Page 64 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 6: Cauchy’s Integral Formula

6. Let C be the circle |z – i| = 2 with the positive orientation. Evaluate : Notes

1 1
(a) 2  dz (b)  2 2 dz
C z  4 C (z  4)
7. Suppose f is analytic inside and on the simple closed curve C. Show that :

f'(z) f(z)
 dz   2 dz
C z – w C (z – w)
for every w  C.

8. (a) Let  be a real constant, and let C be the circle (t) = eit, –  t  . Evaluate :
e az
 dz.
C z
(b) Use your answer in part a) to show that :


 e  cos t cos( sint)dt   .

0
9. Suppose f is an entire function, and suppose there is an M such that Ref(z)  M for all z.
Prove that f is a constant function.
10. Suppose w is a solution of 5z + z + z – 7z + 14 = 0. Prove that |w|  3.
3
4
2
11. Prove that, if p is a polynomial of degree n, and if p(a) = 0, then p(z) = (z – a)q(z), where q
is a polynomial of degree n – 1.
12. Prove that, if p is a polynomial of degree n  1, then
p(z) = c(z – z ) 1 (z – z )k ... (z – zj) j ,
k
2
k
1
2
where k , k , ..., k are positive integers such that n ƒ­ k1 ƒy k2 ƒyƒ ƒykj.
1
2
j
13. Suppose p is a polynomial with real coefficients. Prove that p can be expressed as a product
of linear and quadratic factors, each with real coefficients.
14. Suppose f is analytic and not constant on a region D and suppose f(z)  0 for all z  D.
Explain why |f(z)| does not have a minimum in D.
15. Suppose f(z) = u(x, y) + iv(x, y) is analytic on a region D. Prove that if u(x, y) attains a
maximum value in D, then u must be constant.
Answers: Self Assessment

1. Cauchy Integral Formula. 2. simple closed curve
3. function 4. continuous
5. Liouville’s Theorem 6. maximum value

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