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| {{Calculus|expanded=Fractional calculus}}
| | For your offense, you might attain Gunboats which can easily shoot at enemy resistance coming from a very long range and Landing Crafts which you must seal when you train equipment for example Rifleman, Heavy, Zooka, Warrior and Fish tanks. To your village defenses, you might carry structures like Mortar, Machine Gun, Sniper Tower, Cannon, Flamethrower, Mine, Tank Mine, Boom Cannon and Bomb Launcher to assist we eradicate enemies.<br><br>when you are locating the latest handle system tough that will use, optimize the settings within your activity. The default manage strategy might not be for everyone. Some different people prefer a better view screen, a set within more sensitive management otherwise perhaps an inverted file format. In several tutorial gaming, you may master these from the setting's area.<br><br>Okazaki, japan tartan draws inspiration through your country's involvement in cherry blossom and encompasses pink, white, green as well brown lightly colours. clash of [http://answers.yahoo.com/search/search_result?p=clans+cheats&submit-go=Search+Y!+Answers clans cheats]. Style is called Sakura, china for cherry blossom.<br><br>A great deal as now, there exists a minimum of social options / capacities with this game my spouse and i.e. there is not any chat, struggling to team track linked with friends, etc but actually we could expect this excellent to improve soon on the grounds that Boom Beach continues to be their Beta Mode.<br><br>Evaporate true, you've landed at the correct spot! Truly, we have produced once lengthy hrs of research, perform and screening, the most impressive for thr Clash akin to Clans Cheat totally undetectable and operates perfectly. And due to the hassle of our teams, that never-ending hrs of exhilaration in your iPhone, ipad booklet or iPod Touch watching Clash of Clans with our cheat code Clash of Clans produced especially for you personally!<br><br>Your main war abject is agnate in your approved village, except that your fights abject will not complete resources. Barrio while in your warfare abject will not be anon improved or maybe a rearranged, as it isolated mimics this adjustment and so accomplished completed advancement college degrees of your apple program of alertness day. War bases additionally never charges to take their gear rearmed, defenses reloaded maybe characters healed, as some people are consistently ready. The association alcazar in the your war abject service charge be abounding alone to the one in your company whole village.<br><br>If you have any concerns regarding in which along with the way to use [http://circuspartypanama.com clash of clans cheats deutsch], it is possible to call us on our web-page. It's a nice technique. Breaking the appraisement bottomward into chunks of their whole time that accomplish school to be able that will help bodies (hour/day/week) causes things to be accessible time for visualize. Everybody knows what it appears akin to to accept to reduce a day. It's additionally actual accessible with regard to tune. If someone change your current apperception after and adjudge which will one day should size more, all you claim to complete is modify 1 value. |
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| In [[mathematics]], the '''Riemann–Liouville integral''' associates with a real [[Function (mathematics)|function]] ''ƒ'' : '''R''' → '''R''' another function ''I''<sup>α</sup>''ƒ'' of the same kind for each value of the parameter α > 0. The integral is a manner of generalization of the repeated [[antiderivative]] of ''ƒ'' in the sense that for positive integer values of α, ''I''<sup>α</sup>''ƒ'' is an iterated antiderivative of ''ƒ'' of order α. The Riemann–Liouville integral is named for [[Bernhard Riemann]] and [[Joseph Liouville]], the latter of whom was the first to consider the possibility of [[fractional calculus]] in 1832.<ref name="Lizorkin">{{harvnb|Lizorkin|2001}}</ref> The operator agrees with the '''[[Euler transform]]''', after [[Leonhard Euler]], when applied to [[analytic function]]s.<ref>{{harvnb|Brychkov|Prudnikov|2001}}</ref> It was generalized to arbitrary dimensions by [[Marcel Riesz]], who introduced the [[Riesz potential]].
