Chapter 1: Q2P (page 1)

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

Short Answer

Expert verified

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

The graph is shown in the image.

Step by step solution

To find the solution

The function given is,

$w = \frac{z + 1}{2 i} \cdot \cdot \cdot \cdot \cdot \cdot (1)$

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

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

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:

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To sketch the graph of solution

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