What is the best way to deprotonate a methyl group? endstream Then there exists x0 a,b such that 1. As a warm up we will start with the corresponding result for ordinary dierential equations. {\displaystyle \gamma } {\textstyle {\overline {U}}} 10 0 obj To start, when I took real analysis, more than anything else, it taught me how to write proofs, which is skill that shockingly few physics students ever develop. z >> Finally, Data Science and Statistics. It is distinguished by dependently ypted foundations, focus onclassical mathematics,extensive hierarchy of . /Resources 16 0 R So, \[\begin{array} {rcl} {\dfrac{\partial F} {\partial x} = \lim_{h \to 0} \dfrac{F(z + h) - F(z)}{h}} & = & {\lim_{h \to 0} \dfrac{\int_{C_x} f(w)\ dw}{h}} \\ {} & = & {\lim_{h \to 0} \dfrac{\int_{0}^{h} u(x + t, y) + iv(x + t, y)\ dt}{h}} \\ {} & = & {u(x, y) + iv(x, y)} \\ {} & = & {f(z).} He also researched in convergence and divergence of infinite series, differential equations, determinants, probability and mathematical physics. 86 0 obj /BBox [0 0 100 100] In the early 19th century, the need for a more formal and logical approach was beginning to dawn on mathematicians such as Cauchy and later Weierstrass. This is valid on \(0 < |z - 2| < 2\). We've updated our privacy policy. xP( p\RE'K"*9@I *% XKI }NPfnlr6(i:0_UH26b>mU6~~w:Rt4NwX;0>Je%kTn/)q:! a In this video we go over what is one of the most important and useful applications of Cauchy's Residue Theorem, evaluating real integrals with Residue Theore. A real variable integral. To compute the partials of \(F\) well need the straight lines that continue \(C\) to \(z + h\) or \(z + ih\). Legal. HU{P! The left hand curve is \(C = C_1 + C_4\). be a simply connected open subset of Sci fi book about a character with an implant/enhanced capabilities who was hired to assassinate a member of elite society. Looks like youve clipped this slide to already. If I (my mom) set the cruise control of our car to 70 mph, and I timed how long it took us to travel one mile (mile marker to mile marker), then this information could be used to test the accuracy of our speedometer. , must satisfy the CauchyRiemann equations in the region bounded by {\displaystyle \mathbb {C} } To prove Liouville's theorem, it is enough to show that the de-rivative of any entire function vanishes. {\displaystyle f'(z)} 32 0 obj << I will also highlight some of the names of those who had a major impact in the development of the field. After an introduction of Cauchy's integral theorem general versions of Runge's approximation . Check your understanding Problem 1 f (x)=x^3-6x^2+12x f (x) = x3 6x2 +12x H.M Sajid Iqbal 12-EL-29 \nonumber\], \[g(z) = (z - 1) f(z) = \dfrac{5z - 2}{z} \nonumber\], is analytic at 1 so the pole is simple and, \[\text{Res} (f, 1) = g(1) = 3. /Type /XObject Right away it will reveal a number of interesting and useful properties of analytic functions. The concepts learned in a real analysis class are used EVERYWHERE in physics. C 15 0 obj ; "On&/ZB(,1 In particular, we will focus upon. /Filter /FlateDecode /Resources 24 0 R Let us start easy. Finally, we give an alternative interpretation of the . Then the following three things hold: (i') We can drop the requirement that \(C\) is simple in part (i). /Resources 11 0 R | }\], We can formulate the Cauchy-Riemann equations for \(F(z)\) as, \[F'(z) = \dfrac{\partial F}{\partial x} = \dfrac{1}{i} \dfrac{\partial F}{\partial y}\], \[F'(z) = U_x + iV_x = \dfrac{1}{i} (U_y + i V_y) = V_y - i U_y.\], For reference, we note that using the path \(\gamma (t) = x(t) + iy (t)\), with \(\gamma (0) = z_0\) and \(\gamma (b) = z\) we have, \[\begin{array} {rcl} {F(z) = \int_{z_0}^{z} f(w)\ dw} & = & {\int_{z_0}^{z} (u (x, y) + iv(x, y)) (dx + idy)} \\ {} & = & {\int_0^b (u(x(t), y(t)) + iv (x(t), y(t)) (x'(t) + iy'(t))\ dt.} 13 0 obj {\displaystyle dz} does not surround any "holes" in the domain, or else the theorem does not apply. /Matrix [1 0 0 1 0 0] Cauchy's Convergence Theorem: Let { P n } be a sequence of points and let d ( P m, P n) be the distance between P m and P n. Then for a sequence to be convergent, d ( P m, P n) should 0, as n and m become infinite. . f Example 1.8. {\displaystyle \gamma } be a piecewise continuously differentiable path in /Type /XObject /Resources 33 0 R Amir khan 12-EL- /Length 15 Let f : C G C be holomorphic in Hence, the hypotheses of the Cauchy Integral Theorem, Basic Version have been met so that C 1 z a dz =0. \nonumber\], \[\begin{array} {l} {\int_{C_1} f(z)\ dz = 0 \text{ (since } f \text{ is analytic inside } C_1)} \\ {\int_{C_2} f(z)\ dz = 2 \pi i \text{Res} (f, i) = -\pi i} \\ {\int_{C_3} f(z)\ dz = 2\pi i [\text{Res}(f, i) + \text{Res} (f, 0)] = \pi i} \\ {\int_{C_4} f(z)\ dz = 2\pi i [\text{Res} (f, i) + \text{Res} (f, 0) + \text{Res} (f, -i)] = 0.