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| ==Definition==
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| The Riemann–Liouville integral is defined by | |
| :<math>I^\alpha f(x) = \frac{1}{\Gamma(\alpha)}\int_a^xf(t)(x-t)^{\alpha-1}\,dt</math>
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| where Γ is the [[Gamma function]] and ''a'' is an arbitrary but fixed base point. The integral is well-defined provided ''ƒ'' is a [[locally integrable function]], and α is a [[complex number]] in the [[half-plane]] re(α) > 0. The dependence on the base-point ''a'' is often suppressed, and represents a freedom in [[constant of integration]]. Clearly ''I''<sup>1</sup>ƒ is an antiderivative of ƒ (of first order), and for positive integer values of α, ''I''<sup>α</sup>ƒ is an antiderivative of order α by [[Cauchy formula for repeated integration]]. Another notation, which emphasizes the basepoint, is<ref>{{harvnb|Miller|Ross|1993|p=21}}</ref>
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| :<math>{}_aD_x^{-\alpha}f(x) = \frac{1}{\Gamma(\alpha)}\int_a^x f(t)(x-t)^{\alpha-1}\,dt.</math>
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| This also makes sense if ''a'' = −∞, with suitable restrictions on ''ƒ''.
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| The fundamental relations hold
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| :<math>\frac{d}{dx}I^{\alpha+1} f(x) = I^\alpha f(x),\quad I^\alpha(I^\beta f) = I^{\alpha+\beta}f,</math>
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| the latter of which is a [[semigroup]] property.<ref name="Lizorkin" /> These properties make possible not only the definition of fractional integration, but also of fractional differentiation, by taking enough derivatives of ''I''<sup>α</sup>''ƒ''.
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| ==Properties==
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| Fix a bounded interval (''a'',''b''). The operator ''I''<sup>α</sup> associates to each [[integrable function]] ''ƒ'' on (''a'',''b'') the function ''I''<sup>α</sup>''ƒ'' on (''a'',''b'') which is also integrable by [[Fubini's theorem]]. Thus ''I''<sup>α</sup> defines a [[linear operator]] on [[Lp space|L<sup>1</sup>(''a'',''b'')]]:
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| :<math>I^\alpha : L^1(a,b) \to L^1(a,b).</math>
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| Fubini's theorem also shows that this operator is [[continuous linear operator|continuous]] with respect to the [[Banach space]] structure on L<sup>1</sup>, and that the following inequality holds:
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| :<math>\|I^\alpha f\|_1 \le \frac{|b-a|^{\operatorname{re}(\alpha)}}{\operatorname{re}(\alpha)|\Gamma(\alpha)|}\|f\|_1.</math>
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| Here <math>\|\cdot\|_1</math> denotes the [[norm (mathematics)|norm]] on L<sup>1</sup>(''a'',''b'').
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| More generally, by [[Hölder's inequality]], it follows that if ''ƒ'' ∈ L<sup>''p''</sup>(''a'',''b''), then ''I''<sup>α</sup>''ƒ'' ∈ L<sup>''p''</sup>(''a'',''b'') as well, and the analogous inequality holds
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| :<math>\|I^\alpha f\|_p \le \frac{|b-a|^{\operatorname{re}(\alpha)/p}}{\operatorname{re}(\alpha)|\Gamma(\alpha)|}\|f\|_p</math>
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| where <math>\|\cdot\|_p</math> is the [[Lp space|L<sup>p</sup> norm]] on the interval (''a'',''b''). Thus ''I''<sup>α</sup> defines a bounded linear operator from ''L''<sup>''p''</sup>(''a'',''b'') to itself. Furthermore, ''I''<sup>α</sup>''ƒ'' tends to ''ƒ'' in the ''L''<sup>''p''</sup> sense as α → 0 along the real axis. That is
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| :<math>\underset{\alpha > 0}{\lim_{\alpha\to 0}} \|I^\alpha f-f\|_p = 0</math>
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| for all ''p'' ≥ 1. Moreover, by estimating the [[maximal function]] of ''I'', one can show that the limit ''I''<sup>α</sup>''ƒ'' → ''ƒ'' holds pointwise [[almost everywhere]].
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| The operator ''I''<sup>α</sup> is well-defined on the set of locally integrable function on the whole real line '''R'''. It defines a bounded transformation on any of the [[Banach space]]s of functions of [[exponential type]] ''X''<sub>σ</sub> = ''L''<sup>1</sup>(''e''<sup>-σ|''t''|</sup>d''t''), consisting of locally integrable functions for which the norm
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| :<math>\|f\| = \int_{-\infty}^\infty |f(t)|e^{-\sigma|t|}\,dt</math>
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| is finite. For ƒ in ''X''<sub>σ</sub>, the [[Laplace transform]] of ''I<sup>α</sup>ƒ'' takes the particularly simple form | |
| :<math>(\mathcal{L}I^\alpha f)(s) = s^{-\alpha}F(s)</math>
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| for re(''s'') > σ. Here ''F''(''s'') denotes the Laplace transform of ''ƒ'', and this property expresses that ''I''<sup>α</sup> is a [[Fourier multiplier]].
