Absolutely convergent improper integral
2020 Mathematics Subject Classification: Primary: 28A06 [MSN][ZBL]
Definition
Let $f:[a,b[\to \mathbb R$ be a function which is Riemann-integrable on every subinterval $[a,\beta]\subset [a,b[$ (where we also allow the case $b=\infty$). If \begin{equation}\label{e:abs_conv} \lim_{\beta\to b} \int_a^\beta |f(x)|\, dx < \infty\, , \end{equation} then the improper integral \[ \lim_{\beta \to b} \int_a^\beta f(x)\, dx \] exists and is finite and it is said to be absolutely convergent. Similar variants define absolutely convergent improper integrals on $]a,b]$ and $]a,b[$.
Since Riemann-integrable functions are Lebesgue integrable, it actually follows from the definition that $f$ is absolutely integrable and the improper integral of $f$ is just its Lebesgue integral: the notion of absolutely convergent improper integral is therefore seldomly used nowadays.
The existence of the improper integral does not guarantee its absolute convergence, as for instance is the case for the function $\frac{sin x}{x}$ on $]0, \infty$[.
Criteria
Since the absolute convergence of the improper integral of $f$ is guaranteed by the convergence of the improper integral of $|f|$, one can derive classical from those for the [[Improper integral|improper integral. In particular we have
Comparison principle If $g$ is Riemann-integrable, $|g|\leq f$ and the improper integral of $f$ converges, the improper integral of $g$ converges absolutely.
Cauchy criterion The improper integral of $f$ converges if and only if for every $\varepsilon >0$ there exists $\eta\in [a, b[$ such that \[ \int_\eta^\beta |g(x)|\, dx < \varepsilon \qquad \forall \beta\in ]\eta, b[\, . \]
Higher dimensions
The concept of improper integral is difficult to generalize to integrals of several variables, given a certain arbitrariness on how to approximate a general domain. Instead the absence of a canonical approximating sequence of sets is irrelevant for absolutely convergent improper integrals, since one can use the theory of Lebesgue. Indeed, consider any Lebesgue measurable set $\Omega$ and a function $f: \Omega\to \R$ which is Lebesgue integrable on a sequence of measurable sets $\Omega_k\uparrow \Omega$. If \[ \lim_{k\to\infty} \int_{\Omega_k} |f| < \infty \] then $f$ is summable and \[ \lim_{k\to\infty} \int_{\Omega_k} f \] is the Lebesgue integral of $f$ over $\Omega$ (and hence does not depend on the particular sequence $\Omega_k$).
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 |
Absolutely convergent improper integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolutely_convergent_improper_integral&oldid=29917