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Chapter 14: Q4P (page 710)

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 and several values of as was done for in Figure 9.3. The curves u = const. should be orthogonal to the curves v = const.

w = ez

Short Answer

Expert verified

The solutions are, u = ex cos y, v = ey sin y.

The graph is shown in the below image:

Step by step solution

01

To find the solution

The given function is, w = ez

Now, solve the given function by using z = x + iy

w=ez=ex+iy=ex.eiy=excosy+isiny=excosy+iexsiny=u+iv

Hence,u = ex cos y, v = ey sin y , where u is the real function and v is an imaginary function.

02

To sketch the graph of solution

If u =const. , v = const. and for example 0,1,2,3,-1,-2,-3, the graph of the function is given as follows:

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