Queation 1Question 2 1.Let a function f be analytic everywhere in a domain D. Suppose that f(z) is pureimaginary 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' - 3xy. Show that u and v are harmonicfunctions but that their product uv is not harmonic.Show that u(x,y) = 2x-x' +3xy is harmonic and find a harmonic conjugatev(x, y).4.Show that exp(z) s exp(z|") for all z e C.5.6.Show that Log[(-1+i)*]# 2Log(-1+i).7.Find all roots of the equation log(z) = ri/28.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.prove that Sin] z+(Do not use any other trigonometry identities for question 9)10.Evaluate (3t-i)² dt

Name: Queation 1Question 2 1.Let a function f be analytic everywhere in a do
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Description: Queation 1Question 2 1.Let a function f be analytic everywhere in a domain D. Suppose that f(z) is pureimaginary 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