Lebesgue point: Difference between revisions

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== at the foot of an escape Tissot appear automatically. ==
{{Refimprove|date=December 2009}}
In [[mathematics]], a '''nonelementary antiderivative''' is an [[antiderivative]] for which it can be shown that there exists no formula in terms of ''[[elementary function]]s'' (i.e. involving polynomials, and the standard functions sin, cos, exp, and so on).<ref name= Weisstein>Weisstein, Eric W. [http://mathworld.wolfram.com/ElementaryFunction.html "Elementary Function."] From [[MathWorld]]--A Wolfram Web Resource. Accessed 7 Dec 2012.</ref>  A [[Liouville's theorem (differential algebra)|theorem by Liouville]] in 1835 provided the first proof that nonelementary antiderivatives exist.<ref>{{cite book| title=The Calculus Gallery| first=William| last=Dunham |isbn=978-0-691-13626-4 |publisher=Princeton |year=2005| page=119}}</ref> This theorem also provides a basis for determining (with difficulty) which expressions are [[integrable]]. It can be shown that, if one is given a function of any complexity, the probability that it will have an elementary [[antiderivative]] are very low.{{Citation needed|date=July 2010}}


Some examples of such functions are:
Because more than enough points, even death, loss half points. Still able to top 1001.<br><br>'hands now?' magic hammer staring Feng Luo.<br><br>'Yes.' Luo Feng nodded, 'Now!'<br><br>'Haha, I'll look at, so many [http://www.nrcil.net/fancybox/lib/rakuten_LV_46.html ルイヴィトン バッグ モノグラム] talented fear 'madman' how strong in the end.' That magic hammer vertical face with eyes crazy.<br><br>Luo Feng suspended from half a meter, at the foot of an escape Tissot appear automatically.<br>On<br>street, two distant [http://www.nrcil.net/fancybox/lib/rakuten_LV_16.html ルイヴィトン バッグ ダミエ] relative.<br><br>'Boom!' magic hammer right leg suddenly stepped on the ground, so that the ground suddenly cracked open [http://www.nrcil.net/fancybox/lib/rakuten_LV_15.html ルイヴィトン コレクション] street, [http://www.nrcil.net/fancybox/lib/rakuten_LV_78.html ルイヴィトン デニム] and his whole [http://www.nrcil.net/fancybox/lib/rakuten_LV_55.html ルイヴィトンルイヴィトン] body had been before he went to Luo Feng flashed that all four arms holding the angular Black Hammer, a series of lightning hit coming towards Feng Luo.<br><br>Luo Feng body floating almost backwards, with the foot on a golden escape from Tissot shot, [http://www.nrcil.net/fancybox/lib/rakuten_LV_90.html ルイヴィトン 財布 マルチカラー] and then directly [http://www.nrcil.net/fancybox/lib/rakuten_LV_100.html ルイヴィトン 中古品] into a [http://www.nrcil.net/fancybox/lib/rakuten_LV_100.html ルイヴィトン 新作 2014] gold fish.<br><br>'Go!' Luo Feng eyes of a condensate.<br><br>golden fish swim merrily unto magic hammer
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<ul>
 
  <li>[http://www.vaswim.org/cgi-bin/rcalc.cgi http://www.vaswim.org/cgi-bin/rcalc.cgi]</li>
 
  <li>[http://www.ultraloong.cn/hbepi/show.php?aid=971446 http://www.ultraloong.cn/hbepi/show.php?aid=971446]</li>
 
  <li>[http://www.changwanba.com/home.php?mod=space&uid=88678 http://www.changwanba.com/home.php?mod=space&uid=88678]</li>
 
</ul>


*<math>\sqrt{1 - x^4}</math><ref name= Weisstein/>
== 'interstellar warships to limit my martial arts headquarters ==
*<math>\ln(\ln x) \,</math>
*<math>\frac{1}{\ln x}</math><ref>[http://www.claymath.org/programs/outreach/academy/LectureNotes05/Conrad.pdf Impossibility theorems for elementary integration;] Brian Conrad. [[Clay Mathematics Institute]]: 2005 Academy Colloquium Series. Accessed 7 Dec 2012.</ref>
*<math>e^{e^x} \,</math>
*<math>e^{-\frac{x^2}{2}} \,</math><ref name= Weisstein/> (see [[Normal distribution]])


The evaluation of nonelementary antiderivatives can often be done using [[Taylor series]]. This is because [[Analytic function|Taylor series can ''always'' be integrated]] as one would an ordinary [[polynomial]] (using the fact that any Taylor series is uniformly convergent within its radius of convergence), even if there is no elementary antiderivative of the function that generated the Taylor series.
Observation corridors. And now it has entered the flood a chamber [http://www.nrcil.net/fancybox/lib/rakuten_LV_137.html 新作ルイヴィトン] which, Luo Feng follow to enter.<br><br>secret [http://www.nrcil.net/fancybox/lib/rakuten_LV_68.html ルイヴィトン 長財布 中古] room [http://www.nrcil.net/fancybox/lib/rakuten_LV_131.html ルイヴィトン ショルダーバッグ] that only a large display.<br><br>'This is a damaged starship.' flood plain face, 'when I found it, it has been damaged and can no longer fly into the universe, but ...... even if it is an empty shell of interstellar warships, defense degree far more than any other base on Earth Municipal Building. '<br><br>'Then I have the ability, I will get it out of the ground.'<br><br>'interstellar warships to limit [http://www.nrcil.net/fancybox/lib/rakuten_LV_4.html ルイヴィトン スピーディ] my [http://www.nrcil.net/fancybox/lib/rakuten_LV_137.html 新作ルイヴィトン] martial arts headquarters! Even [http://www.nrcil.net/fancybox/lib/rakuten_LV_17.html ルイヴィトン キャップ] laser cannon, can never expect to break interstellar warships.' Hung point display screen suddenly lights up the sky picture emerges of the universe. 'This is not our universe ...... Many ordinary people believe the Earth is so simple.'<br><br>'Our Earth humans sometimes very arrogant.'<br><br>'even [http://www.nrcil.net/fancybox/lib/rakuten_LV_99.html ルイヴィトン コレクション] a variety [http://www.nrcil.net/fancybox/lib/rakuten_LV_128.html ルイヴィトン最新財布] of animals on land, plant species and many never noticed. ocean
 
