Analytic and Harmonic Functions; the Cauchy-Riemann Equations 15. Let f be analytic on a domain D and suppose that f'(z) = 0 for all z = D. Show that is constant on D. 16. Find the derivative of the linear fractional transformation T(z) = (az + b)/(cz + d), ad bc. In what way does the condition ad - bc # 0 enter? Conclude that T'(z) is never zero, z -d/c. 17. Suppose that f is analytic on a domain D and f'(z) = af(z), z = D, where a is a constant. Show that f(z) = C exp(az), C a constant. (Hint: Consider g(z) = eazf(z) and use Exercise 15 for g.) 18. Show that h(z) = 7 is not analytic on any domain. (Hint: Check the Cauchy- Riemann equations.) 20. Let f u+iv be analytic. In each of the following, find v given u. x u = x² + y² "15 it is not necessary to answer, is just there cause is a hint for exercise 17"