By Richard Beals

Appropriate for a - or three-semester undergraduate direction, this textbook presents an creation to mathematical research. Beals (mathematics, Yale U.) starts off with a dialogue of the houses of the true numbers and the speculation of sequence and one-variable calculus. different subject matters contain degree thought, Fourier research, and differential equations. it really is assumed that the reader already has an exceptional operating wisdom of calculus. approximately 500 workouts (with tricks given on the finish of every) support scholars to check their realizing and perform mathematical exposition.

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Let f (z) = (z − i)/(z + i) for complex z = −i. Show that, for each complex number w = 1, there is a unique z such that f (z) = w. Show that the image of the real line f (IR) is the unit circle {z : |z| = 1}. Find the image of the upper half-plane {z : Im z > 0}. 10. (a) Suppose that A and C are real and B is complex. Prove that the set of complex z’s ¯ + B z¯ + C = 0 is either a circle, a straight line, or the empty set. such that Az z¯ + Bz (b) Prove conversely that any circle or straight line in the complex plane can be described by an equation of this form.

Show that |eit | = 1 and eit+is = eit eis . 14. Use Exercises 3, 12, and 13 to show that each complex z = 0 can be written as r eit , where r > 0, t ∈ IR. Relate this to polar coordinates in the plane. 15. Show that for any n ∈ IN there are exactly n complex solutions to the equation z n = 1. 16. Show that for any complex w = 0 and any n ∈ IN there are exactly n complex solutions to the equation z n = w. P1: IwX 0521840724c03 CY492/Beals 0 521 84072 4 June 18, 2004 14:35 Char Count= 0 3 Real and Complex Sequences The goal of this chapter is to establish the basic definitions and results concerning the convergence of real and complex sequences.

7. 718 . . : n! n 100 n n/2 2n n 2 n nn . P1: IwX 0521840724c03 CY492/Beals 0 521 84072 4 June 18, 2004 42 14:35 Char Count= 0 Real and Complex Sequences 8. Suppose that an , and bn are positive, all n ∈ IN. Suppose that there are positive constants r and N with r < 1 such that an+1 bn+1 ≤r an bn if n ≥ N . Show that limn→∞ an /bn = 0. 9. Suppose that {xn }∞ 1 is a real sequence such that x n+m ≤ x n + x m for each pair of inxn dices n and m. Prove that either the sequence {x/n}∞ n=1 converges or else limn→∞ n = −∞.