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Improper integral

From Wikipedia, the free encyclopedia
Concept in mathematical analysis
An improper Riemann integral of the first kind, where the region in the plane implied by the integral is infinite in extent horizontally. The area of such a region, which the integral represents, may be finite (as here) or infinite.
An improper Riemann integral of the second kind, where the implied region is infinite vertically. The region may have either finite (as here) or infinite area.
Part of a series of articles about
Calculus
∫abf′(t)dt=f(b)−f(a){\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}

Inmathematical analysis, animproper integral is an extension of the notion of adefinite integral to cases that violate the usual assumptions for that kind of integral.[1] In the context ofRiemann integrals (or, equivalently,Darboux integrals), this typically involves unboundedness, either of the set over which the integral is taken or of the integrand (the function being integrated), or both. It may also involve bounded but not closed sets or bounded but notcontinuous functions. While an improper integral is typically written symbolically just like a standard definite integral, it actually represents alimit of a definite integral or a sum of such limits; thus improper integrals are said to converge or diverge.[2][1] If a regular definite integral (which mayretronymically be called aproper integral) is worked out as if it is improper, the same answer will result.

In the simplest case of a real-valued function of a single variable integrated in the sense of Riemann (or Darboux) over a single interval, improper integrals may be in any of the following forms:

  1. ∫a∞f(x)dx{\displaystyle \int _{a}^{\infty }f(x)\,dx}
  2. ∫−∞bf(x)dx{\displaystyle \int _{-\infty }^{b}f(x)\,dx}
  3. ∫−∞∞f(x)dx{\displaystyle \int _{-\infty }^{\infty }f(x)\,dx}
  4. ∫abf(x)dx{\displaystyle \int _{a}^{b}f(x)\,dx}, wheref(x){\displaystyle f(x)} is undefined or discontinuous somewhere on[a,b]{\displaystyle [a,b]}

The first three forms are improper because the integrals are taken over an unbounded interval. (They may be improper for other reasons, as well, as explained below.) Such an integral is sometimes described as being of the "first" type or kind if the integrand otherwise satisfies the assumptions of integration.[2] Integrals in the fourth form that are improper becausef(x){\displaystyle f(x)} has avertical asymptote somewhere on the interval[a,b]{\displaystyle [a,b]} may be described as being of the "second" type or kind.[2] Integrals that combine aspects of both types are sometimes described as being of the "third" type or kind.[2]

In each case above, the improper integral must be rewritten using one or more limits, depending on what is causing the integral to be improper. For example, in case 1, iff(x){\displaystyle f(x)} is continuous on the entire interval[a,∞){\displaystyle [a,\infty )}, then

∫a∞f(x)dx=limb→∞∫abf(x)dx.{\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{b\to \infty }\int _{a}^{b}f(x)\,dx.}

The limit on the right is taken to be the definition of the integral notation on the left.

Iff(x){\displaystyle f(x)} is only continuous on(a,∞){\displaystyle (a,\infty )} and not ata{\displaystyle a} itself, then typically this is rewritten as

∫a∞f(x)dx=limt→a+∫tcf(x)dx+limb→∞∫cbf(x)dx,{\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{t\to a^{+}}\int _{t}^{c}f(x)\,dx+\lim _{b\to \infty }\int _{c}^{b}f(x)\,dx,}

for any choice ofc>a{\displaystyle c>a}. Here both limits must converge to a finite value for the improper integral to be said to converge. This requirement avoids the ambiguous case of adding positive and negative infinities (i.e., the "∞−∞{\displaystyle \infty -\infty }"indeterminate form). Alternatively, aniterated limit could be used or a single limit based on theCauchy principal value.

Iff(x){\displaystyle f(x)} is continuous on[a,d){\displaystyle [a,d)} and(d,∞){\displaystyle (d,\infty )}, with adiscontinuity of any kind atd{\displaystyle d}, then

∫a∞f(x)dx=limt→d−∫atf(x)dx+limu→d+∫ucf(x)dx+limb→∞∫cbf(x)dx,{\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{t\to d^{-}}\int _{a}^{t}f(x)\,dx+\lim _{u\to d^{+}}\int _{u}^{c}f(x)\,dx+\lim _{b\to \infty }\int _{c}^{b}f(x)\,dx,}

for any choice ofc>d{\displaystyle c>d}. The previous remarks about indeterminate forms, iterated limits, and the Cauchy principal value also apply here.

The functionf(x){\displaystyle f(x)} can have more discontinuities, in which case even more limits would be required (or a more complicated principal value expression).

Cases 2–4 are handled similarly. See the examples below.