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However, in some cases it is not possible to rely on Taylor series. For example, if the function is not ''[[infinitely differentiable]]'', one cannot generate a Taylor series. Even if a Taylor series can be generated, there is a good possibility that it will [[divergent series|diverge]] and ''not'' represent the function one is attempting to antidifferentiate. Many functions which are infinitely differentiable have higher order [[derivative]]s that are unmanageable by hand. In these cases, it is not possible to evaluate [[indefinite integral]]s, but [[definite integral]]s can be evaluated numerically, for instance by [[Simpson's rule]].  There are yet other cases where definite integrals can be evaluated exactly without numerical methods, but indefinite integrals cannot, for lack of an elementary antiderivative.
<ul>
 
 
The integrals for many of these functions can be written down if one allows so-called "special" (nonelementary) functions. For example, the first example's integral is expressible using [[Elliptic integral#Incomplete elliptic integral of the first kind|incomplete elliptic integrals of the first kind]], the second and third use the [[logarithmic integral]], the fourth the [[exponential integral]], and the fifth the [[error function]]. Still, there exist functions, such as <math>x^x</math> and <math>\sin(\sin(x))</math> for which no notation currently exists to describe their integrals (other than the use of the integrals themselves).
  <li>[http://www.oakleighsongs.com/cgi-bin/mojo/mojo.cgi http://www.oakleighsongs.com/cgi-bin/mojo/mojo.cgi]</li>
 
 
==See also==
  <li>[http://www.rguerra.com/cgi-bin/guestbook/guestbook.cgi http://www.rguerra.com/cgi-bin/guestbook/guestbook.cgi]</li>
* [[List of integrals]]
 
* [[Derivative]]s
  <li>[http://www.rhhid.com/plus/feedback.php?aid=35 http://www.rhhid.com/plus/feedback.php?aid=35]</li>
* [[Risch algorithm]]
 
* [[Symbolic integration]]
</ul>
* [[Algebraic function|Algebraic]] and [[transcendental function]]s
 
==References==
<references/>
* [http://www.sosmath.com/calculus/integration/fant/fant.html Integration of Nonelementary Functions], S.O.S MATHematics.com; accessed 7 Dec 2012.
 
==Further reading==
* Williams, Dana P., [http://www.math.dartmouth.edu/~dana/bookspapers/elementary.pdf NONELEMENTARY ANTIDERIVATIVES], 1 Dec 1993. Accessed January 24, 2014.
 
{{DEFAULTSORT:Nonelementary Integral}}
[[Category:Integral calculus]]
[[Category:Integrals]]

Revision as of 17:01, 4 February 2014

at the foot of an escape Tissot appear automatically.

Because more than enough points, even death, loss half points. Still able to top 1001.

'hands now?' magic hammer staring Feng Luo.

'Yes.' Luo Feng nodded, 'Now!'

'Haha, I'll look at, so many ルイヴィトン バッグ モノグラム talented fear 'madman' how strong in the end.' That magic hammer vertical face with eyes crazy.

Luo Feng suspended from half a meter, at the foot of an escape Tissot appear automatically.
On
street, two distant ルイヴィトン バッグ ダミエ relative.

'Boom!' magic hammer right leg suddenly stepped on the ground, so that the ground suddenly cracked open ルイヴィトン コレクション street, ルイヴィトン デニム and his whole ルイヴィトンルイヴィトン body had been before he went to Luo Feng flashed that all four arms holding the angular Black Hammer, a series of lightning hit coming towards Feng Luo.

Luo Feng body floating almost backwards, with the foot on a golden escape from Tissot shot, ルイヴィトン 財布 マルチカラー and then directly ルイヴィトン 中古品 into a ルイヴィトン 新作 2014 gold fish.

'Go!' Luo Feng eyes of a condensate.

golden fish swim merrily unto magic hammer 相关的主题文章：

'interstellar warships to limit my martial arts headquarters

Observation corridors. And now it has entered the flood a chamber 新作ルイヴィトン which, Luo Feng follow to enter.

secret ルイヴィトン 長財布 中古 room ルイヴィトン ショルダーバッグ that only a large display.

'This is a damaged starship.' flood plain face, 'when I found it, it has been damaged and can no longer fly into the universe, but ...... even if it is an empty shell of interstellar warships, defense degree far more than any other base on Earth Municipal Building. '

'Then I have the ability, I will get it out of the ground.'

'interstellar warships to limit ルイヴィトン スピーディ my 新作ルイヴィトン martial arts headquarters! Even ルイヴィトン キャップ laser cannon, can never expect to break interstellar warships.' Hung point display screen suddenly lights up the sky picture emerges of the universe. 'This is not our universe ...... Many ordinary people believe the Earth is so simple.'

'Our Earth humans sometimes very arrogant.'

'even ルイヴィトン コレクション a variety ルイヴィトン最新財布 of animals on land, plant species and many never noticed. ocean 相关的主题文章：

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