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Conformal Mapping and its Applications - IISER Pune

Conformal Mapping and its ApplicationsSuman Ganguli11 Department of Physics, University of Tennessee, Knoxville, TN 37996(Dated: November 20, 2008) Conformal (Same form or shape) Mapping is an important technique used in complex analysisand has many Applications in different physical the function is harmonic (ie it satisfiesLaplace s equation 2f= 0 )then the transformation of such functions via Conformal Mapping is alsoharmonic. So equations pertaining to any field that can be represented by a potential function (allconservative fields) can be solved via Conformal the physical problem can be representedby complex functions but the geometric structure becomes inconvenient then by an appropriatemapping it can be transferred to a problem with much more convenient geometry.

Conformal Mapping and its Applications Suman Ganguli1 1Department of Physics, University of Tennessee, Knoxville, TN 37996 (Dated: November 20, 2008) Conformal (Same form or shape) mapping is an important technique used in complex analysis

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Transcription of Conformal Mapping and its Applications - IISER Pune

1 Conformal Mapping and its ApplicationsSuman Ganguli11 Department of Physics, University of Tennessee, Knoxville, TN 37996(Dated: November 20, 2008) Conformal (Same form or shape) Mapping is an important technique used in complex analysisand has many Applications in different physical the function is harmonic (ie it satisfiesLaplace s equation 2f= 0 )then the transformation of such functions via Conformal Mapping is alsoharmonic. So equations pertaining to any field that can be represented by a potential function (allconservative fields) can be solved via Conformal the physical problem can be representedby complex functions but the geometric structure becomes inconvenient then by an appropriatemapping it can be transferred to a problem with much more convenient geometry.

2 This article givesa brief introduction to Conformal mappings and some of its Applications in physical numbers:I. INTRODUCTIONA Conformal map is a function which preserves map preserves both angles and shapeof infinitesimal small figures but not necessarily formally, a mapw=f(z)(1)is called Conformal (or angle-preserving) atz0if it pre-serves oriented angles between curves throughz0, as wellas their orientation, important family of examples of Conformal mapscomes from complex analysis. If U is an open subset ofthe complex plane, , then a functionf :U Cis Conformal if and only if it is holomorphic and itsderivative is everywhere non-zero on U. If f is antiholo-morphic (that is, the conjugate to a holomorphic func-tion), it still preserves angles, but it reverses their Riemann Mapping theorem, states that any non-empty open simply connected proper subset of C admitsa bijective Conformal map to the open unit disk(the openunit disk around P (where P is a given point in the plane),is the set of points whose distance from P is less than 1)in complex plane C ie if U is a simply connected opensubset in complex plane C, which is not all of C,thenthere exists a bijective ie one-to-one Mapping f from Uto open unit disk :U DwhereD ={z C.}

3 |z|<1}As f is a bijective map it is map of the extended complex plane (which is con-formally equivalent to a sphere) onto itself is conformalif and only if it is a Mobius transformation ie a transfor-mation leading to a rational function of the form f(z) =az+bcz+d. Again, for the conjugate, angles are preserved, butorientation is 1: Mapping of graphII. BASIC THEORYLet us consider a functionw=f(z)(2)wherez=x+iyandw=u+ivWe find thatdz=dx+idy, anddw=du+idv,|dz|2=dx2+dy2,(3)and|dw|2=d u2+dv2(4)Then the square of the length element in (x,y) plane isds2=dx2+dy2(5)and square of the length element in (u,v) plane isdS2=du2+dv2(6)From equations (3) , (4), (5), (6) we find that,dS/ds=|dw/dz|.(7)ie the ratio of arc lengths of two planes remains essentiallyconstant in the neighborhood of each point in z plane pro-videdw(z) is analytic and have a nonzero or finite slopeat that implies the linear dimensions in twoplanes are proportional and the net result of this transfor-mation is to change the dimensions in equal proportionsand rotate each infinitesimal area in the neighborhood ofthat point.

4 Thus the angle (which is represented as theratio of linear dimensions) is preserved although shape ina large scale will not be preserved in general as the valueof|dw/dz|will vary considerably at different points inthe plane. Due to this property such transformations arecalled Conformal . This leads to the following :Assume that f (z) is analytic and not con-stant in a domain D of the complex z plane. For any pointz D for which f(z)6=0, this Mapping is Conformal ,that is, it preserves the angle between two : Let D be the rectangular region in the zplane bounded by x = 0, y = 0, x = 2 and y = 1. Theimage of D under the transformation w = (1 + )z +(1 +2 ) is given by the rectangular region D of the w planebounded by u + v = 3, u - v = -1, u + v = 7 and u - v= w = u + v, where u, v R, then u = x - y + 1, v =x + y + 2.

