Q14P - Mathematical Methods in Physical Sciences

Question

We have discussed the fact that a conformal transformation magnifies and rotates an infinitesimal geometrical figure. We showed that $\left|dw/dz\right|$ is the magnification factor. Show that the angle of $\mathit{d}\mathit{w}\mathbf{/}\mathit{d}\mathit{z}$  is the rotation angle. Hint: Consider the rotation and magnification of an arc $\mathit{d}\mathit{z}\mathbf{=}\mathit{d}\mathit{x}\mathbf{+}\mathit{i}\mathit{d}\mathit{y}$  (of length   and angle arctan  $\mathit{d}\mathit{y}\mathbf{/}\mathit{d}\mathit{x}$ which is required to obtain the image of dz , namely dw.

Step-by-Step Solution

Answer

The answer is  ${\varphi }_0={\theta }_0+\alpha ={\theta }_0+arg f\text{'}\left(z_0\right)$.

Step 1: Cauchy Riemann theorem

Expression from Cauchy Riemann theorem:

w = u + iv And z = x + iy, w = f(z) and z(x,y)n, u(x,y) and u(x,y)  .

Implicit formula:

 $\frac{dw}{dt}=\frac{dw}{dz}.\frac{dz}{dt}=f\left(z\right)\frac{dz}{dt}$

Step 2: Prove that under this transformation the tangent at z 0 is rotated through the angle argf (z 0 )

Consider the transformation w = f(z) , where f(z)  is analytic at z₀  and $f\left(z_0\right)\ne 0$.

To prove that under this transformation the tangent at z₀  is rotated through the angle argf(z₀)  .

As a point moves from z₀  to $z_0+∆z$  along C  the image point moves along in the w-plane from w₀  to $w_0+∆w$  . 

If the parameter used to describe the curve is t, then the corresponding to the path          z = z(t)   in the  z-plane, the path w = w(t)  in the w-plane. the derivatives $\frac{dz}{dt}·\frac{dw}{dt}$  represent the tangent vectors to corresponding on  C.

Now  $\frac{dw}{dt}=\frac{dw}{dz}·\frac{dz}{dt}=f\left(z\right)\frac{dz}{dt}$ and in particular at  z₀ and  w₀.

   ${\overline{)\frac{dw}{dt}}}_{w=w_0}=f\text{'}\left(z_0\right){\overline{)\frac{dz}{dt}}}_{z=z_0}$                                                                                          ...... (1)

Providing f(x)   is analytic at  z = z₀.

Obtain:

$$\begin{gathered} {\overline{)\frac{dw}{dt}}}_{w=w_0}={\rho }_0{e}^{i\varphi }f\text{'}\left(z\right)=R{e}^{i\infty } \\ {\overline{)\frac{dz}{dt}}}_{z=z_0}=r_0{e}^{i{\theta }_0}{\rho }_0{e}^{i\varphi } \\ {\overline{)\frac{dz}{dt}}}_{z=z_0}=R{r}_0{e}^{i\left({\theta }_0+\infty \right)} \end{gathered}$$                                                                                      .....(2)

So that as required:

 ${\varphi }_0={\theta }_0+\alpha ={\theta }_0+argf\text{'}\left(z_0\right)$                                                                                     ...... (3)

Note that  f(z₀) = 0 then $\alpha$  is indeterminate.

Points where f(z) = 0  are called critical points.