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Lecture22 - 17 Conformal Mapping We now go back to the main...

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17 Conformal Mapping We now go back to the “main road”, which leads from Chapter 12 (PDEs), via Chapters 13, 14 & 17, to Chapter 18 (Potential Theory in 2D). In this chapter, we interpret complex functions f : C C as mappings from R 2 to R 2 , and thus consider a geometric approach to complex anal- ysis. This new approach gives new insight on holomorphic functions; its importance is similar to the study of curves { ( x, f ( x )) R 2 | x R } for real functions. Conformal mapping will also yield a standard method for solving bound- ary value problems in two-dimensional potantial theory, by transforming a complicated region into a simple one. 17.1 Geometry of Holomorphic Functions: Conformal Mapping We interpret a complex function f as a mapping of its domain of definition (in C R 2 ) onto its image (in C R 2 ): w = f ( z ) = u ( x, y ) + iv ( x, y ) , z = x + iy. (979) We shall refer to such a mapping as “the mapping w = f ( z )”, and we refer to the “ z -plane” and the “ w -plane” to distinguish between the spaces containing the domain and image of f . We use cartesian coordinates ( x, y ) and ( u, v ) or polar coordinates ( r, ϑ ) and ( R, ϕ ) to represent points in the z - and w -planes, respectively. Example: The mapping w = z 2 . Using polar forms z = re in the z -plane and w = Re in the w -plane, we have w = z 2 = r 2 e 2 , and therefore R = r 2 , ϕ = 2 ϑ (cf. Section 13.2). Hence circles | z | = r = r 0 in the z -plane are mapped onto circles | w | = R = r 2 0 in the w -plane and rays arg z = ϑ = ϑ 0 in the z -plane onto rays arg w = ϕ = 2 ϑ 0 in the w -plane. In Cartesian coordinates we have z = x + iy and u = Re w = Re( z 2 ) = x 2 - y 2 , v = Im w = Im( z 2 ) = 2 xy. (980) Hence vertical lines x = c = const . in the z -plane are mapped onto u = c 2 - y 2 , v = 2 cy . So we obtain v 2 = 4 c 2 y 2 = 4 c 2 ( c 2 - u ) , (981) 174
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which defines for each c R a parabola in the w -plane that opens to the left. Similarly, horizontal lines y = k = const . in the z -plane are mapped onto parabolas in the w -plane, v 2 = 4 k 2 ( k 2 + u ) , (982) which are parabolas opening to the right, for each k R . Conformal Mapping A mapping w = f ( z ) is called conformal if it pre- serves angles between oriented curves, in magnitude as well as in sense. Theorem 34 The mapping w = f ( z ) by a holomorphic function f is con- formal, except at critical points, that is, points at which the derivative of f vanishes. Proof: We consider a curve C = { γ ( t ) | t [ a, b ] } in the z -plane, with γ : [ a, b ] C (cf. Section 14.1). The tangent to C at z 0 = γ ( t 0 ) C is given by ˙ γ ( t 0 ) C , which is to be understood as a vector in the z -plane, attached to z 0 . The image of C under f is given by the curve C
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