In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
integrals
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
of
inverse functions
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X ...
can be computed by means of a formula that expresses the
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
s of the inverse
of a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
and
invertible function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X\t ...
in terms of
and an antiderivative of This formula was published in 1905 by
Charles-Ange Laisant
Charles-Ange Laisant (1 November 1841 – 5 May 1920), French politician and mathematician, was born at Indre, near Nantes on 1 November 1841, and was educated at the École Polytechnique as a military engineer. He was a Freemason and a liberta ...
.
Statement of the theorem
Let
and
be two
intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets
* A statistical level of measurement
* Interval e ...
of
Assume that
is a continuous and invertible function. It follows from the
intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two import ...
that
is
strictly monotone. Consequently,
maps intervals to intervals, so is an open map and thus a homeomorphism. Since
and the inverse function
are continuous, they have antiderivatives by the
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
.
Laisant proved that if
is an antiderivative of then the antiderivatives of
are:
:
where
is an arbitrary real number. Note that it is not assumed that
is differentiable.
![FunktionUmkehrIntegral2](https://upload.wikimedia.org/wikipedia/commons/5/59/FunktionUmkehrIntegral2.svg)
In his 1905 article, Laisant gave three proofs. First, under the additional hypothesis that
is
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
, one may differentiate the above formula, which completes the proof immediately. His second proof was geometric. If
and the theorem can be written:
:
The figure on the right is a
proof without words
In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered mor ...
of this formula. Laisant does not discuss the hypotheses necessary to make this proof rigorous, but this can be proved if
is just assumed to be strictly monotone (but not necessarily continuous, let alone differentiable). In this case, both
and
are Riemann integrable and the identity follows from a bijection between lower/upper
Darboux sums of
and upper/lower Darboux sums of
The antiderivative version of the theorem then follows from the fundamental theorem of calculus in the case when
is also assumed to be continuous. Laisant's third proof uses the additional hypothesis that
is differentiable. Beginning with one multiplies by
and integrates both sides. The right-hand side is calculated using integration by parts to be and the formula follows.
Nevertheless, it can be shown that this theorem holds even if
or
is not differentiable:
it suffices, for example, to use the Stieltjes integral in the previous argument. On the other hand, even though general monotonic functions are differentiable almost everywhere, the proof of the general formula does not follow, unless is absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central oper ...
.[
It is also possible to check that for every in the derivative of the function is equal to In other words:
:
To this end, it suffices to apply the ]mean value theorem
In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...
to between and taking into account that is monotonic.
Examples
#Assume that hence The formula above gives immediately
#Similarly, with and
#With and
History
Apparently, this theorem of integration was discovered for the first time in 1905 by Charles-Ange Laisant
Charles-Ange Laisant (1 November 1841 – 5 May 1920), French politician and mathematician, was born at Indre, near Nantes on 1 November 1841, and was educated at the École Polytechnique as a military engineer. He was a Freemason and a liberta ...
, who "could hardly believe that this theorem is new", and hoped its use would henceforth spread out among students and teachers. This result was published independently in 1912 by an Italian engineer, Alberto Caprilli, in an opuscule entitled "Nuove formole d'integrazione".[Read online](_blank)
/ref> It was rediscovered in 1955 by Parker, and by a number of mathematicians following him.[It is equally possible that some or all of them simply recalled this result in their paper, without referring to previous authors.] Nevertheless, they all assume that or is differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
.
The general version of the theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
, free from this additional assumption, was proposed by Michael Spivak in 1965, as an exercise in the ''Calculus'',Michael Spivak
Michael David Spivak (25 May 19401 October 2020)Biographical sketch in Notices of the AMS', Vol. 32, 1985, p. 576. was an American mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Per ...
, ''Calculus'' (1967), chap. 13, pp. 235. and a fairly complete proof following the same lines was published by Eric Key in 1994.
This proof relies on the very definition of the Darboux integral
In the branch of mathematics known as real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a functi ...
, and consists in showing that the upper Darboux sums of the function are in 1-1 correspondence with the lower Darboux sums of .
In 2013, Michael Bensimhoun, estimating that the general theorem was still insufficiently known, gave two other proofs: The second proof, based on the Stieltjes integral
Thomas Joannes Stieltjes (, 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics at ...
and on its formulae of integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
and of homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
change of variables
Change or Changing may refer to:
Alteration
* Impermanence, a difference in a state of affairs at different points in time
* Menopause, also referred to as "the change", the permanent cessation of the menstrual period
* Metamorphosis, or change, ...
, is the most suitable to establish more complex formulae.
Generalization to holomorphic functions
The above theorem generalizes in the obvious way to holomorphic functions:
Let and be two open and simply connected sets of and assume that is a biholomorphism
In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic.
Formal definit ...
. Then and have antiderivatives, and if is an antiderivative of the general antiderivative of is
:
Because all holomorphic functions are differentiable, the proof is immediate by complex differentiation.
See also
* Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
* Legendre transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions of ...
* Young's inequality for products In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers. The inequality is named after William Henry Young and should not be confused with Young's convolution inequality.
Young's inequality f ...
References
*
{{Integrals
Calculus
Theorems in analysis
Theorems in calculus