Question

Question 2. (a) [7 marks] Evaluate the integral ?_C dz / (1 + z^2) where C is the circle with centre -2i and radius 2 traversed anti-clockwise. (b) [8 marks] Let a be a fixed complex number different from zero. Let CR denote the circle with centre at 0 and radius R ? |a| traversed anti-clockwise. What are the possible values of the integral ?_CR (sin z / (z - a)^3) dz? Justify your answer.

          Question 2.
(a) [7 marks] Evaluate the integral
?_C dz / (1 + z^2)
where C is the circle with centre -2i and radius 2 traversed anti-clockwise.
(b) [8 marks] Let a be a fixed complex number different from zero. Let CR denote the circle with centre at 0 and radius R ? |a| traversed anti-clockwise. What are the possible values of the integral
?_CR (sin z / (z - a)^3) dz?
Justify your answer.

Question 2.
(a) [7 marks] Evaluate the integral
?C dz / (1 + z^2)
where C is the circle with centre -2i and radius 2 traversed anti-clockwise.
(b) [8 marks] Let a be a fixed complex number different from zero. Let CR denote the circle with centre at 0 and radius R ? |a| traversed anti-clockwise. What are the possible values of the integral
?CR (sin z / (z - a)^3) dz?
Justify your answer.

Added by Guillermo R.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

10/05/2023

Step 1

So we have: f(2i) = (1/2πi) ∫C 1/(z+2) dz To apply this formula, we need to parameterize the contour C. Since C is a circle with centre 2i and radius 2, we can use the parametrization: z = 2i + 2e^(it), 0 ≤ t ≤ 2π Then dz = 2ie^(it) dt, and we have: f(2i) = Show more…

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Question 2. (a) [7 marks] Evaluate the integral ∫_C dz / (1 + z^2) where C is the circle with centre -2i and radius 2 traversed anti-clockwise. (b) [8 marks] Let a be a fixed complex number different from zero. Let CR denote the circle with centre at 0 and radius R ≠ |a| traversed anti-clockwise. What are the possible values of the integral ∫_CR (sin z / (z - a)^3) dz? Justify your answer.