1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

3.3 • HARMONIC FUNCTIONS 121

Suppose that ( xo , y 0 ) is a point common to the two curves <f> ( x, y) = c 1 and

..p ( x , y) = c2. Use t he gradients of <f> and ..P to show that the nor mals to the curves

ar:e perpendicular:.

15. 
    

    We introduce the logarithmic function in Chapter 5. For now, let F (z) = Logz =
    In I zl + iArg z. Here we have (x, y) = In I zl and ..p (x, y) = Arg z. Sketch the
    equipot entials = 0, In 2, In 3, In 4 and the streamlines ..p = k; for k = 0, 1,... , 7.

    
    


16. 
    

    Discuss and compare the statements "u (x, y) is harmonic" and "u (x, y) is the
    imaginary part of an analytic function."