Improper integrals can also be evaluated in the context of complex numbers, in higher dimensions, and in other theoretical frameworks such asLebesgue integration orHenstock–Kurzweil integration. Integrals that are considered improper in one framework may not be in others.

Examples

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The original definition of theRiemann integral does not apply to a function such as1/x2{\displaystyle 1/{x^{2}}} on the interval[1, ∞), because in this case the domain of integration isunbounded. However, the Riemann integral can often be extended bycontinuity, by defining the improper integral instead as alimit

∫1∞dxx2=limb→∞∫1bdxx2=limb→∞(−1b+11)=1.{\displaystyle \int _{1}^{\infty }{\frac {dx}{x^{2}}}=\lim _{b\to \infty }\int _{1}^{b}{\frac {dx}{x^{2}}}=\lim _{b\to \infty }\left(-{\frac {1}{b}}+{\frac {1}{1}}\right)=1.}

The narrow definition of the Riemann integral also does not cover the function1/x{\textstyle 1/{\sqrt {x}}} on the interval[0, 1]. The problem here is that the integrand isunbounded in the domain of integration. In other words,the definition of the Riemann integral requires that both the domain of integration and the integrand be bounded. However, the improper integral does exist if understood as the limit

∫01dxx=lima→0+∫a1dxx=lima→0+(2−2a)=2.{\displaystyle \int _{0}^{1}{\frac {dx}{\sqrt {x}}}=\lim _{a\to 0^{+}}\int _{a}^{1}{\frac {dx}{\sqrt {x}}}=\lim _{a\to 0^{+}}\left(2-2{\sqrt {a}}\right)=2.}
The improper integral
∫0∞dx(x+1)x=π{\displaystyle \int _{0}^{\infty }{\frac {dx}{(x+1){\sqrt {x}}}}=\pi }
has unbounded intervals for both domain and range.

Sometimes integrals may have two singularities where they are improper. Consider, for example, the function1/((x + 1)√x) integrated from 0 to∞ (shown right). At the lower bound of the integration domain, asx 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 ofπ/6. To integrate from 1 to∞, a Riemann sum is not possible. However, any finite upper bound, sayt (witht > 1), gives a well-defined result,2 arctan(√t) − π/2. This has a finite limit ast goes to infinity, namelyπ/2. Similarly, the integral from 1/3 to 1 allows a Riemann sum as well, coincidentally again producingπ/6. Replacing 1/3 by an arbitrary positive values (withs < 1) is equally safe, givingπ/2 − 2 arctan(√s). This, too, has a finite limit ass goes to zero, namelyπ/2. Combining the limits of the two fragments, the result of this improper integral is

∫0∞dx(1+x)x=lims→0+∫s1dx(1+x)x+limt→∞∫1tdx(1+x)x=lims→0+(π2−2arctan⁡s)+limt→∞(2arctan⁡t−π2)=π2+(π−π2)=π.{\displaystyle {\begin{aligned}\int _{0}^{\infty }{\frac {dx}{(1+x){\sqrt {x}}}}&{}=\lim _{s\to 0^{+}}\int _{s}^{1}{\frac {dx}{(1+x){\sqrt {x}}}}+\lim _{t\to \infty }\int _{1}^{t}{\frac {dx}{(1+x){\sqrt {x}}}}\\&{}=\lim _{s\to 0^{+}}\left({\frac {\pi }{2}}-2\arctan {\sqrt {s}}\right)+\lim _{t\to \infty }\left(2\arctan {\sqrt {t}}-{\frac {\pi }{2}}\right)\\&{}={\frac {\pi }{2}}+\left(\pi -{\frac {\pi }{2}}\right)\\&{}=\pi .\end{aligned}}}

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 of1/x does not converge; and over the unbounded interval from 1 to∞ the integral of1/√x does not converge.

The improper integral
∫−11dxx23=6{\displaystyle \int _{-1}^{1}{\frac {dx}{\sqrt[{3}]{x^{2}}}}=6}
converges, since both left and right limits exist, though the integrand is unbounded near an interior point.

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:

∫−11dxx23=lims→0−∫−1sdxx23+limt→0+∫t1dxx23=lims→0−3(1−s3)+limt→0+3(1−t3)=3+3=6.{\displaystyle {\begin{aligned}\int _{-1}^{1}{\frac {dx}{\sqrt[{3}]{x^{2}}}}&{}=\lim _{s\to 0^{-}}\int _{-1}^{s}{\frac {dx}{\sqrt[{3}]{x^{2}}}}+\lim _{t\to 0^{+}}\int _{t}^{1}{\frac {dx}{\sqrt[{3}]{x^{2}}}}\\&{}=\lim _{s\to 0^{-}}3\left(1-{\sqrt[{3}]{s}}\right)+\lim _{t\to 0^{+}}3\left(1-{\sqrt[{3}]{t}}\right)\\&{}=3+3\\&{}=6.\end{aligned}}}

But the similar integral

∫−11dxx{\displaystyle \int _{-1}^{1}{\frac {dx}{x}}}

cannot be assigned a value in this way, as the integrals above and below zero in the integral domain do not independently converge. (However, seeCauchy principal value.)

Convergence of the integral

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An improper integral converges if the limit defining it exists. Thus for example one says that the improper integral

limt→∞∫atf(x) dx{\displaystyle \lim _{t\to \infty }\int _{a}^{t}f(x)\ dx}

exists and is equal toL if the integrals under the limit exist for all sufficiently larget, and the value of the limit is equal toL.

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

limb→∞∫1bdxx=∞.{\displaystyle \lim _{b\to \infty }\int _{1}^{b}{\frac {dx}{x}}=\infty .}

However, other improper integrals may simply diverge in no particular direction, such as

limb→∞∫1bxsin⁡(x)dx,{\displaystyle \lim _{b\to \infty }\int _{1}^{b}x\sin(x)\,dx,}

which does not exist, even as anextended real number. 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

∫−∞∞f(x)dx{\displaystyle \int _{-\infty }^{\infty }f(x)\,dx}

can be defined by taking two separate limits; to which

∫−∞∞f(x)dx=lima→−∞limb→∞∫abf(x)dx{\displaystyle \int _{-\infty }^{\infty }f(x)\,dx=\lim _{a\to -\infty }\lim _{b\to \infty }\int _{a}^{b}f(x)\,dx}

provided the double limit is finite. It can also be defined as a pair of distinct improper integrals of the first kind:

lima→−∞∫acf(x)dx+limb→∞∫cbf(x)dx{\displaystyle \lim _{a\to -\infty }\int _{a}^{c}f(x)\,dx+\lim _{b\to \infty }\int _{c}^{b}f(x)\,dx}

wherec 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 theGaussian integral∫−∞∞e−x2dx=π{\textstyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}}. An example which evaluates to infinity is∫−∞∞exdx{\textstyle \int _{-\infty }^{\infty }e^{x}\,dx}. But one cannot even define other integrals of this kind unambiguously, such as∫−∞∞xdx{\textstyle \int _{-\infty }^{\infty }x\,dx}, since the double limit is infinite and the two-integral method

lima→−∞∫acxdx+limb→∞∫cbxdx{\displaystyle \lim _{a\to -\infty }\int _{a}^{c}x\,dx+\lim _{b\to \infty }\int _{c}^{b}x\,dx}

yields anindeterminate form,∞−∞{\displaystyle \infty -\infty }. In this case, one can however define an improper integral in the sense ofCauchy principal value:

p.v.⁡∫−∞∞xdx=limb→∞∫−bbxdx=0.{\displaystyle \operatorname {p.v.} \int _{-\infty }^{\infty }x\,dx=\lim _{b\to \infty }\int _{-b}^{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 ofmathematical analysis. The second one can be addressed by calculus techniques, but also in some cases bycontour integration,Fourier transforms and other more advanced methods.

Types of integrals

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There is more than one theory ofintegration. From the point of view of calculus, theRiemann integral 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 theDarboux integral, which is equivalent to it), improper integration is necessaryboth 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).
  • TheLebesgue integral 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∫1∞dxx2{\textstyle \int _{1}^{\infty }{\frac {dx}{x^{2}}}}. On the other hand, there are also integrals that have an improper Riemann integral but do not have a (proper) Lebesgue integral, such as∫0∞sin⁡xxdx{\textstyle \int _{0}^{\infty }{\frac {\sin x}{x}}\,dx}. The Lebesgue theory does not see this as a deficiency: from the point of view ofmeasure theory,∫0∞sin⁡xxdx=∞−∞{\textstyle \int _{0}^{\infty }{\frac {\sin x}{x}}\,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 theCauchy principal value. The Lebesgue integral is more or less essential in the theoretical treatment of theFourier transform, with pervasive use of integrals over the whole real line.
  • For theHenstock–Kurzweil integral, improper integrationis 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

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Figure 1
Figure 2

In some cases, the integral

∫acf(x) dx{\displaystyle \int _{a}^{c}f(x)\ dx}

can be defined as an integral (aLebesgue integral, for instance) without reference to the limit

limb→c−∫abf(x)dx{\displaystyle \lim _{b\to c^{-}}\int _{a}^{b}f(x)\,dx}

but cannot otherwise be conveniently computed. This often happens when the functionf being integrated froma toc has avertical asymptote atc, or ifc = ∞ (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 (Apostol 1974, Theorem 10.33):

