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2014-10-28T15:36:50Z

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en>Michael Hardy: /* Multidimensional Fourier transform */

2014-01-03T23:27:40Z

Multidimensional Fourier transform

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In [[mathematics]], '''[[integrals]] of [[inverse functions]]''' can be computed by mean of a formula that expresses the [[antiderivative]]s of the inverse <math>f^{-1}</math> of a [[continuous function|continuous]] and [[inverse functions|invertible function]] <math>f</math>, in terms of <math>f^{-1}</math> and an antiderivative of <math>f</math>.

==Statement of the theorem==
Let <math>I_1</math> and <math>I_2</math> be two [[interval (mathematics)|intervals]] of <math>\mathbb{R}</math>.
Assume that <math>f: I_1\to I_2</math> is a continuous and invertible function, and let <math>f^{-1}</math> denote its inverse <math>I_2\to I_1</math>.
Then <math>f</math> and <math>f^{-1}</math> have antiderivatives, and if <math>F</math> is an antiderivative of <math>f</math>, the possible antiderivatives of <math>f^{-1}</math> are:
:<math>\int f^{-1}(y)\,dy= x f^{-1}(y)-F\circ f^{-1}(y)+C,</math>
where <math>C</math> is an arbitrary real number.

[[File:FunktionUmkehrIntegral2.svg|thumb|Proof without words of the theorem]]

If <math>f</math> is assumed to be [[differentiable]], then the proof of the above formula follows immediately by differentiation. Under this assumption, another direct derivation of the formula can be obtained by performing the [[substitution (algebra)|substitution]] <math>y=f(x)</math>, followed by an [[integration by parts]].<ref name=Laisant/> Nevertheless, it can be shown that this theorem holds even if <math>f</math> is not differentiable:<ref name=Key/><ref name=Bensimhoun>{{cite journal|last=Bensimhoun|first=Michael|title=On the antiderivative of inverse functions|year=2013|journal=Arxiv.org e-Print Archive|arxiv=1312.3839|bibcode=2013arXiv1312.3839B|volume=1312|pages=3839}}</ref> it suffices, for example, to use the Stieltjes integral in the previous argument.<ref name=Bensimhoun/> On the other hand, even thought general monotonic functions are differentiable almost everywhere, the proof of the general formula does not follow, unless <math>f^{-1}</math> is [[absolutely continuous]].<ref>As pointed out by Michael Bensimhoun at the end of the aforementioned article.</ref>

If <math>f(a)=c</math> and <math>f(b)=d</math>, the theorem can be written:
:<math>\int_c^df^{-1}(y)\,dy+\int_a^bf(x)\,dx=bd-ac.</math>
The figure on the right is a [[proof without words]] of this formula. It can be made explicit with the help of the [[Darboux integral]]<ref name=Key/> (or eventually using [[Fubini's theorem]]<ref name=Bensimhoun/> if a demonstration based on the Lebesgue integral is desired).

It is also possible to check that for every <math>y</math> in <math>I_2</math>, the derivative of the function <math>y \to y f^{-1}(y) -F(f^{-1}(y))</math> is equal to <math>f^{-1}(y)</math>.<ref>This very simple proof of the general theorem, the only one that does not make use of integrals, was communicated by the French mathematician and Wikipedian Anne Bauval in the corresponding pages in French. It seems to have escaped the persons who published proofs of this result.</ref> In other words:
:<math>\forall x\in I_1\quad\lim_{h\to 0}\frac{(x+h)f(x+h)-xf(x)-\left(F(x+h)-F(x)\right)}{f(x+h)-f(x)}=x.</math>
To this end, it suffices to apply the [[mean value theorem]] to <math>F</math> between <math>x</math> and <math>x+h</math>, taking into account that <math>f</math> is monotonic.

