Question

For each of the following functions w = f(z) = u + iv, find u and v as functions of x and y . Sketch the graph in(x,y) plane of the images of u = const. and v = const. for several values of u and several values of v as was done for w = z²in Figure 9.3. The curves u = const. should be orthogonal to the curves v = const.

$\mathit{w}\mathbf{=}\frac{\mathbf{1}}{\mathbf{z}}$

Short answer

The solutions are,

$u=\frac{x}{x^2+y^2},v=-\frac{y}{x^2+y^2}$.

The graph is shown in the below image:

Step-by-step solution

To find the solution

The given function is,

$\begin{array}{rcl}w& =& \frac{1}{z}······\left(1\right)\\ & =& \frac{1}{x+iy}\\ & & \\ & =& \frac{1}{x+iy}·\frac{x-iy}{x-iy}\\ & =& \frac{x-iy}{x^2+y^2}\\ & =& \frac{x}{x^2+y^2}+i\left(-\frac{y}{x^2+y^2}\right)\\ & =& u+iv\end{array}$

Thus, $\begin{array}{rcl}u& =& \frac{x}{x^2+y^2},v=\left(-\frac{y}{x^2+y^2}\right)\end{array}$, where u is the real function and v is an imaginary function.

To sketch the graph of solution

If u = const.,v = const. and u,v take different values, then the graph of the functions is as follows: