Chapter 9. Conformal Mapping Weiqi Luo (骆伟祺) School of Software Sun Yat-Sen University Email:weiqi.luo@yahoo.com Office:# A313 Chapter 9: Conformal Mapping Preservation of Angles Scale Factors Local Inverses Harmonic Conjugates 2 School of Software 101. Preservation of Angles Preservation of Angles Let C be a smooth arc, represented by the equation z z (t ),(a t b) and let f(z) be a function defined at all points z on C. The equation w f [ z (t )],(a t b) is a parametric representation of the image Г of C under the transformation w=f(z). 3 School of Software 101. Preservation of Angles Suppose that C passes through a point z0=z(t0) (a<t0<b) at which f is analytic and f’(z0)≠0. According to the chain rule, if w(t)=f[z(t)], then w '(t0 ) f '[ z(t0 )]z '(t0 ) arg w '(t0 ) arg f '[ z(t0 )] arg z '(t0 ) 0 0 0 0 arg f '( z0 ) 4 School of Software 101. Preservation of Angles Consider two intersectant arcs C1 and C2 1 0 1 2 0 2 0 1 1 2 2 1 2 1 2 from C1 to C2 from Г1 to Г 2 Note that both magnitude and sense are the same. 5 School of Software 101. Preservation of Angles Conformal A transformation w=f(z) is said to be conformal at a point z0 if f is analytic there and f’(z0)≠0. Note that such a transformation is actually conformal at each point in some neighborhood of z0. For it must be analytic in a neighborhood of z0; and since its derivative f’ is continuous in that neighborhood, Theorem 2 in Sec. 18 tells us that there is also a neighborhood of z0 throughout which f (z) ≠ 0. 6 School of Software 101. Preservation of Angles Conformal Mapping A transformation w = f (z), defined on a domain D, is referred to as a conformal transformation, or conformal mapping, when it is conformal at each point in D. That is, the mapping is conformal in D if f is analytic in D and its derivative f has no zeros there. 7 School of Software 101. Preservation of Angles Example 1 The mapping w = ez is conformal throughout the entire z plane since (ez)’ = ez ≠ 0 for each z. Consider any two lines x = c1 and y = c2 in the z plane, the first directed upward and the second directed to the right. 8 School of Software 101. Preservation of Angles Isogonal mapping A mapping that preserves the magnitude of the angle between two smooth arcs but not necessarily the sense is called an isogonal mapping. Example 3 The transformation w z , which is a reflection in the real axis, is isogonal but not conformal. If it is followed by a conformal transformation, the resulting transformation w f ( z ) is also isogonal but not conformal. 9 School of Software 101. Preservation of Angles Critical Point Suppose that f is not a constant function and is analytic at a point z0 . If, in addition, f ’(z0) = 0, then z0 is called a critical point of the transformation w = f (z). Example 4. The point z0 = 0 is a critical point of the transformation w = 1 + z2, which is a composition of the mappings Z = z2 and w = 1 + Z. A ray θ = α from the point z0 = 0 is evidently mapped onto the ray from the point w0 = 1 whose angle of inclination is 2α, and the angle between any two rays drawn from z0 = 0 is doubled by the transformation. 10 School of Software 102. Scale Factors Scale factor From the definition of derivative, we know that f ( z ) f ( z0 ) | f '( z0 ) || lim | z z0 z z0 | f ( z ) f ( z0 ) | lim z z0 | z z0 | Exercise 7, Sec. 18 Now |z-z0| is the length of a line segment joining z0 and z, and |f(z)-f(z0)| is the length of the line segment joining the point f(z0) and f(z) in the w plane. Expansion:| f '( z0 ) | 1 Contraction:| 11 f '( z0 ) | 1 School of Software 102. Scale Factors Example When f(z)=z2, the transformation w f ( z) x y i 2xy 2 2 is conformal at the point z=1+i, where the half lines y x,( x 0) & x 1,( y 1) intersect. 