By Jane P. Gilman, Irwin Kra, Rubi E. Rodriguez

The authors' objective is to offer an exact and concise therapy of these components of complicated research that are meant to be normal to each study mathematician. They persist with a course within the culture of Ahlfors and Bers by means of dedicating the ebook to a really special aim: the assertion and facts of the elemental Theorem for capabilities of 1 complicated variable. They talk about the numerous similar methods of realizing the idea that of analyticity, and supply a rest exploration of fascinating effects and purposes. Readers must have had undergraduate classes in complex calculus, linear algebra, and a few summary algebra. No heritage in advanced research is needed.

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4. Suppose z = x + ı y. Deﬁne f (z) = xy 2 (x + ı y) , x2 + y 4 for z = 0, and f (0) = 0. Show that f (z) − f (0) =0 z as z → 0 along any straight line. Show that as z → 0 along the curve x = y 2 , the limit of the diﬀerence quotient is 12 , thus showing that f (0) does not exist. 5. Does there exist a holomorphic function f on C whose real part is (a) u (x, y) = ex ? (b) or u (x, y) = ex (x cos y − y sin y)? Justify your answer. That is, if yes, exhibit the holomorphic function(s); if not, prove it.

6. Prove the Fundamental Theorem of Algebra: If a0 , . . , an−1 are complex numbers (n ≥ 1) and p(z) = z n + an−1 z n−1 + . . + a0 , then there exists a number z0 ∈ C such that p(z0 ) = 0. Hints: (a) Show there is an M > 0 and an R > 0 so that |p(z)| ≥ M for |z| ≥ R. (b) Show next that there is a z0 ∈ C such that |p(z0 )| = min{|p(z)| ; z ∈ C} . (c) By the change of variable p(z + z0 ) = g(z), it suﬃces to show that g(0) = 0. 20 2. FOUNDATIONS (d) Write g(z) = α + z m (β + c1 z + . . + cn−m z n−m ) with β = 0.

27. It should be observed that the functions sin and cos deﬁned above agree for real values of the independent variable z with the familiar real-valued functions with the same names. The easiest way to conclude this is from the power series expansions of these functions at z = 0. Also note that sin z and cos z form a basis for the power series solutions to the ordinary diﬀerential equation f (z) + f (z) = 0. Similarly, using either this last characterization of the sine and cosine functions or the additivity of the exponential function, one establishes that for all z and ζ ∈ C, cos(z + ζ) = cos z cos ζ − sin z sin ζ and sin(z + ζ) = sin z cos ζ + cos z sin ζ .