By Mario Gonzalez
Textual content at the conception of capabilities of 1 advanced variable comprises, with many gildings, the topic of the classes and seminars provided by way of the writer over a interval of forty years, and may be thought of a resource from which a number of classes may be drawn. as well as the fundamental issues within the cl
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Additional resources for Classical Complex Analysis
7. If lul 1 and u =:f -1, prove that u can be expressed in the form u (1+ki) / (1- ki) , where k is a real number. Deduce that unimodular and = = 25 Complex Numbers numbers other than -1 can be put into one-to-one correspondence with the real numbers. 8. If u and v are unimodular, show that (u+v)/(l+uv) 9. If lzkl < 1, rk > 0, and E�=l rk = 1, prove that 10. If IPI (a ) 1 1. 1 and lql < 1, p- q <1 1 -pq < (uv-:/= - 1 ) is real. prove that: ( b) IPl-lql < p- q 1 -pq 1 -IPIIqI Show that p -q =1 1 -pq 1 2.
20. 21. 22. = and if z1 + z2 + Z3 + Z4 0, then = ( b ) lz1 - z212 + lz2 - zal2 + lza - z412 + lz4 - z112 lz312 + lz412) where equality holds iff Z3 = -z1 and Z4 = > 2(lz112 + lz212 + -z2. Remark (a ) and (b ) are special cases of Schoenberg's inequality: If L:;==l Zk = 0, then n L lzk+t - zkl2 k=l > ( 4 sin2 n �) L lzkl2 k=l where Zn+l = z1. ] 23. Let z1, z2, and w be complex numbers. Then w satisfies iff w = c1z1 + c2 z2 with lc1 I= lc2l = 1. [H. A. MacLean, Amer. Math. Monthly, 85 (1978), 105) 24.
When P' lies inside the unit circle the steps in the foregoing construction are reversed, and when P' lies on the circle, P' is its own inverse. z1/lzil2 shows that 1/z1 is in the direction of Since 1/z1 zi/z1z1 and has a length l/lz1 j, we may perform the construction of 1 /z1 as = z1 = in Fig. 11. y 0 ' \ x \ \ \ \ \ \ I I \1 P'(Z1) Fig. 10 Chapter 40 1 y 0 x Fig. 4 1. Do all operations geometrically. (a) (c) (e) (g) (i) (- 3,4)+(-2, -5) (4+i)+(1+3i)+(-1+2i) (-2+i) - (3 - 4i) (2+ 2i)(l + Vai) 2 e i7r/6 5i (k) 1+2i 2.