Tham khảo tài liệu 'mathematical method in science and engineering episode 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | MAPPINGS 313 Fig. Conformal mapping Conformal Mappings To see an interesting property of analytic functions we differentiate w f z at Z where the modulus and the arguments of the derivative are given as and a respectively. We now use polar coordinates to write the modulus z I ZQ lim AzAo Aw Ă7 and the argument Fig. as arg 72 dz . Aw a arg lim I aAzAo Az a Jim arg Aw Jim0 arg Az . Since this function f z maps a curve c2 in the z-plane into another curve cw in the w-plane from the arguments Eq. it is seen that if the slope of c2 at Zq is 00 then the slope of cw at Wo is a ffo- For a pair of curves intersecting at Zo the angle between their tangents in the w- and z-planes will be equal that is ự 2 - ỉ ĩ Ỡ2 a - Ổ1 a 2 1- Hence analytic functions preserve angles between the curves they map Fig. . For this reason they are also called conformal mappings or transformations. 314 COMPLEX VARIABLES AND FUNCTIONS Electrostatics and Conformal Mappings Conformal mappings are very useful in electrostatic and laminar irrotational flow problems where the Laplace equation must be solved. Even though the method is restricted to cases with one translational symmetry it allows one to solve analytically some complex boundary value problems. Example . Conformal mappings and electrostatics Let US consider two conductors held at potentials Vj and V2 with hyperbolic cross sections X2 y2 Cj and x2 y2 C2- We want to find the equipotentials and the electric field lines. In the complex z-plane the problem can be shown as in Figure . We use the conformal mapping w z2 x2 -y2 i 2xy to map these hyperbolae to the straight lines u Cj and u C 2 in the w-plane Fig. . The problem is now reduced to finding the equipotentials and the electric field lines between two infinitely long MAPPINGS 315 Fig. Equipotentiais and electric field lines in the w-plane parallel plates