5 Thus the points a, b, c, and d are mapped tothe points (0,3), (1,2), (3,4), and (2,5), respectively. Theline x = 0 is mapped to u = -y+1, v = y+2, or u+v = 3;similarly for the other sides of the rectangle (Fig 2). Therectangle D is translated by (1 + 2 ), rotated by an angle /4 in the counterclockwise direction, and dilated by afactor (2). In general, a linear transformation f(z) = z + , translates by , rotates by arg(/ ), and dilates(or contracts) by| |. Because f(z) = 6= 0, a lineartransformation is always 2: Mapping of a rectangleThe below theorem (stated without proof), related toinverse Mapping , is an important property of conformalmapping as it states that inverse Mapping also preservesthe :Assume that f (z) is analytic atz0and thatf (z)6=0.

6 Then f (z) is univalent in the neighborhood ofz0. More precisely, f has a unique analytic inverse F inthe neighborhood ofw0 f(z0); that is, if z is sufficientlynearz0, then z = F(w), where w f(z). Similarly, if w issufficiently nearw0and z F(w), then w = f (z). Fur-thermore, f (z)F (w) = 1, which implies that the inversemap is uniqueness and Conformal property of inversemapping allows us to map the solution obtained in w-plane to Points: If f (z0) = 0, then the analytic trans-formation f(z) ceases to be Conformal . Such a point iscalled a critical point of f . Because critical points arezeroes of the analytic function f , they are APPLICATIONSA large number of problems arising in fluid mechanics,electrostatics, heat conduction, and many other physicalsituations can be mathematically formulated in terms ofLaplaces equation.

7 Ie, all these physical problems reduceto solving the equation xx+ yy= 0.(8)in a certain region D of the z plane. The function (x,y), in addition to satisfying this equation also satisfiescertain boundary conditions on the boundary C of theregion D. From the theory of analytic functions we knowthat the real and the imaginary parts of an analytic func-tion satisfy Laplaces equation. It follows that solving theabove problem reduces to finding a function that is ana-lytic in D and that satisfies certain boundary conditionson C. It turns out thatthe solution of this problem can begreatly simplified if the region D is either the upper halfof the z plane or the unit : Consider two infinite parallel flat plates,separated by a distanced and maintained at zero poten-tial.

8 A line of charge q per unit length is located betweenthe two planes at a distance a from the lower plate .Theproblem is to find the electrostatic potential in the shadedregion of the z Conformal Mapping w = exp( z/d) maps theshaded strip of the z plane onto the upper half of thew plane. So the point z = a is mapped to the pointw0= exp( a/d); the points on the lower plate, z = x, andon the upper plate, z = x + d, map to the real axis w= u for u>0 and u<0, respectively. Let us considera line of charge q atw0and a line of charge -q the associated complex potential (w) = 2 log(w w0)+2qlog(w w0) = 2qlog(w w0w w0)(9)2 FIG. 3: Mapping of two infinite parallel conducting plate witha charge in betweenCallingCqa closed contour around the charge q, wesee that Gauss law is satisfied, CqEnds=Im Cq Edz=Im Cq (w) = 4 q(10)where Cqis the image ofCqin the w-plane.

9 Then,calling = + , we see that is zero on the realaxis of the w plane. Consequently, we have satisfied theboundary condition = 0 on the plates, and hence theelectrostatic potential at any point of the shaded regionof the z plane is given by = 2qlog(w e vw e v)(11)wherev= a/dConformal mappings are invaluable for solving prob-lems in engineering and physics that can be expressedin terms of functions of a complex variable, but that ex-hibit inconvenient geometries. By choosing an appropri-ate Mapping , the analyst can transform the inconvenientgeometry into a much more convenient one. For exam-ple, one may be desirous of calculating the electric field,E(z), arising from a point charge located near the cornerof two conducting planes making a certain angle (wherez is the complex coordinate of a point in 2-space).

10 Thisproblem is quite clumsy to solve in closed , by employing a very simple Conformal map-ping, the inconvenient angle is mapped to one of pre-cisely pi radians, meaning that the corner of two planesis transformed to a straight line. In this new domain, theproblem, that of calculating the electric field impressedby a point charge located near a conducting wall, is quiteeasy to solution is obtained in this domain, E(w), andthen mapped back to the original domain by noting thatw was obtained as a function (viz., the composition ofE and w) of z, whence E(w) can be viewed as E(w(z)),which is a function of z, the original coordinate this application is not a contradiction to the factthat Conformal mappings preserve angles, they do so onlyfor points in the interior of their domain, and not at 4: Two semiinfinite plane conductors meet at an angle0< < /2 and are charged at constant potentials 1and 2 FIG.


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