# Improper integral

2010 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

 [Ap1] T. M. Apostol, "Calculus" , 1–2 , Blaisdell (1969) [Ap1] T. M. Apostol, "Mathematical analysis" , Addison-Wesley (1963) [Ru1] R.C. Buck, "Advanced calculus" , McGraw-Hill (1965) [Ru1] G.H. Hardy, "A course of pure mathematics" , Cambridge Univ. Press (1975) [IP] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian) [Ku] L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) (In Russian) MR1617334 MR1070567 MR1070566 MR1070565 MR0866891 MR0767983 MR0767982 MR0628614 MR0619214 Zbl 1080.00002 Zbl 1080.00001 Zbl 1060.26002 Zbl 0869.00003 Zbl 0696.26002 Zbl 0703.26001 Zbl 0609.00001 Zbl 0632.26001 Zbl 0485.26002 Zbl 0485.26001 [Nik] S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian) Zbl 0397.00003 Zbl 0384.00004 [Ru] W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98 MR0210528 Zbl 0142.01701 [Sch] L. Schwartz, L. Schwartz, "Méthodes mathématiques pour les sciences physiques" , Hermann (1965) [Sh] G.E. Shilov, "Mathematical analysis" , 1–2 , M.I.T. (1974) (Translated from Russian) [Roy] G. Valiron, "Théorie des fonctions" , Masson (1948) [Zaa] A.C. Zaanen, "Integration" , North-Holland (1967) MR0222234 Zbl 0175.05002
How to Cite This Entry:
Improper integral. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Improper_integral&oldid=29898
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article