12 School of Software 102. Scale Factors Illustrations w f ( z ) x 2 y 2 i 2 xy u( x, y) iv( x, y) C1: y=x, x ≥ 0 Г1: u=0, v=2x2 ,x≥0 C2: x=1,y ≥ 0 Г2: u=1-y2, v=2y C3: y=0,x ≥ 0 Г3: u=x2, v=0 13 f '( z) 2 z 2( x iy) f '(1) 2 1&| f '(1 i) | 2 2 1 School of Software 103. Local Inverses Local Inverse A transformation w = f (z) that is conformal at a point z0 has a local inverse there. That is, if w0 = f (z0), then there exists a unique transformation z = g(w), which is defined and analytic in a neighborhood N of w0, such that g(w0) = z0 and f [g(w)] = w for all points w in N. The derivative of g(w) is, moreover, 1 g '( w) f '( z ) Note that the transformation z=g(w) is itself conformal at w0. Refer to pp. 360-361 for the proof! 14 School of Software 103. Local Inverses Example If f(z)=ez, the transformation w=f(z) is conformal everywhere in the z plane and, in particular at the point z0=2πi. The image of this choice of z0 is the point w0=1. When points in the w plane are expressed in the form w = ρ exp(iφ), the local inverse at z0 can be obtained by writing g(w) = logw, where logw denotes the branch g (w) log w ln i ,( 0, 3 ) Why? Not contain the origin g (1) 2 i & g '(w) 1/ w 15 School of Software 103. Local Inverses Example (Cont’) If the point z0=0 is chosen, one can use the principal branch g (w) L og w ln i ,( 0, ) g (1) ln1 i0 0 16 School of Software 103. Homework pp. 362-363 Ex. 1, Ex. 6 17 School of Software 104. Harmonic Conjugates Suppose f ( z) u( x, y) iv( x, y) is analytic in a domain D, then the real-valued functions u and v are harmonic in that domain. That is uxx uyy 0& vxx vyy 0 pp. 79 Theorem 1 According to the Cauchy-Riemann equations ux uy & uy vx And v is called a harmonic conjugate of u. 18 School of Software 104. Harmonic Conjugates Properties If u(x,y) is any given harmonic function defined on a simply connected domain D, then u(x,y) always has a harmonic conjugate v(x,y) in D. Proof: Suppose that P(x, y) and Q(x, y) have continuous firstorder partial derivatives in a simply connected domain D of the xy plane, and let (x0, y0) and (x, y) be any two points in D. If Py = Qx everywhere in D, then the line integral P( s, t )ds Q(s, t )dt C from (x0, y0) to (x, y) is independent of the contour C that is taken as long as the contour lies entirely in D. 19 School of Software 104. Harmonic Conjugates The integral is a single-valued function with the parameters x and y ( x, y ) F ( x, y ) P( s, t )ds Q( s, t )dt ( x0 , y0 ) Furthermore, we have Fx ( x, y) P( x, y) & Fy ( x, y) Q( x, y) P Since u(x,y) is harmonic Q uxx uyy 0 (uy ) y (ux ) x ( x, y ) v ( x, y ) ut ( s, t ) ds us ( s, t )dt +C,C R ( x0 , y0 ) vx ( x, y) uy ( x, y)&v y ( x, y) ux ( x, y) Therefore v is the harmonic conjugate of u. 20 School of Software 104. Harmonic Conjugates Example Consider the function u(x, y) = xy, which is harmonic throughout the entire xy plane. Find a harmonic conjugate of u(x,y). Way #1: (pp.81) y2 ux y v y v ( x) 2 x2 ( x) C , C R u y vx x '( x) 2 x2 y 2 v C, C R 2 21 School of Software 104. Harmonic Conjugates Example (Cont’) Way #2: us t & ut s u ( s, t ) st ( x, y ) v ( x, y ) ( x, y ) ut ds us dt (0,0) t (x,0) sds tdt (0,0) 1 2 1 2 v ( x, y ) x y 2 2 (x,y) O +C s 22 School of Software 104. Homework pp. 81 Ex. 1 (using method #2) 23 School of Software