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Resources for MATH628, Spring 2017

MATH628 is an introduction to the theory of functions of a complex variable.

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Suggested problems from book: page 5, #1, 4, 11. page 8: 1, 5, 6. page 12: 1, 3, 4, 5, 6. page 14-15: 1, 2, 9, 10, 13 (hint for 13: Consider (z-z_0) bar(z-z_0), where bar means complex conjugate), 14. pages 22-23: 1, 2, 4, 5(c), 6, 8. Pages 29-30: 3, 4, 5(a), 6, 7, 9. Page 33: 1, 2, 4, 7, 8, 10. pages 37-38: 1, 3, 4. pages 44-45: 1, 2, 3, 5, 8. pages 55-56: 1c, 2c, 3, 5, 6b, 7, 9, 10, 13. pages 62-63: 1, 3, 4, 8, 9. pages 71-73: 1, 2a, 2b, 3b, 3c, 4b, 5, 6, 9a, 10. pages 77-78: 1a,1d, 2a, 4a,4c, 5, 6, 7. pages 81-82: 1b, 2, 3, 6, 7, 8. pages 87-88: 4, 5. pages 92: 1a, 1c, 3, 4, 5, 7, 8c, 11, 13. pages 108-109: 1, 4b (for this you will need a Lemma that we haven't covered yet, but that is ok), 7, 8a, 9, 10b, 11, 12, 14, 16. pages 111-112: 1, 2b, 5, 6b, 12, 13, 14, 15b. pages 87-88: 1,2, 3 (hint for 3: use uniqueness of analytic continuation). pages 97-98: 1b, 2c, 3, 4, 5, 8, 9, 10 (Hint for 10: Consier the function Log(z^2). Where is it analytic?), 11. page 100: 1, 2, 4, 6. page 104: 1a, 3, 5, 6, 7, 8a. pages 114-115: 1a, 2, 5. page 121: 1a, 2a, 2c, 3, 4, 5. pages 125-126: 2, 5, 6 (I recommend 6b). pages 135-136: 1, 3, 4, 6 (see example 1 on page 133 of the 8th edition, or example 4 of page 100 of the 6th edition, for hints/help on this), 10a. pages 140-142: for these questions, you will need the reverse triangle inequality: | z - w | >= | |z| - |w| |. 1, 4, 5, 6, 8b. page 149: 2, 3, 4, 5. pages 160-163: 1c, 1d, 1f, 2, 3, 5, 6 (IMPORTANT), 7. pages 170-172: 1a,1c, 2a, 3, 7 (hint: use that sin(asin(-theta)) = -sin(asin(theta)), and so therefore some integral is zero by using symmetry), 8b, 8c, 10. pages 178-179: 1, 2, 3, 6, 7. pages 188-189: 2, 3, 6, 9, and read the example from Section 56 of the 8th edition, since this example is important. pages 195-197: 1, 2, 6, 10, 12, 13. pages 205-208: 2 (Hint: Use formula for the coefficients in the Laurent series), 3, 4, 6, 9, 11. pages 219-222: 3, 4 (Hint: Find the Laurent series for sin(z)/z when z is not equal to 0, by writing sin(z) as a Taylor series, then multiplying by 1/z. Turns out this Laurent series has no negative powers of z, so is a regular power series. This power series equals 1 at z = 0, and so the power series converges to f(z) for all complex numbers z. Therefore, by Corollary of Section 49 of 6th edition or Corollary of Section 65 of 8th edition, it is analytic everywhere), 5, 6, 7 (Hint: using exercise 6, show that f(z) is given by a power series, when z is not equal to 1. This power series equals 1 when z = 1. So the power series agrees with f(z) whenever |z - 1| < 1, again by exercise 6, and therefore f(z) is analytic on |z-1| < 1. On the other hand, it is clear, by definition of Log, that f(z) is analytic everywhere else in the complex plane, except the negative real axis and the origin. Done.), 11. pages 225-227: 1, 4 (Hint: by Exercise 10 on page 221 8th edition, one can integrate term by term, since C is within .....), 8. pages 239-240: 1a, 1b, 1e, 2a, 2b, 2d, 3a, 4, 6 (Hint: Write 1/z^2 P(1/z) / Q(1/z) as a quotient of two polynomials, for z not equal to 0. Then show why z = 0 is a removable singularity of this quotient.) pages 243-244: 1a, 1d, 1e, 2a, 2c, 3, 5. p. 248: 1, 2, 3a, 4, 6c. p. 255-257: 1, 2a, 3a, 4a, 5, 7, 8a, 9, 10, 11 (Hint for 11: consider the function g = 1/f. Suppose there are infinitely many poles. By problem 10, there is an accumulation point, call it a. The goal is to show that f has a pole at a. First, there must then be a neighborhood U of a such that f has no zeroes, since otherwise, we could apply problem 10 to show that f(a) = 0, clearly contradicting that a is an accumulation point of poles. Moreover, if f has a pole at z, then g(z) = 0. Therefore, since f is analytic inside C except possibly just poles, then 1/f is analytic on U. By problem 10 again, g = 1/f has a zero at a. That is, g(a) = 0. Therefore, by a lemma/theorem in the text, g(z) = (z-a)^m h(z), for some h which is analytic and nonzero at a. Thus, solving for f, we get that f has a pole at a, finally. But poles are isolated, contradiction)