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- 3. Show that z is a real number if and only if Z = z. 4. If Z and w are complex numbers, prove the following equations: |z + w/2 = |z|² + 2Rezw+ 1w/2. 12-WW2 = - 12+W12 + |z-w|² = 2(|z|²+|w|²). o. Let R(z) be a rational function of z. Show that R(z) = R(z) if all the coefficients in R(z) are real. 3.3 12,+22+...t2, Also useful is the inequality 3.4 (121-1m)-512-m Now that we have given a geometric interpretation of the absolute value let us see what taking a complex conjugate does to a point in the plane. This is also easy; in fact, z is the point obtained by reflecting z across the x-axis (i.e., the real axis). Exercises 1 2. Show that equality occurs in (3.3) if and only if N 0 for any integers k and I, 1 s k, / = n, for which 21 * 0. 3. Let a € R and c > 0 be fixed. Describe the set of points z satisfying 12-a)-12+al=2c for every possible choice of a and c. Now let a be any complex number and, using a rotation of the plane, describe the locus of points satisfying the above equation. 2m 5. Let Z = cis for an integer n 2. Show that 1 +z+. 7. If Z e C and Re(z") >0 for every positive integer n, show that Z is a non-negative real number. Exercise 1. Let C be the circle {z: r > 0; let a = ctr cis a and put LB - I = z: Im (7)-0} where b = cis p. Find necessary and sufficient conditions in terms of B that LB be tangent to C at a. 2. Which of the following subsets of C are open and which are closed: (a) {z:|z|<1} ; (b) the real axis; (c) { Z : = 1 for some integer n > 1); (d) { Z E C:z is real and 052<1); (e) { z E C: Z is real and 0<z< 1)?<br="">ball is in fact an onen 4. Show that the union of a finite number of compact sets is compact. </z<>1. Show that f(z) = ||² = x2 + y2 has a derivative only at the origin. 2. Prove that if bn, an are real and positive and 0 < b = lim bn < 80, a = lim sup an then ab = lim sup Does this remain true if the re- quirement of positivity is dropped? 3. Show that lim n¹/" = 1. 44 Elementary Properties and Examples of Analytic Functions 4. Show that (cos z)' = - -sin Z and (sin z)' = cos z. 5. Derive formulas (2.14). 6. Describe the following sets: {z: e² = i), {z: e = -1), {2: e2 = - -i), {z: cos Z = 0}, , {z: sin z = 0}. 7. Prove formulas for cos (z+ w) and sin (z+w). sin Z 8. Define tan Z = cos Z ; where is this function defined and analytic? 9. Suppose that 2n, Z € G = C- - {z: Z S 0} and 2n = ree , Z = reie where < < 7. Prove that if 2n Z then On 0 and "n r.

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