  • If a functionf is Riemann integrable on [a,b] for everyb ≥ a, and the partial integrals
∫ab|f(x)|dx{\displaystyle \int _{a}^{b}|f(x)|\,dx}
are bounded asb → ∞, then the improper Riemann integrals
∫a∞f(x)dx,and ∫a∞|f(x)|dx{\displaystyle \int _{a}^{\infty }f(x)\,dx,\quad {\mbox{and }}\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

∫0∞dx1+x2{\displaystyle \int _{0}^{\infty }{\frac {dx}{1+x^{2}}}}

can be interpreted alternatively as the improper integral

limb→∞∫0bdx1+x2=limb→∞arctan⁡b=π2,{\displaystyle \lim _{b\to \infty }\int _{0}^{b}{\frac {dx}{1+x^{2}}}=\lim _{b\to \infty }\arctan {b}={\frac {\pi }{2}},}

or it may be interpreted instead as aLebesgue integral 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 off 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,

∫0∞sin⁡(x)xdx{\displaystyle \int _{0}^{\infty }{\frac {\sin(x)}{x}}\,dx}

cannot be interpreted as a Lebesgue integral, since

∫0∞|sin⁡(x)x|dx=∞.{\displaystyle \int _{0}^{\infty }\left|{\frac {\sin(x)}{x}}\right|\,dx=\infty .}

Butf(x)=sin⁡(x)x{\displaystyle f(x)={\frac {\sin(x)}{x}}} is nevertheless integrable between any two finite endpoints, and its integral between 0 and ∞ is usually understood as the limit of the integral:

∫0∞sin⁡(x)xdx=limb→∞∫0bsin⁡(x)xdx=π2.{\displaystyle \int _{0}^{\infty }{\frac {\sin(x)}{x}}\,dx=\lim _{b\to \infty }\int _{0}^{b}{\frac {\sin(x)}{x}}\,dx={\frac {\pi }{2}}.}

Singularities

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One can speak of thesingularities of an improper integral, meaning those points of theextended real number line at which limits are used.

Cauchy principal value

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Main article:Cauchy principal value

Consider the difference in values of two limits:

lima→0+(∫−1−adxx+∫a1dxx)=0,{\displaystyle \lim _{a\to 0^{+}}\left(\int _{-1}^{-a}{\frac {dx}{x}}+\int _{a}^{1}{\frac {dx}{x}}\right)=0,}
lima→0+(∫−1−adxx+∫2a1dxx)=−ln⁡2.{\displaystyle \lim _{a\to 0^{+}}\left(\int _{-1}^{-a}{\frac {dx}{x}}+\int _{2a}^{1}{\frac {dx}{x}}\right)=-\ln 2.}

The former is the Cauchy principal value of the otherwise ill-defined expression

∫−11dxx (which gives −∞+∞).{\displaystyle \int _{-1}^{1}{\frac {dx}{x}}{\ }\left({\mbox{which}}\ {\mbox{gives}}\ -\infty +\infty \right).}

Similarly, we have

lima→∞∫−aa2xdxx2+1=0,{\displaystyle \lim _{a\to \infty }\int _{-a}^{a}{\frac {2x\,dx}{x^{2}+1}}=0,}

but

lima→∞∫−2aa2xdxx2+1=−ln⁡4.{\displaystyle \lim _{a\to \infty }\int _{-2a}^{a}{\frac {2x\,dx}{x^{2}+1}}=-\ln 4.}

The former is the principal value of the otherwise ill-defined expression

∫−∞∞2xdxx2+1 (which gives −∞+∞).{\displaystyle \int _{-\infty }^{\infty }{\frac {2x\,dx}{x^{2}+1}}{\ }\left({\mbox{which}}\ {\mbox{gives}}\ -\infty +\infty \right).}

All of the above limits are cases of theindeterminate form∞−∞{\displaystyle \infty -\infty }.

Thesepathologies do not affect "Lebesgue-integrable" functions, that is, functions the integrals of whoseabsolute values are finite.

Summability

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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 calledsummability methods.

One summability method, popular inFourier analysis, is that ofCesàro summation. The integral

∫0∞f(x)dx{\displaystyle \int _{0}^{\infty }f(x)\,dx}

is Cesàro summable (C, α) if

limλ→∞∫0λ(1−xλ)αf(x) dx{\displaystyle \lim _{\lambda \to \infty }\int _{0}^{\lambda }\left(1-{\frac {x}{\lambda }}\right)^{\alpha }f(x)\ dx}

exists and is finite (Titchmarsh 1948, §1.15). 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

∫0∞sin⁡xdx{\displaystyle \int _{0}^{\infty }\sin x\,dx}

which fails to exist as an improper integral, but is (C,α) summable for everyα > 0. This is an integral version ofGrandi's series.

