Transcribed TextTranscribed Text

- 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.

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:

    for this solution

    or FREE if you
    register a new account!

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Complex Analysis Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats