 # Problem 1 A function f : S C is said to be bounded from below if t...

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Problem 1 A function f : S C is said to be bounded from below if there exists a constant M > 0, such that If *(z) M for every z E S. Prove, using the definition of limit and of bounded from below, that if f and g are functions defined in a neighborhood of 20 E C, except perhaps at 20, if limz 20 f (z) = 00, and if g is bounded from below, then lim2 20 f(z)g(z) = 80. Problem 2 Prove: If P is a nonconstant polynomial, then limz 8 P(z) = 80. You may not use the Fundemental Theorem of Algebra, nor the result that a polynomial can be written as a product of linear factors. Problem 3 Suppose that f = ut iv and that f'(zo) exists. Show that: (a) Ux (zo) = vy(zo) and Uy (zo) = -Ux (zo) (b) 1,5' (zo) 2 = + (UI(20))2 = (ux (zo))² + (uz(20))2. Suggestions: Use the connections among f', fa,fy we found in class, and the facts that fx = Ux +ivx, fy = uly + ivy, provide the decompositions of fx and fy into real and imaginary parts. Problem 4 Suppose that f = u + iv is defined in a neighborhood of ZO E C. Prove: If f' (zo) and (f) , (zo) both exist, then f' (zo) = 0. Problem 5 Suppose that Z = T(x1,22, x3), where I: S2 C* is the inverse stereographic projection. (a) Using, if you like, the formula Z = x1 - + ix2 , (x1,x2,x3) E s21((0,0,1)}, 1 X3 express x3 as a function of r = 121, and express r as a function of x3. (b) Verify that x(xo, 2) = (1+12(2)1/2 2 and that x(0,2) =1*, where X is the chordal metric on C* and !* is an expression you need to supply. Problem 6 Do not turn this problem in Find the real and imaginary parts, and the absolute value, of Z = 1 (3+1)(1+4i)

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