Multivariable improper integrals

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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 asR2{\displaystyle \mathbb {R} ^{2}}, or is integrating a function with singularities, likef(x,y)=log⁡(x2+y2){\displaystyle f(x,y)=\log \left(x^{2}+y^{2}\right)}.

Improper integrals over arbitrary domains

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Iff:Rn→R{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is a non-negative function that is Riemann integrable over every compact cube of the form[−a,a]n{\displaystyle [-a,a]^{n}}, fora>0{\displaystyle a>0}, then the improper integral off overRn{\displaystyle \mathbb {R} ^{n}} is defined to be the limit

lima→∞∫[−a,a]nf,{\displaystyle \lim _{a\to \infty }\int _{[-a,a]^{n}}f,}

provided it exists.

A function on an arbitrary domainA inRn{\displaystyle \mathbb {R} ^{n}} is extended to a functionf~{\displaystyle {\tilde {f}}} onRn{\displaystyle \mathbb {R} ^{n}} by zero outside ofA:

f~(x)={f(x)x∈A0x∉A{\displaystyle {\tilde {f}}(x)={\begin{cases}f(x)&x\in A\\0&x\not \in A\end{cases}}}

The Riemann integral of a function over a bounded domainA is then defined as the integral of the extended functionf~{\displaystyle {\tilde {f}}} over a cube[−a,a]n{\displaystyle [-a,a]^{n}} containingA:

∫Af=∫[−a,a]nf~.{\displaystyle \int _{A}f=\int _{[-a,a]^{n}}{\tilde {f}}.}

More generally, ifA is unbounded, then the improper Riemann integral over an arbitrary domain inRn{\displaystyle \mathbb {R} ^{n}} is defined as the limit:

∫Af=lima→∞∫A∩[−a,a]nf=lima→∞∫[−a,a]nf~.{\displaystyle \int _{A}f=\lim _{a\to \infty }\int _{A\cap [-a,a]^{n}}f=\lim _{a\to \infty }\int _{[-a,a]^{n}}{\tilde {f}}.}

Improper integrals with singularities

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Iff is a non-negative function which is unbounded in a domainA, then the improper integral off is defined by truncatingf at some cutoffM, integrating the resulting function, and then taking the limit asM tends to infinity. That is forM>0{\displaystyle M>0}, setfM=min{f,M}{\displaystyle f_{M}=\min\{f,M\}}. Then define

∫Af=limM→∞∫AfM{\displaystyle \int _{A}f=\lim _{M\to \infty }\int _{A}f_{M}}

provided this limit exists.

Functions with both positive and negative values

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These definitions apply for functions that are non-negative. A more general functionf can be decomposed as a difference of its positive partf+=max{f,0}{\displaystyle f_{+}=\max\{f,0\}} and negative partf−=max{−f,0}{\displaystyle f_{-}=\max\{-f,0\}}, so

f=f+−f−{\displaystyle f=f_{+}-f_{-}}

withf+{\displaystyle f_{+}} andf−{\displaystyle f_{-}} both non-negative functions. The functionf has an improper Riemann integral if each off+{\displaystyle f_{+}} andf−{\displaystyle f_{-}} has one, in which case the value of that improper integral is defined by

∫Af=∫Af+−∫Af−.{\displaystyle \int _{A}f=\int _{A}f_{+}-\int _{A}f_{-}.}

In order to exist in this sense, the improper integral necessarily converges absolutely, since

∫A|f|=∫Af++∫Af−.{\displaystyle \int _{A}|f|=\int _{A}f_{+}+\int _{A}f_{-}.}[3][4]

Notes

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  1. ^abBuck, R. Creighton (1965).Advanced Calculus (2nd ed.). McGraw-Hill. pp. 133–134.
  2. ^abcdSpiegel, Murray R. (1963).Schaum's Outline of Theory and Problems of Advanced Calculus. McGraw-Hill. p. 260.ISBN 0-07-060229-8.{{cite book}}:ISBN / Date incompatibility (help)
  3. ^Cooper 2005, p. 538: "We need to make this stronger definition of convergence in terms of |f(x)| because cancellation in the integrals can occur in so many different ways in higher dimensions."
  4. ^Ghorpade & Limaye 2010, p. 448: "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."

Bibliography

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External links

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