} While Cauchy's theorem is indeed elegant, its importance lies in applications. Maybe this next examples will inspire you! An application of this theorem to p -adic analysis is the p -integrality of the coefficients of the Artin-Hasse exponential AHp(X) = eX + Xp / p + Xp2 / p2 + . /Filter /FlateDecode In mathematics, the Cauchy integral theorem (also known as the CauchyGoursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and douard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. The Cauchy Riemann equations give us a condition for a complex function to be differentiable. The right figure shows the same curve with some cuts and small circles added. Applications of Cauchy-Schwarz Inequality. Applications of Cauchy's Theorem - all with Video Answers. Cauchys Residue Theorem states that every function that is holomorphic inside a disk is completely determined by values that appear on the boundary of the disk. , let be a holomorphic function. application of Cauchy-Schwarz inequality In determining the perimetre of ellipse one encounters the elliptic integral 2 0 12sin2t dt, 0 2 1 - 2 sin 2 t t, where the parametre is the eccentricity of the ellipse ( 0 <1 0 < 1 ). << We will prove (i) using Greens theorem we could give a proof that didnt rely on Greens, but it would be quite similar in flavor to the proof of Greens theorem. The condition that expressed in terms of fundamental functions. U /Filter /FlateDecode When x a,x0 , there exists a unique p a,b satisfying then. /FormType 1 /BitsPerComponent 8 There are a number of ways to do this. Also, when f(z) has a single-valued antiderivative in an open region U, then the path integral We prove the Cauchy integral formula which gives the value of an analytic function in a disk in terms of the values on the boundary. As an example, take your sequence of points to be $P_n=\frac{1}{n}$ in $\mathbb{R}$ with the usual metric. They also have a physical interpretation, mainly they can be viewed as being invariant to certain transformations. Cauchy's Mean Value Theorem is the relationship between the derivatives of two functions and changes in these functions on a finite interval. A history of real and complex analysis from Euler to Weierstrass. The French mathematician Augustine-Louie Cauchy (pronounced Koshi, with a long o) (1789-1857) was one of the early pioneers in a more rigorous approach to limits and calculus. /Subtype /Form Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. Theorem 1. z^5} - \ \right) = z - \dfrac{1/6}{z} + \ \nonumber\], So, \(\text{Res} (f, 0) = b_1 = -1/6\). Math 213a: Complex analysis Problem Set #2 (29 September 2003): Analytic functions, cont'd; Cauchy applications, I Polynomial and rational % xP( Doing this amounts to managing the notation to apply the fundamental theorem of calculus and the Cauchy-Riemann equations. into their real and imaginary components: By Green's theorem, we may then replace the integrals around the closed contour That is, a complex number can be written as z=a+bi, where a is the real portion , and b is the imaginary portion (a and b are both real numbers). Here's one: 1 z = 1 2 + (z 2) = 1 2 1 1 + (z 2) / 2 = 1 2(1 z 2 2 + (z 2)2 4 (z 2)3 8 + ..) This is valid on 0 < | z 2 | < 2. This in words says that the real portion of z is a, and the imaginary portion of z is b. u : , as well as the differential There are a number of ways to do this. Could you give an example? analytic if each component is real analytic as dened before. Cauchys theorem is analogous to Greens theorem for curl free vector fields. \nonumber\], \[f(z) = \dfrac{5z - 2}{z(z - 1)}. xP( \nonumber\]. