1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

120 Cl!APTE!l 3 • ANALYTIC ANO HARMONIC FUNCTIONS

2. Does an analytic function f (z) = u(x,y) +iv (x,y) exist for which v (x,y) =

x^3 + y^3? Why or why not?

3. Let a, b, and c be real constants. Determine a relation among the coefficients that
   will guarantee that the function cf> ( x , y) = ax^2 + bxy + er/ is harmonic.


4. Let v (x, y) = arctan~ for x f O. Compute t he partial derivatives of v, and verify
   that v satisfies Laplace's equation.


5. Find an analytic function f (z) = u (x, y) +iv (x, y) for the following expressions.

(a) u (x, y) = y^3 - 3x^2 y.

(b) u (x, y) = sin y sinh x.

(c) v(x,y) =e^11 sin x.

(d) v(x,y) = sinxcoshy.

6. Let u1 (x,y) = x^2 - y^2 and u2 = x^3 - 3xy^2. Show that u 1 and u 2 are harmonic
   functions but that their product u 1 (x,y)u 2 (x,y) is not a harmonic function.


7. Assume that u (x, y) is harmonic on a region D that is symmetric a bout t he li ne
   y = O. Show that U(x, y) = u (x, -y) is harmonic on D. Hint: Use the chain rule
   for differentiation of real functions and note that u (x, -y) is really t he function
   u(g(x,y)), where g (x , y) = (x,- y).


8. Let v be a harmonic conjugate of u. Show that - u is a harmonic conjugate of v.


9. Let v be a harmonic conjugate of u. Show that h = u^2 - v^2 is a harmonic function.


10. Suppose that v is a harmonic conjugate of u and that u is a harmonic conjugate
    of v. Show that u and v must be const a nt functions.


11. Let f (z) = f (re'^9 ) = u (r , Ii) +iv (r, Ii) be analytic on a domain D that does
    not contain the origin. Use the polar form of the Cauchy-Riemann equations
    ue = -rvr and Ve = rur. Differentiate them first with respect to Ii and then with
    respect to r. Use the results to establish the polar form of Laplace's equation:

r^2 urr (r, Ii)+ rur (r, Ii)+ U99 (r, 8) = 0.

12. Use the polar form of Laplace's equation given in Exercise 11 to show t hat the
    following functions a re harmonic.

(a) u (r,8) = (r +~)cos Ii and v (r,IJ) = (r - ~)sin Ii.

{b) u (r, 8) = r " cosnli and v (r, 8) = r" sin n 8.

13. The function F (z) = ± is used to determine a field known as a dipole.

(a.) Express F (z) in the form F (z) =cf> (x, y) + i,P (x, y).

(b) Sketch the equipotentials cf>= 1, !, ~and the streamlines .P = 1, ~. ~-

14. Assume that F (z) = cf> (x, y) + i,P (x, y) is analytic on the domain D a nd t hat
    F' ( z) f 0 on D. Consider t he families of level curves {cf> ( x, y) = const a nt} and
    {.P(x,y) =constant}, which are the equipotentials and streamlines for the fluid
    flow V (x, y) = F' (z). Prove that the two families of curves a re orthogonal. Hint: