1. Let a functionf be analytic everywhere in a domain D. Suppose that f(z) is pure imaginary for all z in D. What can we conclude about the values of f(z)? (Hint: Use the theorem or the first...

Question 3 Question 4

Extracted text: 1. Let a functionf be analytic everywhere in a domain D. Suppose that f(z) is pure imaginary for all z in D. What can we conclude about the values of f(z)? (Hint: Use the theorem or the first corollary presented in Lecture 10) Let f and g be analytic functions in a domain D. If f'(z) = g'(z) for all z in D, then show that f(z)= g(z)+c, where c is a complex constant. 2. 3. Let u(x, y) = x2 - y² and v(x, y) = x' - 3xy². Show that u and v are harmonic functions but that their product uv is not harmonic. Show that u(x, y) = 2x-x'+3xy is harmonic and find a harmonic conjugate v(x, y). 4. Show that exp(z) s exp(=|") for all z E C. 5. 6. Show that Log[(-1+i)*]#2Log(-1+i). 7. Find all roots of the equation log(z) = ni / 2 8. Find the principal value of (1+ i)' . 9. Use the definitions of sin(z) and cos(z) given in Lecture 13 at the 13:30 mark to cos(z) for all z e C. %3D prove that Sin z+ (Do not use any other trigonometry identities for question 9) 1 10. Evaluate (3t-i)dt