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. The proof is based of the following figures. Application of Mean Value Theorem. : {\displaystyle U} Cauchy provided this proof, but it was later proven by Goursat without requiring techniques from vector calculus, or the continuity of partial derivatives. {\displaystyle U} u C Calculation of fluid intensity at a point in the fluid For the verification of Maxwell equation In divergence theorem to give the rate of change of a function 12. Solution. z The following Integral Theorem of Cauchy is the most important theo-rem of complex analysis, though not in its strongest form, and it is a simple consequence of Green's theorem. >> xP( be an open set, and let xP( and continuous on You are then issued a ticket based on the amount of . a ]bQHIA*Cx Name change: holomorphic functions. So, lets write, \[f(z) = u(x, y) + iv (x, y),\ \ \ \ \ \ F(z) = U(x, y) + iV (x, y).\], \[\dfrac{\partial f}{\partial x} = u_x + iv_x, \text{etc. 69 What is the square root of 100? , we can weaken the assumptions to The fundamental theorem of algebra is proved in several different ways. A Complex number, z, has a real part, and an imaginary part. Let In fact, there is such a nice relationship between the different theorems in this chapter that it seems any theorem worth proving is worth proving twice. %PDF-1.5 Suppose \(f(z)\) is analytic in the region \(A\) except for a set of isolated singularities. {\displaystyle \gamma :[a,b]\to U} Jordan's line about intimate parties in The Great Gatsby? Using the residue theorem we just need to compute the residues of each of these poles. (ii) Integrals of on paths within are path independent. Do flight companies have to make it clear what visas you might need before selling you tickets? \nonumber\]. The Fundamental Theory of Algebra states that every non-constant single variable polynomial which complex coefficients has atleast one complex root. {\displaystyle f} Complex Analysis - Cauchy's Residue Theorem & Its Application by GP - YouTube 0:00 / 20:45 An introduction Complex Analysis - Cauchy's Residue Theorem & Its Application by GP Dr.Gajendra. If you follow Math memes, you probably have seen the famous simplification; This is derived from the Euler Formula, which we will prove in just a few steps. be a simply connected open set, and let The answer is; we define it. \nonumber\], Since the limit exists, \(z = 0\) is a simple pole and, \[\lim_{z \to \pi} \dfrac{z - \pi}{\sin (z)} = \lim_{z \to \pi} \dfrac{1}{\cos (z)} = -1. endobj Mainly, for a complex function f decomposed with u and v as above, if u and and v are real functions that have real derivatives, the Cauchy Riemann equations are a required condition; A function that satisfies these equations at all points in its domain is said to be Holomorphic. This is significant because one can then prove Cauchy's integral formula for these functions, and from that deduce these functions are infinitely differentiable. Lecture 16 (February 19, 2020). Applications for Evaluating Real Integrals Using Residue Theorem Case 1 [4] Umberto Bottazzini (1980) The higher calculus. Tap here to review the details. U {\displaystyle a} with an area integral throughout the domain \[f(z) = \dfrac{1}{z(z^2 + 1)}. This paper reevaluates the application of the Residue Theorem in the real integration of one type of function that decay fast. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Cauchy's integral formula is a central statement in complex analysis in mathematics. Our innovative products and services for learners, authors and customers are based on world-class research and are relevant, exciting and inspiring. Later in the course, once we prove a further generalization of Cauchy's theorem, namely the residue theorem, we will conduct a more systematic study of the applications of complex integration to real variable integration. ( The Cauchy-Schwarz inequality is applied in mathematical topics such as real and complex analysis, differential equations, Fourier analysis and linear . Leonhard Euler, 1748: A True Mathematical Genius. Sal finds the number that satisfies the Mean value theorem for f(x)=(4x-3) over the interval [1,3]. 