Q4P - Mathematical Methods in Physical Sciences

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 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 = e^z

Step-by-Step Solution

Answer

The solutions are, u = e^x cos y, v = e^y sin y. 

The graph is shown in the below image:

Step 1: To find the solution

The given function is,  w = e^z

 

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

$\begin{array}{rcl}w& =& e^z\\ & =& e^{x+iy}\\ & =& e^x.e^{iy}\\ & =& e^x\left(\cos y +i \sin y\right)\\ & =& e^x\cos y+i{e}^x\sin y\\ & =& u+iv\end{array}$ 

 

 

Hence,u = e^x cos y, v = e^y sin y , where u is the real function and v is an imaginary function.

 

Step 2: 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: