Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX VARIABLESysi Pw = g(z)Q R S TxR ′ P ′ Q ′T ′S ′rFigure 24.4 Transforming the upper half of the z-plane into the interior ofthe unit circle in the w-plane, in such a way that z = i is mapped onto w =0and the points x = ±∞ are mapped onto w =1.Further, suppose that (say) Re f(z) =φ is constant over a boundary C in thez-plane; then Re F(w) = Φ is constant over C in the z-plane. But this is the sameas saying that Re F(w) is constant over the boundary C ′ in the w-plane, C ′ beingthe curve into which C is transformed by the conformal transformation w = g(z).This result is exploited extensively in the next chapter to solve Laplace’s equationfor a variety of two-dimensional geometries.Examples of useful conformal transformations are numerous. For instance,w = z + b, w =(expiφ)z and w = az correspond, respectively, to a translation byb, a rotation through an angle φ and a stretching (or contraction) in the radialdirection (for a real). These three examples can be combined into the generallinear transformation w = az +b, where,ingeneral,a and b are complex. Anotherexample is the inversion mapping w =1/z, which maps the interior of the unitcircle to the exterior and vice versa. Other, more complicated, examples also exist.◮Show that if the point z 0 lies in the upper half of the z-plane then the transformationw =(expiφ) z − z 0z − z0∗maps the upper half of the z-plane into the interior of the unit circle in the w-plane. Hencefind a similar transformation that maps the point z = i onto w =0and the points x = ±∞onto w =1.Taking the modulus of w, we have|w| =∣ (exp iφ) z − z ∣ 0∣∣∣ z − z0∗ ∣ = z − z 0z − z0∗ ∣ .However, since the complex conjugate z0 ∗ is the reflection of z 0 in the real axis, if z and z 0both lie in the upper half of the z-plane then |z − z 0 |≤|z − z0 ∗ |; thus |w| ≤1, as required.We also note that (i) the equality holds only when z lies on the real axis, and so this axisis mapped onto the boundary of the unit circle in the w-plane; (ii) the point z 0 is mappedonto w = 0, the origin of the w-plane.By fixing the images of two points in the z-plane, the constants z 0 and φ can also befixed. Since we require the point z = i to be mapped onto w = 0, we have immediately842