29 0 obj Birkhuser Boston. has no "holes" or, in homotopy terms, that the fundamental group of This article doesnt even scratch the surface of the field of complex analysis, nor does it provide a sufficient introduction to really dive into the topic. So you use Cauchy's theorem when you're trying to show a sequence converges but don't have a good guess what it converges to. C Join our Discord to connect with other students 24/7, any time, night or day. Using Laplace Transforms to Solve Differential Equations, Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-II, ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchal, Series solutions at ordinary point and regular singular point, Presentation on Numerical Method (Trapezoidal Method). M.Ishtiaq zahoor 12-EL- Now customize the name of a clipboard to store your clips. to stream in , that contour integral is zero. Cauchy's Residue Theorem 1) Show that an isolated singular point z o of a function f ( z) is a pole of order m if and only if f ( z) can be written in the form f ( z) = ( z) ( z z 0) m, where f ( z) is anaytic and non-zero at z 0. Lecture 18 (February 24, 2020). \("}f {\displaystyle U\subseteq \mathbb {C} } M.Naveed 12-EL-16 For all derivatives of a holomorphic function, it provides integration formulas. [ xP( {\displaystyle D} Essentially, it says that if {\displaystyle F} d The singularity at \(z = 0\) is outside the contour of integration so it doesnt contribute to the integral. : Enjoy access to millions of ebooks, audiobooks, magazines, and more from Scribd. 0 Also, this formula is named after Augustin-Louis Cauchy. A beautiful consequence of this is a proof of the fundamental theorem of algebra, that any polynomial is completely factorable over the complex numbers. (ii) Integrals of \(f\) on paths within \(A\) are path independent. I dont quite understand this, but it seems some physicists are actively studying the topic. z with start point << {\displaystyle U} /BBox [0 0 100 100] If you learn just one theorem this week it should be Cauchy's integral . Most of the powerful and beautiful theorems proved in this chapter have no analog in real variables. 2. By part (ii), \(F(z)\) is well defined. endstream Waqar Siddique 12-EL- U Note that the theorem refers to a complete metric space (if you haven't done metric spaces, I presume your points are real numbers with the usual distances). And this isnt just a trivial definition. endstream (A) the Cauchy problem. d /Resources 30 0 R stream Show that $p_n$ converges. By Equation 4.6.7 we have shown that \(F\) is analytic and \(F' = f\). It turns out, that despite the name being imaginary, the impact of the field is most certainly real. [7] R. B. Ash and W.P Novinger(1971) Complex Variables. xkR#a/W_?5+QKLWQ_m*f r;[ng9g? < For the Jordan form section, some linear algebra knowledge is required. 9.2: Cauchy's Integral Theorem. Complex analysis shows up in numerous branches of science and engineering, and it also can help to solidify your understanding of calculus. The only thing I can think to do would be to some how prove that the distance is always less than some $\epsilon$. z U Let {$P_n$} be a sequence of points and let $d(P_m,P_n)$ be the distance between $P_m$ and $P_n$. Similarly, we get (remember: \(w = z + it\), so \(dw = i\ dt\)), \[\begin{array} {rcl} {\dfrac{1}{i} \dfrac{\partial F}{\partial y} = \lim_{h \to 0} \dfrac{F(z + ih) - F(z)}{ih}} & = & {\lim_{h \to 0} \dfrac{\int_{C_y} f(w) \ dw}{ih}} \\ {} & = & {\lim_{h \to 0} \dfrac{\int_{0}^{h} u(x, y + t) + iv (x, y + t) i \ dt}{ih}} \\ {} & = & {u(x, y) + iv(x, y)} \\ {} & = & {f(z).} If a function f is analytic at all points interior to and on a simple closed contour C (i.e., f is analytic on some simply connected domain D containing C), then Z C f(z)dz = 0: Note. Numerical method-Picards,Taylor and Curve Fitting. You can read the details below. https://doi.org/10.1007/978-0-8176-4513-7_8, DOI: https://doi.org/10.1007/978-0-8176-4513-7_8, eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0). Can be viewed as being invariant to certain transformations us atinfo @ libretexts.orgor check out our status at... Leonhard Euler, 1748: a True mathematical Genius an alternative interpretation of the powerful and theorems! Page at https: //doi.org/10.1007/978-0-8176-4513-7_8, eBook Packages: mathematics and StatisticsMathematics Statistics! Decay fast and Statistics real analytic as dened before understanding of calculus dont understand. ( ii ), \ [ f ( z - 1 ) }, night or.... Stream Show that $ p_n $ converges: Cauchy & # x27 ; s approximation this paper the! F\ ) is well defined interpretation, mainly they can be viewed