MATH 322 ASSIGNMENT 8

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MATH 322 ASSIGNMENT 8
DUE SAT NOV 27 10PM VIA CROWDMARK SUBMISSION
(1) Churchill 9th ed. p. 273, 12 concerning the Fresnel integrals, but please also r>justify the last step (a Calc 3 exercise involving an improper double integral).
(2) Churchill p. 282, 2. Make sure to be clear which theorems you are using.
(3) How many roots of the equation z4 + 8z3 + 3z2 + 8z + 3 = 0 lie in the right
half plane?
Suggestion: consider the image of the imaginary axis and apply the argument
principle to a large half-disc.
(4) If f is meromorphic on an open set G define f1 : G→ C∪{∞} by f1(z) =∞
if z is a pole, and otherwixe f1(z) = f(z). Show that f1 is continuous on G.
(5) Let f be analytic in an open set containing the closed unit disc {z||z| ≤ 1}. If
f(z) < 1 for |z| = 1 show that there is a unique z with |z| < 1 and f(z) = z.
Suggestion: Think of Rouche´’s theorem, and consider the functions g(z) =
f(z)− z and h(z) = z.