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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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