Transformaciones que conservan ángulos

1. Transformaciones que conservan ángulos w = f(z) definida en un dominio D se llama conforme en z = z0 en D cuando f preserva el ángulo entre dos curvas en D que se intersectan en z0.

2. Transformación conforme Si f(z) es analítica en el dominio D y f’(z)  0, entonces f es conforme en z = z0. Demostración Si una curva C en D está definida por z = z(t), entonces w = f(z(t)) es la imagen de la curva en el plano w. Tenemos Si C1 y C2 intersectan en z = z0, entonces:

3. Puesto que f (z0)  0, tenemos que: Ejemplos: (a) f(z) = ez es conforme en todos los puntos del plano complejo, ya que f (z) = ez no es nunca cero. (b) La función g(z) = z2 es conforme en todos los puntos del plano complejo excepto z = 0, ya que g(z) = 2z 0, para z  0.

4. Example 2 • The vertical strip −/2  x  /2 is called the fundamental region of the trigonometric function w = sin z. A vertical line x = a in the interior of the region can be described by z = a + it, −  t  . We find that sin z = sin x cosh y + i cos x sinh yand so u + iv = sin (a + it) = sin a cosh t + i cos a sinh t.

5. Since cosh2t − sinh2t = 1, thenThe image of the vertical line x = a is a hyperbola with  sin a as u-intercepts and since −/2 < a < /2, the hyperbola crosses the u-axis between u = −1 and u = 1. Note if a = −/2, then w = −cosh t, the line x = −/2 is mapped onto the interval (−, −1]. Likewise, the line x = /2 is mapped onto the interval [1, ).

6. The complex function f(z) = z + 1/z is conformal at all points except z = 1 and z = 0. In particular, the function is conformal at all points in the upper half-plane satisfying |z| > 1. If z = rei, then w = rei+ (1/r)e-i, and soNote if r = 1, then v = 0 and u = 2 cos  .Thus the semicircle z = eit, 0  t  , is mapped onto [−2, 2] on the u-axis. If r > 1, the semicircle z = reit, 0  t  , is mapped onto the upper half of the ellipse u2/a2 + v2/b2 = 1, where a = r + 1/r, b = r − 1/r. See Fig 20.12.

7. For a fixed value of , the ray tei, for t  1, is mapped to the point u2/cos2−v2/sin2 = 4 in the upper half-plane v  0. This follows from (3) since Since f is conformal for |z| > 1 and a ray  = 0 intersects a circle |z| = r at a right angle, the hyperbolas and ellipses in the w-plane are orthogonal.

8. Contents • 20.1 Complex Functions as Mappings • 20.2 Conformal Mappings • 20.3 Linear Fractional Transformations • 20.4 Schwarz-Christoffel Transformations • 20.5 Poisson Integral Formulas • 20.6 Applications

9. 20.1 Complex Functions as Mappings • IntroductionThe complex function w = f(z) = u(x, y) + iv(x, y) may be considered as the planar transformation. We also call w = f(z) is the image of z under f. See Fig 20.1.

10. Fig 20.1

11. Example 1 • Consider the function f(z) = ez. If z = a + it, 0  t  , w = f(z) = eaeit. Thus this is a semicircle with center w = 0 and radius r = ea. If z = t + ib, − t  , w = f(z) = eteib. Thus this is a ray with Arg w = b, |w| = et. See Fig 20.2.

12. Fig 20.2

13. Example 2 • The complex function f = 1/z has domain z 0 and

14. Example 2 (2) • Likewise v(x, y) = b, b  0 can be written asSee Fig 20.3.

15. Fig 20.3

16. Translation and Rotation • The function f(z) = z + z0 is interpreted as a translation. The function is interpreted as a rotation. See Fig 20.4.

17. Example 3 Find a complex function that maps −1  y  1 onto 2  x  4. SolutionSee Fig 20.5. We find that −1  y  1 is first rotated through 90 and shifted 3 units to the right. Thus the mapping is

18. Fig 20.5

19. Magnification • A magnification is the function f(z) = z, where  is a fixed positive real number. Note that |w| = |z| = |z|. If g(z) = az + b and then the vector is rotated through 0, magnified by a factor r0, and then translated using b.

20. Example 4 Find a complex function that maps the disk |z|  1 onto the disk |w – (1 + i)|  ½. Solution Magnified by ½ and translated to 1 + i, we can have the desired function as w = f(z) = ½z + (1 + i).

21. Power Functions • A complex function f(z) = zwhere  is a fixed positive number, is called a real power function. See Fig 20.6. If z = rei, then w = f(z) = rei.

22. Example 5 Find a complex function that maps the upper half-plane y 0 onto the wedge 0  Arg w  /4. Solution The upper half-plane can also be described by 0  Arg w  . Thus f(z) = z1/4 will mapthe upper half-plane onto the wedge 0  Arg w  /4.

23. Successive Mapping • See Fig 20.7. If  = f(z) maps R onto R, and w = g() maps R onto R, w = g(f(z)) maps R onto R.

24. Fig 20.7

25. Example 6 Find a complex function that maps 0  y   onto the wedge 0  Arg w  /4. Solution We have shown that f(z) = ezmaps0  y   onto to 0  Arg    and g() =  1/4 maps0  Arg   onto 0  Arg w  /4. Thus the desired mapping is w = g(f(z)) = g(ez) = ez/4.

26. Example 7 Find a complex function that maps /4  Arg z  3/4 onto the upper half-plane v  0. Solution First rotate /4  Arg z  3/4 by  = f(z) = e-i/4z. Then magnify it by 2, w = g() =  2.Thus the desired mapping is w = g(f(z)) = (e-i/4z)2 = -iz2.