#### Complex Analysis

5. (9 points) Show that the function u(x, y) = x³ − 3xy² + y is harmonic and determine it's harmonic conjugate.

(2) (a) Describe the domain on which the function f(z) = Log(z + i) is analytic and sketch this domain. (b) Describe the domain on which the function g(z) = alytic and sketch this domain. Log(z+i) z²+1 is an-

(3) Show that the function u(x, y) = ln(x² + y²) is harmonic on any domain that does not contain (0,0).

(4) Suppose that the function f(z) = u(x, y) + iv(x, y) is analytic on a domain D. Explain why the functions U(x, y) = e(y) cos(v(x, y)), V(x, y) = e¹(x,y) sin(v(x, y)), are harmonic on the domain D.

Determine whether the function f(2)= 3x+y+L(y-x) is analytic or not. [Page - 76 E2.-1.@ of the book]

9.2 Show that f'(2) does not exist at any pt. where FC2) = 2-2 [Page-70. Ex. 1.6 of the book]

9.4 Show that Cosliz) = Cos (17) for all z Sin (12) - Sin (iz) if and only if 2. mi (n = 0,1112_) Page-108 Ex-14 of the book)

(1) Solve cos(z)=-i.

9.5 Evaluate the integral S(+-i) ³d+ (Page-119 Ex- 2. of the book)

Find the principal value of (l-i)4i (page - 103 ex-2(c) of the book)