==Examples==
#Assume that <math>f(x)=\exp(x)</math>, hence <math>f^{-1}(y)=\ln(y)</math>. The formula above gives immediately{{break}}<math>\int \ln(y) \, dy = y\ln(y)-y + C.</math>
#Similarly, with <math>f(x)=\cos(x)</math> and <math>f^{-1}(y)=\arccos(y)</math>,{{break}}<math>\quad\quad\int \arccos(y) \, dy = y\arccos(y) - \sin(\arccos(y))+C.</math>
#With <math>\quad\quad f(x) = \tan(x)</math> and <math> f^{-1}(y) = \arctan(y)</math>,{{break}}<math>\quad\quad\int \arctan(y) \, dy = y\arctan(y) + \ln|\cos(\arctan(y))| + C. </math>

==History==
Apparently, this theorem of integration has been discovered for the first time in 1905 by [[Charles-Ange Laisant]],<ref name=Laisant>{{cite journal|last=Laisant|first=C.-A.|title=Intégration des fonctions inverses|year=1905|pages = 253–257|issue=4|volume=5|journal=[[Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale]]}}</ref> 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 intitled "Nuove formole d'integrazione".<ref name="Caprilli">[http://ebooks.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math&cc=math&idno=00420001&view=image&seq=5&size=100 Read online]</ref> It was rediscovered in 1955 by Parker,<ref name=Parker>{{cite journal|last=Parker|first=F. D.|title=Integrals of inverse functions|year=1955|month=Jun. and Jul.|pages = 439–440|volume=62|journal=[[The American Mathematical Monthly]]|doi=10.2307/2307006|issue=6}}</ref> and by a number of mathematicians following him. Nevertheless, they all assume that {{math|''f''}} is [[differentiable]]. It seems that the first proof of the correctness of the general version of the [[theorem]], free from this additional assumption, was given by Eric Key in 1994.<ref name=Key>{{cite journal|last=Key|first=E. |title=Disks, Shells, and Integrals of Inverse Functions|date=Mar 1994|pages = 136–138|volume=25|journal=[[The College Mathematics Journal]]|issue=2|doi=10.2307/2687137}}</ref>
His proof, based on the very definition of the [[Darboux integral]], consists in showing that the upper [[Darboux integral|Darboux sums]] of the function <math>f</math> are in 1-1 correspondence with the lower Darboux sums of <math>f^{-1}</math>.
In 2013, Michael Bensimhoun, estimating that the general theorem was still widely unknown, gave two other proofs:<ref name="Bensimhoun"/> The second proof, based on the [[Stieltjes integral]] and on its formulae of [[integration by parts]] and of [[homeomorphic]] [[integration by substitution|change of variables]], is the most suitable to establish more complex formulae.<ref>See for instance the formula at the end of his article.</ref>

==Generalization to holomorphic functions==
The above theorem generalizes in the obvious way to holomorphic functions:
Let <math>U</math> and <math>V</math> be two open and simply connected sets of <math>\mathbb{C}</math>, and assume that <math>f: U\to V</math> is a biholomorphism. Then <math>f</math> and <math>f^{-1}</math> have antiderivatives, and if <math>F</math> is an antiderivative of <math>f</math>, the general antiderivative of <math>f^{-1}</math> is
:<math>G(z)= z f^{-1}(z)-F\circ f^{-1}(z)+C.</math>

Because all holomorphic functions are differentiable, the proof is immediate by complex differentiation.

== References ==
{{Reflist}}

{{refbegin}}
* {{ Cite journal | first = J. H. | last = Staib | title=The Integration of Inverse Functions | journal = [[Mathematics magazine]]|date=Sept 1966 | volume=39 | pages=223–224 | issue=4 | postscript = {{inconsistent citations}} | doi = 10.2307/2688087}}
{{refend}}

==See also==

* [[Young inequality]]

[[Category:Calculus]]

Multidimensional transform - Revision history

en>Tjhuston225: Added category

en>Michael Hardy: /* Multidimensional Fourier transform */