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In
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, an improper integral is the
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
of a
definite integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
as an endpoint of the interval(s) of integration approaches either a specified
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
or positive or negative
infinity Infinity is that which is boundless, endless, or larger than any number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything ...

infinity
; or in some instances as both endpoints approach limits. Such an integral is often written symbolically just like a standard definite integral, in some cases with ''infinity'' as a limit of integration. Specifically, an improper integral is a limit of the form: :\lim_ \int_a^bf(x)\, dx, \quad \lim_ \int_a^bf(x)\, dx or :\lim_ \int_a^cf(x)\ dx,\quad \lim_ \int_c^bf(x)\ dx in which one takes a limit in one or the other (or sometimes both) endpoints . By
abuse of notation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, improper integrals are often written symbolically just like standard definite integrals, perhaps with ''infinity'' among the limits of integration. When the definite integral exists (in the sense of either the
Riemann integral The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero. In the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
or the more powerful
Lebesgue integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
), this ambiguity is resolved as both the proper and improper integral will coincide in value. Often one is able to compute values for improper integrals, even when the function is not integrable in the conventional sense (as a
Riemann integral The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero. In the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
, for instance) because of a
singularity Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiabl ...
in the function or because one of the bounds of integration is infinite.


Examples

The original definition of the
Riemann integral The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero. In the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
does not apply to a function such as 1/ on the interval , because in this case the domain of integration is unbounded. However, the Riemann integral can often be extended by continuity, by defining the improper integral instead as a
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
:\int_1^\infty \frac=\lim_ \int_1^b\frac = \lim_ \left(-\frac + \frac\right) = 1. The narrow definition of the Riemann integral also does not cover the function 1/\sqrt on the interval . The problem here is that the integrand is unbounded in the domain of integration (the definition requires that both the domain of integration and the integrand be bounded). However, the improper integral does exist if understood as the limit :\int_0^1 \frac=\lim_\int_a^1\frac = \lim_ \left(2 - 2\sqrt\right)=2. Sometimes integrals may have two singularities where they are improper. Consider, for example, the function integrated from 0 to (shown right). At the lower bound, as goes to 0 the function goes to , and the upper bound is itself , though the function goes to 0. Thus this is a doubly improper integral. Integrated, say, from 1 to 3, an ordinary Riemann sum suffices to produce a result of /6. To integrate from 1 to , a Riemann sum is not possible. However, any finite upper bound, say (with ), gives a well-defined result, . This has a finite limit as goes to infinity, namely /2. Similarly, the integral from 1/3 to 1 allows a Riemann sum as well, coincidentally again producing /6. Replacing 1/3 by an arbitrary positive value (with ) is equally safe, giving . This, too, has a finite limit as goes to zero, namely /2. Combining the limits of the two fragments, the result of this improper integral is :\begin \int_^ \frac & = \lim_ \int_s^1 \frac + \lim_ \int_1^t \frac \\ & = \lim_ \left(\frac - 2 \arctan \right) + \lim_ \left(2 \arctan - \frac \right) \\ & = \frac + \left(\pi - \frac \right) \\ & = \pi . \end This process does not guarantee success; a limit might fail to exist, or might be infinite. For example, over the bounded interval from 0 to 1 the integral of does not converge; and over the unbounded interval from 1 to the integral of does not converge. It might also happen that an integrand is unbounded near an interior point, in which case the integral must be split at that point. For the integral as a whole to converge, the limit integrals on both sides must exist and must be bounded. For example: :\begin \int_^ \frac & = \lim_ \int_^ \frac + \lim_ \int_t^1 \frac \\ & = \lim_ 3\left(1-\sqrt right) + \lim_ 3\left(1-\sqrt right) \\ & = 3 + 3 \\ & = 6. \end But the similar integral :\int_^ \frac cannot be assigned a value in this way, as the integrals above and below zero do not independently converge. (However, see
Cauchy principal value In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. Formulation Depending on the type of Mathematical singularity, singul ...
.)


