Improper integral

2020 Mathematics Subject Classification: Primary: 28A06 [MSN][ZBL]

Definition

The term usually denotes a limiting process which yields a definition of integral of an unbounded function or of a function over an unbounded set, even when the function is not summable.

Assume that $f$ is a function defined on an half-open interval $[a, b[\subset \mathbb R$, where $b$ is allowed to take the value $+\infty$. If $f$ is Riemann- (or Lebesgue-) integrable on every interval $[a, \beta]\subset [a,b]$ and the limit \[ \lim_{\beta\uparrow b} \int_a^b f(x)\, dx \] exists, then such limit is called the improper integral of $f$ over $[a,b[$. If the limit exists and is finite, then one says that the improper integral converges and if not, that it diverges. A similar definition is possible for the cases $]a,b]$ and $]a,b[$. In the latter the improper integral is the sum of the limits \[ \lim_{\beta\uparrow b} \int_c^\beta f(x)\, dx \] and \[ \lim_{\alpha\downarrow a} \int_\alpha^c f(x)\, dx\, \] which are assumed to exist for some point $c\in ]a,b[$ and not to give the indeterminate form $+\infty-\infty$. Under these assumptions the result is independent of the point $c$.

Some generalizations are used even for functions defined on domains of type $]a_0, a_1[\cup ]a_1, a_2[\cup \ldots \cup ]a_{k-1}, a_k[$. In this case it is required that the improper integral exists on every separate interval and that in the resulting $k$ values $+\infty$ and $-\infty$ do not both appear.

Comparison with Riemann- and Lebesgue- integrals

If the function $f$ is Riemann-integrable over $[a,b]$, then the improper integral coincides with the Riemann integral. The same holds with the Lebesgue integral if $f$ is Lebesgue-integrable over $[a,b[$. A partial converse to the last statement holds: if $f$ is Lebesgue-measurable for every $[a,\beta]\subset [a,b[$ and the improper integral of $|f|$ exists and is finite, then $f$ is summable and the improper integral coincides with the Lebesgue integral. However the improper integral might exist even when $f$ is not summable, as it is the case of \[ \int_0^\infty \frac{\sin x}{x}\, dx\, . \]

Properties

The general properties of integrals carry over to improper integrals: linearity, additivity with respect to the intervals over which the integration proceeds, the rule for integrating inequalities, the mean-value theorems, integration by parts, change of variable, and the Newton-Leibniz formula. For example, if $f$ coincides almost-everywhere on $[a,b[$ with the derivative of a function $F$ that is absolutely continuous on every $[a,\beta]\subset [a,b[$ then \[ \int_a^b f(x)\, dx = F(b)-F(a)\, . \]

Criteria

To decide about the convergence of the indefinite integral of functions of constant sign one uses the comparison test. That is, if $0\leq f\leq g$ and the improper integral of $g$ converges, then so does the improper integral of $f$.

A useful general criterion is that of Cauchy: the improper integral of $f$ on $[a,b[$ converges if and only if for every $\varepsilon>0$ $\exists \eta\in [a,b[$ such that \[ \left|\int_\alpha^\beta f(x)\, dx\right| < \varepsilon \qquad \forall \beta>\alpha> \eta\, . \]

The convergence of the improper integral can be turned into deciding the convergence of certain series: the improper integral of $f$ over $[a,b[$ converges if and only if for every sequence $b_n \uparrow b$ the corresponding series \[ \sum_{i=1}^\infty \int_{b_{i-1}}^{b_i} f(x)\, dx \] converges.

Higher dimensions and Cauchy principal value

The concept of improper integral can be generalized to integrals of several variables. However such generalization hinges on deciding for a given domain $\Omega$ in which way it should be approximated by a sequence of cannonical domains and this is not so clear in more than one variable. Moreover, the fact that the higher-dimensional version of the Riemann integral is quite involved has made some definitions of improper integral seldomly used.

A popular version of integrating functions with a point singularity, which is of uttermost importance in potential theory, harmonic analysis and partial differential equations, leads to the Cauchy principal value. Assume $f: \Omega \to \mathbb R$ is a function which is Lebesgue integrable on $\Omega\setminus B_r (x_0)$ for any $r>0$. The Cauchy principal value of the integral of $f$ over $\Omega$, which is denoted by \[ {\rm PV}\, \int_\Omega f \] is given by the limit \[ \lim_{r\downarrow 0} \int_{\Omega \setminus B_r (x_0)} f \] (when it exists).

However the Cauchy principal value is rarely called improper integral, especially in one space dimension. In fact, if we consider the function $\frac{1}{x}$ on $]-1,1[$, given its symmetry it is obvious that \[ {\rm PV}\, \int_{-1}^1 \frac{1}{x}\, dx = 0\, . \] On the other hand most authors say that the improper integral of $\frac{1}{x}$ does not exist, since for the improper integral of $f$ to be well defined for a function which is singular at $0$ it is usually required that both limits \[ \lim_{\alpha\downarrow 0} \int_\alpha^1 f(x)\, dx \] and \[ \lim_{a\uparrow 0} \int_{-1}^a f(x)\, dx \] exist and their sum gives not the indeterminacy $+\infty-\infty$.

References

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