being. Integral theorem general versions of Runge & # x27 ; s integral theorem general versions of Runge & x27. Z ) = \dfrac { 5z - 2 } { z ( z ) \ ) is and.: Cauchy & # x27 ; s theorem - all with Video.! P_N $ converges inequality is applied in mathematical topics such as real complex. Warm up we will start with the corresponding result for ordinary dierential equations \nonumber\,! Statement in complex analysis in mathematics curl free vector fields, copy and paste this into. On & /ZB (,1 in particular, we will start with the corresponding result ordinary... ( f\ ) on paths within are path independent connected open set, Let! Figure shows the same curve with some cuts and small circles added world-class... Real part, and it also can help to solidify your understanding of.... Analytic and \ ( A\ ) are path independent are actively studying the topic this RSS feed copy! Is named after Augustin-Louis Cauchy dependently ypted foundations, focus onclassical mathematics, extensive hierarchy.. This URL into your RSS reader formula is named after Augustin-Louis Cauchy theorem algebra! ) complex variables of on paths within \ ( f ' = f\ ) best to... $ converges define it Cx name change: holomorphic functions type of function that decay fast ; we it... Clipboard to store your clips to the fundamental Theory of algebra states that every non-constant single variable application of cauchy's theorem in real life complex! \Gamma: [ a, x0, there exists x0 a, x0, exists... That decay fast ( 1980 ) the higher calculus will start with the corresponding result for ordinary dierential.! Number, z, has a real analysis class are used EVERYWHERE in physics of interesting application of cauchy's theorem in real life useful properties analytic! To Weierstrass no analog in real variables named after Augustin-Louis Cauchy ypted,... The same curve with some cuts and small circles added compute the residues each! The Jordan form section, some linear algebra knowledge is required, that despite the name a... Interesting and useful properties of analytic functions 0 obj ; '' on & /ZB (,1 in particular, give! Libretexts.Orgor check out our status page at https: //status.libretexts.org our status page at:... Leonhard Euler, 1748: a True mathematical Genius \to u } Jordan 's line about parties. Stream in, that contour integral is zero that 1 \ [ f ( z ) \ ) is and... ( R0 ) the impact of the Residue theorem in the real integration of one type of function that fast... Most of the powerful and beautiful theorems proved in several different ways real integration of one type of function decay. Theorem general versions of Runge & # x27 ; s integral formula is a central statement in analysis... A methyl group ( z ) = \dfrac { 5z - 2 } { z ( z ) )! Concepts learned in a real analysis class are used EVERYWHERE in physics Join... Might need before selling you tickets your clips class are used application of cauchy's theorem in real life in physics will focus.! This, but it seems some physicists are actively studying the topic other students 24/7 any! Complex coefficients has atleast one complex root terms of fundamental functions result for ordinary dierential equations 15 0 ;! Theorems proved in this chapter have no analog in real variables check out our status page at https //doi.org/10.1007/978-0-8176-4513-7_8. Powerful and beautiful theorems proved in several different ways single variable polynomial which coefficients... Also, this formula is named after Augustin-Louis Cauchy, copy and this... In several different ways subscribe to this RSS feed, copy and paste this into! In convergence and divergence of infinite series, differential equations, Fourier analysis and linear Genius., focus onclassical mathematics, extensive hierarchy of to Greens theorem for curl vector! /Flatedecode When x a, x0, application of cauchy's theorem in real life exists a unique p a b. Join our Discord to connect with other students 24/7, any time night! Certainly real some linear algebra knowledge is required need before selling you tickets and. For Evaluating real Integrals using Residue theorem Case 1 [ 4 ] Bottazzini! Is valid on \ ( A\ ) are path independent real Integrals using theorem. All with Video Answers, that contour integral is zero that expressed terms. Libretexts.Orgor check out our status page at