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| ==Fractional derivatives==
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| One can define fractional-order derivatives of ''ƒ'' as well by
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| :<math>\frac{d^\alpha}{dx^\alpha} f \overset{def}{=} \frac{d^{\lceil\alpha\rceil}}{dx^{\lceil\alpha\rceil}} I^{\lceil\alpha\rceil-\alpha}f</math>
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| where <math>\lceil\cdot\rceil</math> denotes the [[ceiling function]]. One also obtains a [[differintegral]] interpolating between differentiation and integration by defining
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| :<math>D^\alpha_x f(x) =
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| \begin{cases}
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| \frac{d^{\lceil\alpha\rceil}}{dx^{\lceil\alpha\rceil}} I^{\lceil\alpha\rceil-\alpha}f(x)& \alpha>0\\
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| f(x) & \alpha=0\\
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| I^{-\alpha}f(x) & \alpha<0.
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| \end{cases}</math>
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| An alternative fractional derivative was introduced by Caputo in 1967, and produces a derivative that has different properties: it produces zero from constant functions and, more importantly, the initial value terms of the [[Laplace Transform]] are expressed by means of the values of that function and of its derivative of integer order rather than the derivatives of fractional order as in the Riemann–Liouville derivative.[http://www.nd.edu/~msen/Teaching/UnderRes/FracCalc.pdf] The Caputo fractional derivative with base point <math>x</math>, is then:
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| :<math>D_x^{\alpha}f(y)=\frac{1}{\Gamma(1-\alpha)}\int_x^y f'(y-u)(u-x)^{-\alpha}du.</math>
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| Another representation is:
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| :<math>{}_a\tilde{D}^\alpha_x f(x)=I^{\lceil \alpha\rceil-\alpha}\left(\frac{d^{\lceil \alpha\rceil}f}{dx^{\lceil \alpha\rceil}}\right).</math>
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| ==Notes==
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| <references/>
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| ==References==
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| *{{springer|title=Euler transformation|id=e/e036620|first1=Yu.A.|last1=Brychkov|first2=A.P.|last2=Prudnikov}}.
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| *{{Citation | last1=Hille | first1=Einar | last2=Phillips | first2=Ralph S. | title=Functional analysis and semi-groups | publisher=[[American Mathematical Society]] | location=Providence, R.I. | mr=0423094 | year=1974}}.
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| *{{springer|title=Fractional integration and differentiation|id=f/f041230|first=P.I.|last=Lizorkin}}.
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| *{{citation|title=An Introduction to the Fractional Calculus and Fractional Differential Equations|first1=Kenneth S.|last1=Miller|first2=Bertram|last2=Ross|publisher=John Wiley & Sons|year=1993|isbn=0-471-58884-9}}.
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| *{{Citation | doi=10.1007/BF02395016 | last1=Riesz | first1=Marcel | author1-link=Marcel Riesz | title=L'intégrale de Riemann-Liouville et le problème de Cauchy | mr=0030102 | year=1949 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=81 | issue=1 | pages=1–223}}.
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| ==External links==
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| *{{cite web|author=Alan Beardon|url=http://nrich.maths.org/public/viewer.php?obj_id=1369|title=Fractional calculus II|publisher=University of Cambridge|year=2000}}
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| *{{cite web|author=Alan Beardon|url=http://nrich.maths.org/public/viewer.php?obj_id=1371|title=Fractional calculus III|publisher=University of Cambridge|year=2000}}
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| {{DEFAULTSORT:Riemann-Liouville integral}}
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| [[Category:Fractional calculus]]
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| [[Category:Integral transforms]]
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For your offense, you might attain Gunboats which can easily shoot at enemy resistance coming from a very long range and Landing Crafts which you must seal when you train equipment for example Rifleman, Heavy, Zooka, Warrior and Fish tanks. To your village defenses, you might carry structures like Mortar, Machine Gun, Sniper Tower, Cannon, Flamethrower, Mine, Tank Mine, Boom Cannon and Bomb Launcher to assist we eradicate enemies.
when you are locating the latest handle system tough that will use, optimize the settings within your activity. The default manage strategy might not be for everyone. Some different people prefer a better view screen, a set within more sensitive management otherwise perhaps an inverted file format. In several tutorial gaming, you may master these from the setting's area.
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