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 = z² in Figure 9.3. The curves u = const. should be orthogonal to the curves v = const

$\mathit{w}\mathbf{=}\frac{\mathbf{z}\mathbf{+}\mathbf{1}}{\mathbf{2}\mathbf{i}}$

Short answer

The solutions are,$u=\frac{y}{2},v=-\left(\frac{x+1}{2}\right)$

The graph is shown in the image.

Step-by-step solution

To find the solution

The function given is,

$w=\frac{z+1}{2i}······\left(1\right)$

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

Hence, the required solutions are, $\begin{array}{rcl}u& =& \frac{y}{2},v=-\left(\frac{x+1}{2}\right)\\ & & \end{array}$,

where is u the real function and v is an imaginary function.

To sketch the graph of solution

If u = const.,v = const., then if we plot equation (3) for values $u=-3\to 3,v=-1\to 3$the graph of the functions is as follows: