of integration.

In calculus, interchange of the order of integration is a methodology that transforms multiple integrations of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed.

1 Problem statement
2 Relation to integration by parts
3 General examples
4 Principal-value integrals
5 Basic theorems
6 References and notes
7 Outside links

[edit] Problem statement

The problem for examination is evaluation of an integral of the form:

$\int \int_D dx \ dy \ f(x, \ y ) \ ,$

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. A major headache with this interchange is determining the change in description of the domain D.

The method also is applicable to 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.

[edit] Relation to integration by parts

Figure 1: Integration over the triangular area can be done using vertical or horizontal strips as the first step. The sloped line is the curve y = x.

Consider the double integral:

$\int_a^z \ dx\ \int_a^x \ dy \ h(y)$

In the order written above, the strip of width dx is integrated first over the y-direction as shown in the left panel of Figure 1, which is inconvenient especially when function h ( y ) is not easily integrated. The integral can be reduced to a single integration by reversing the order of integration as shown in the right panel of the figure. To accomplish this interchange of variables, the strip of width dy is first integrated from the line x = y to the limit x = z, and then the result is integrated from y = a to y = z, resulting in:

$\int_a^z \ dx\ \int_a^x \ dy \ h(y) = \int_a^z \ dy \ h(y)\ \int_y^z \ dx = \int_a^z \ dy \ \left(z-y\right) h(y) \ .$

This result can be seen to be an example of the formula for integration by parts, as stated below:^[3]

$\int_a^z f(x) g'(x)\, dx = \left[ f(x) g(x) \right]_{a}^{z} - \int_a^z f'(x) g(x)\, dx\!$

Substitute:

$g (x) = \int_a^x \ dy \ h(y) \$ and $f(x) = z-x \ .$

However, compared to using the formula for integration by parts, exchange of the order of integration has the merit that it generates the function f in a natural manner.

[edit] General examples

More complex examples of changing the order of integration can be found at Ron Miech's UCLA Calculus Problems (see Problems 33, 35, 37, 39, 41 & 43) and Duane Nykamp's University of Minnesota website. For a general introduction, see Murthy and Srinivas^[4], Widder^[5], or Johnson.^[6]

[edit] Principal-value integrals

For application to principal-value integrals, see Whittaker and Watson,^[7] , Gakhov,^[8] Lu^[9], or Zwillinger.^[10] See also the discussion of the Poincaré-Bertrand transformation in Obolashvili.^[11] An example where the order of integration cannot be exchanged is given by Kanwal:^[12]

$\frac {1}{(2\pi i )^2} \int_L^* \frac{d{\tau}_1}{{\tau}_1 - t}\ \int_L^*\ g(\tau)\frac{d \tau}{\tau-\tau_1} = \frac{1}{4} g(t) \ ,$

while:

$\frac {1}{(2\pi i )^2} \int_L^* g( \tau ) \ d \tau \left( \int_L^* \frac{d \tau_1 } {\left( \tau_1 - t\right) \left( \tau-\tau_1 \right)} \right) = 0 \ .$

The second form is evaluated using a partial fraction expansion and an evaluation using the Sokhotski-Plemelj formula:^[13]

$\int_L^*\frac{d \tau_1}{\tau_1-t} = \int_L^* \frac {d\tau_1}{\tau_1-t} = \pi\ i \ .$

The notation $\int_L^*$ indicates a Cauchy principal value. See Kanwal.^[12].

[edit] Basic theorems

A good discussion of the basis for reversing the order of integration is found in Kőrner.^[14] 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:

$\int_1^{\infty} \frac {x^2-y^2}{\left(x^2+y^2\right)^2}\ dy = \left[\frac{y}{x^2+y^2}\right]_1^{\infty} = \frac{1}{1+x^2} \ \left[x \ge 1 \right]\ .$

$\int_1^{\infty} \left( \int_1^{\infty}\frac {x^2-y^2}{\left(x^2+y^2\right)^2}\ dy \right)\ dx = \frac{\pi}{4} \ .$

$\int_1^{\infty} \left( \int_1^{\infty}\frac {x^2-y^2}{\left(x^2+y^2\right)^2}\ dx \right)\ dy = -\frac{\pi}{4} \ .$

Two basic theorems governing admissibility of the interchange are quoted below from Chaudhry and Zubair:^[15]

Theorem I: Let f(x, y) be a continuous function of constant sign defined for a ≤ x < ∞, c ≤ y < ∞, and let the integrals $J(y):= \int_a^{\infty} dx \ f(x,\ y)$ and $J^*(x) = \int_c^{\infty}dy\ f(x, \ y)$ regarded as functions of the corresponding parameter be, respectively, continuous for c ≤ y < ∞, a ≤ x < ∞. Then if at least one of the iterated integrals $\int_c^{\infty}dy \ \left(\int_a^{\infty}dx\ f(x,\ y) \right )$ and $\int_a^{\infty}dx \ \left(\int_c^{\infty}dy\ f(x,\ y) \right )$ converges, the other integral also converges and their values coincide.

Theorem II: Let f(x, y) be continuous for a ≤ x < ∞, c ≤ y < ∞, and let the integrals $J(y):= \int_a^{\infty} dx \ f(x,\ y)$ and $J^*(x) = \int_c^{\infty}dy\ f(x, \ y)$ be respectively, uniformly convergent on every finite interval c ≤ y < C and on every finite interval a ≤ x < A. Then if at least one of the iterated integrals $\int_c^{\infty}dy \ \left(\int_a^{\infty}dx\ |f(x,\ y)| \right )$ and $\int_a^{\infty}dx \ \left(\int_c^{\infty}dy\ |f(x,\ y)| \right )$ converges, the iterated integrals $\int_c^{\infty}dy \ \left(\int_a^{\infty}dx\ f(x,\ y) \right )$ and $\int_a^{\infty}dx \ \left(\int_c^{\infty}dy\ f(x,\ y) \right )$ also converge and their values are equal.

The most important theorem for the applications is quoted from Protter and Morrey:^[16]

Suppose F is a region given by $F=\left\{(x,\ y):a \le x \le b, p(x) \le y \le q(x) \right\} \,$ where p and q are continuous and p(x) ≤ q(x) for a ≤ x ≤ b. Suppose that f(x, y) is continuous on F. Then $\iint_F f(x,y) dA = \int_a^b\ \int_{p(x)}^{q(x)} f(x,\ y)dy\ dx \ .$
The corresponding result holds if the closed region F has the representation $F=\left\{(x,\ y):c\le y \le d,\ r(y) \le x \le s(y)\right\}$ where r(y) ≤ s(y) for c ≤ y ≤ d. In such a case, $\iint_F f(x,\ y) dA = \int_c^d \ \int_{r(x)}^{s(x)} f(x,\ y) dx\ dy \ .$ In other words, both iterated integrals, when computable, are equal to the double integral and therefore equal to each other.