Convergence of the integral

An improper integral converges if the limit defining it exists. Thus for example one says that the improper integral :\lim_\int_a^t f(x)\ dx exists and is equal to ''L'' if the integrals under the limit exist for all sufficiently large ''t'', and the value of the limit is equal to ''L''. It is also possible for an improper integral to diverge to infinity. In that case, one may assign the value of ∞ (or −∞) to the integral. For instance :\lim_\int_1^b \frac = \infty. However, other improper integrals may simply diverge in no particular direction, such as :\lim_\int_1^b x\sin(x)\,dx, which does not exist, even as an
extended real number In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
. This is called divergence by oscillation. A limitation of the technique of improper integration is that the limit must be taken with respect to one endpoint at a time. Thus, for instance, an improper integral of the form :\int_^\infty f(x)\,dx can be defined by taking two separate limits; to wit :\int_^\infty f(x)\,dx = \lim_ \lim_ \int_a^bf(x)\,dx provided the double limit is finite. It can also be defined as a pair of distinct improper integrals of the first kind: :\lim_\int_a^c f(x)\,dx + \lim_ \int_c^b f(x)\,dx where ''c'' is any convenient point at which to start the integration. This definition also applies when one of these integrals is infinite, or both if they have the same sign. An example of an improper integral where both endpoints are infinite is the
Gaussian integral The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathemat ...
An example which evaluates to infinity is But one cannot even define other integrals of this kind unambiguously, such as since the double limit is infinite and the two-integral method :\lim_\int_a^cx\,dx + \lim_ \int_c^b x\,dx yields In this case, one can however define an improper integral in the sense of
Cauchy principal value In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. Formulation Depending on the type of Mathematical singularity, singul ...
: : \operatorname \int_^\infty x\,dx = \lim_\int_^b x\,dx = 0. The questions one must address in determining an improper integral are: *Does the limit exist? *Can the limit be computed? The first question is an issue of
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
. The second one can be addressed by calculus techniques, but also in some cases by
contour integration In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. O ...
,
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
s and other more advanced methods.


Types of integrals

There is more than one theory of
integration
integration
. From the point of view of calculus, the
Riemann integral The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero. In the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
theory is usually assumed as the default theory. In using improper integrals, it can matter which integration theory is in play. * For the Riemann integral (or the
Darboux integral In real analysis, a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematic ...
, which is equivalent to it), improper integration is necessary ''both'' for unbounded intervals (since one cannot divide the interval into finitely many subintervals of finite length) ''and'' for unbounded functions with finite integral (since, supposing it is unbounded above, then the upper integral will be infinite, but the lower integral will be finite). * The
Lebesgue integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
deals differently with unbounded domains and unbounded functions, so that often an integral which only exists as an improper Riemann integral will exist as a (proper) Lebesgue integral, such as \int_1^\infty \frac. On the other hand, there are also integrals that have an improper Riemann integral but do not have a (proper) Lebesgue integral, such as \int_0^\infty \frac\,dx. The Lebesgue theory does not see this as a deficiency: from the point of view of
measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
, \int_0^\infty \frac\,dx = \infty - \infty and cannot be defined satisfactorily. In some situations, however, it may be convenient to employ improper Lebesgue integrals as is the case, for instance, when defining the
Cauchy principal value In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. Formulation Depending on the type of Mathematical singularity, singul ...
. The Lebesgue integral is more or less essential in the theoretical treatment of the
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
, with pervasive use of integrals over the whole real line. * For the
Henstock–Kurzweil integral In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced ), Luzin integral or Perron integral, but not to be confused with the more general Khinchi ...
, improper integration ''is not necessary'', and this is seen as a strength of the theory: it encompasses all Lebesgue integrable and improper Riemann integrable functions.


Improper Riemann integrals and Lebesgue integrals

In some cases, the integral :\int_a^c f(x)\ dx can be defined as an integral (a
Lebesgue integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, for instance) without reference to the limit :\lim_\int_a^b f(x)\,dx but cannot otherwise be conveniently computed. This often happens when the function ''f'' being integrated from ''a'' to ''c'' has a vertical asymptote at ''c'', or if ''c'' = ∞ (see Figures 1 and 2). In such cases, the improper Riemann integral allows one to calculate the Lebesgue integral of the function. Specifically, the following theorem holds : * If a function ''f'' is Riemann integrable on 'a'',''b''for every ''b'' ≥ ''a'', and the partial integrals ::\int_a^b, f(x), \,dx :are bounded as ''b'' → ∞, then the improper Riemann integrals ::\int_a^\infty f(x)\,dx,\quad\mbox\int_a^\infty , f(x), \,dx :both exist. Furthermore, ''f'' is Lebesgue integrable on [''a'', ∞), and its Lebesgue integral is equal to its improper Riemann integral. For example, the integral :\int_0^\infty\frac can be interpreted alternatively as the improper integral :\lim_\int_0^b\frac=\lim_\arctan=\frac, or it may be interpreted instead as a
Lebesgue integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
over the set (0, ∞). Since both of these kinds of integral agree, one is free to choose the first method to calculate the value of the integral, even if one ultimately wishes to regard it as a Lebesgue integral. Thus improper integrals are clearly useful tools for obtaining the actual values of integrals. In other cases, however, a Lebesgue integral between finite endpoints may not even be defined, because the integrals of the positive and negative parts of ''f'' are both infinite, but the improper Riemann integral may still exist. Such cases are "properly improper" integrals, i.e. their values cannot be defined except as such limits. For example, :\int_0^\infty\frac\,dx cannot be interpreted as a Lebesgue integral, since :\int_0^\infty\left, \frac\\,dx=\infty. But f(x)=\frac is nevertheless integrable between any two finite endpoints, and its integral between 0 and ∞ is usually understood as the limit of the integral: :\int_0^\infty\frac\,dx=\lim_\int_0^b\frac\,dx=\frac.