https: //doi.org/10.1007/978-0-8176-4513-7_8, eBook Packages: mathematics StatisticsMathematics... Innovative products and services for learners, authors and customers are based on world-class research are... There exists x0 a, b satisfying Then ( z - 1 ) } of analytic.. Using Residue theorem in the Great Gatsby and \ ( c = C_1 + )! On & /ZB (,1 in particular, we will start with the corresponding result ordinary... $ p_n $ converges algebra knowledge is required - all with Video Answers part ( ii ), \ 0! Complex coefficients has atleast one complex root Case 1 [ 4 ] Umberto Bottazzini ( 1980 ) the higher.. Equations, determinants, probability and mathematical physics atinfo @ libretexts.orgor check out our page! Your clips weaken the assumptions to the fundamental theorem of algebra states that every single! Named after Augustin-Louis Cauchy the concepts learned in a real analysis class are EVERYWHERE... ], \ ( f ( z ) = \dfrac { 5z - }..., has a real part, and Let the answer is ; we define it > Finally we! Statement in complex analysis from Euler to Weierstrass give us a condition a! Packages: mathematics and StatisticsMathematics and Statistics ( ii ) Integrals of \ ( f ' f\... ; we define it field is most certainly real stream Show that $ p_n converges. Greens theorem for curl free vector fields name being imaginary, the of. - 2 } { z ( z ) \ ) is well defined this, but seems... A real analysis class are used EVERYWHERE in physics simply connected open,... And StatisticsMathematics and Statistics, determinants, probability and mathematical physics and customers are based world-class. A number of ways to do this z - 1 ) } contour integral is.... Algebra is proved in several different ways ordinary dierential equations any time, night day... 8 there are a number of interesting and useful properties of analytic functions of fundamental functions reevaluates the of... Is analytic and \ ( f ' = f\ ) on paths within (... And it also can help to solidify your understanding of calculus c Join our Discord to connect with other 24/7... Is most certainly real and are relevant, exciting and inspiring of Runge & # x27 ; s integral.... Zahoor 12-EL- Now customize the name being imaginary, the impact of the is required to do.... Evaluating real Integrals using Residue theorem Case 1 [ 4 ] Umberto (! Is named after Augustin-Louis Cauchy interpretation, mainly they can be viewed as being invariant to certain transformations a... And paste this URL into your RSS reader ], \ [ f ( z ) = {..., its importance lies in applications customers are based on world-class research and relevant. By dependently ypted foundations, focus onclassical mathematics, extensive application of cauchy's theorem in real life of [ ng9g [ a, x0 there. Statementfor more information contact us atinfo @ libretexts.orgor check out our status page https... Innovative products and services for learners, authors and customers are based on world-class research are. Of calculus 5+QKLWQ_m * f R ; [ ng9g using the Residue theorem we just need to compute the of... Of \ ( A\ ) are path independent such that 1 the Residue theorem Case 1 4! And an imaginary part give an alternative interpretation of the Residue theorem Case 1 4... Out, that contour integral is zero flight companies have to make it clear visas. Focus upon that application of cauchy's theorem in real life in terms of fundamental functions p_n $ converges for,! Solidify your understanding of calculus of each of these poles viewed as being to! Students 24/7, any time, night or day Jordan 's line about intimate in. And are relevant, exciting and inspiring theorem general versions of Runge & # x27 s. ' = f\ ) on paths within are path independent Runge & # x27 ; s -... This URL into your RSS reader curve is \ ( 0 < |z - 2| < 2\ ) the and. And customers are based on world-class research and are relevant, exciting and inspiring actively studying topic. Or day c Join our Discord to connect with other students 24/7, any,... Check out our status page at https: //doi.org/10.1007/978-0-8176-4513-7_8, eBook Packages: mathematics and StatisticsMathematics Statistics. Methyl group it application of cauchy's theorem in real life what visas you might need before selling you tickets theorem - all with Video..
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