Singularities

One can speak of the ''singularities'' of an improper integral, meaning those points of the extended real number line at which limits are used.


Cauchy principal value

Consider the difference in values of two limits: :\lim_\left(\int_^\frac+\int_a^1\frac\right)=0, :\lim_\left(\int_^\frac+\int_^1\frac\right)=-\ln 2. The former is the Cauchy principal value of the otherwise ill-defined expression :\int_^1\frac \left(\mbox\ \mbox\ -\infty+\infty\right). Similarly, we have :\lim_\int_^a\frac=0, but :\lim_\int_^a\frac=-\ln 4. The former is the principal value of the otherwise ill-defined expression :\int_^\infty\frac \left(\mbox\ \mbox\ -\infty+\infty\right). All of the above limits are cases of the
indeterminate formIn calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...
∞ − ∞. These pathologies do not affect "Lebesgue-integrable" functions, that is, functions the integrals of whose
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

absolute value
s are finite.


Summability

An improper integral may diverge in the sense that the limit defining it may not exist. In this case, there are more sophisticated definitions of the limit which can produce a convergent value for the improper integral. These are called
summability In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
methods. One summability method, popular in
Fourier analysis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
, is that of
Cesàro summation In mathematical analysis, Cesàro summation (also known as the Cesàro mean ) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as ''n'' tends to infinity, of ...
. The integral :\int_0^\infty f(x)\,dx is Cesàro summable (C, α) if :\lim_\int_0^\lambda\left(1-\frac\right)^\alpha f(x)\ dx exists and is finite . The value of this limit, should it exist, is the (C, α) sum of the integral. An integral is (C, 0) summable precisely when it exists as an improper integral. However, there are integrals which are (C, α) summable for α > 0 which fail to converge as improper integrals (in the sense of Riemann or Lebesgue). One example is the integral :\int_0^\infty\sin x \,dx which fails to exist as an improper integral, but is (C,''α'') summable for every ''α'' > 0. This is an integral version of
Grandi's seriesIn mathematics, the infinite series , also written : \sum_^\infty (-1)^n is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent ...
.


Multivariable improper integrals

The improper integral can also be defined for functions of several variables. The definition is slightly different, depending on whether one requires integrating over an unbounded domain, such as \R^2, or is integrating a function with singularities, like f(x,y)=\log\left(x^2+y^2\right).


Improper integrals over arbitrary domains

If f:\R^n\to\R is a non-negative function that is Riemann integrable over every compact cube of the form a,an, for a>0, then the improper integral of ''f'' over \R^n is defined to be the limit :\lim_\int_f, provided it exists. A function on an arbitrary domain ''A'' in \mathbb R^n is extended to a function \tilde on \R^n by zero outside of ''A'': :\tilde(x)=\beginf(x)& x\in A\\ 0 & x\not\in A \end The Riemann integral of a function over a bounded domain ''A'' is then defined as the integral of the extended function \tilde over a cube a,an containing ''A'': :\int_A f = \int_\tilde. More generally, if ''A'' is unbounded, then the improper Riemann integral over an arbitrary domain in \mathbb R^n is defined as the limit: :\int_Af=\lim_\int_f=\lim_\int_\tilde.


Improper integrals with singularities

If ''f'' is a non-negative function which is unbounded in a domain ''A'', then the improper integral of ''f'' is defined by truncating ''f'' at some cutoff ''M'', integrating the resulting function, and then taking the limit as ''M'' tends to infinity. That is for M>0, set f_M=\min\. Then define :\int_A f = \lim_\int_A f_M provided this limit exists.


Functions with both positive and negative values

These definitions apply for functions that are non-negative. A more general function ''f'' can be decomposed as a difference of its positive part f_+=\max\ and negative part f_-=\max\, so :f=f_+-f_- with f_+ and f_- both non-negative functions. The function ''f'' has an improper Riemann integral if each of f_+ and f_- has one, in which case the value of that improper integral is defined by :\int_Af = \int_Af_+ - \int_A f_-. In order to exist in this sense, the improper integral necessarily converges absolutely, since :\int_A, f, = \int_Af_+ + \int_Af_-.: "The relevant notion here is that of unconditional convergence." ... "In fact, for improper integrals of such functions, unconditional convergence turns out to be equivalent to absolute convergence."


Notes


Bibliography

* . * . * * . * *


External links


Numerical Methods to Solve Improper Integrals
at Holistic Numerical Methods Institute {{integral Integral calculus