1. Consider the function f(z) z²³ 2²-2iz+3 (a) Find the poles of this function and state the order of each. (b) Let C₁ b

[answerhappygod]

1 Consider The Function F Z Z 2 2iz 3 A Find The Poles Of This Function And State The Order Of Each B Let C B 1 (37.52 KiB) Viewed 31 times

1 Consider The Function F Z Z 2 2iz 3 A Find The Poles Of This Function And State The Order Of Each B Let C B 2 (28.02 KiB) Viewed 31 times

1 Consider The Function F Z Z 2 2iz 3 A Find The Poles Of This Function And State The Order Of Each B Let C B 3 (29.6 KiB) Viewed 31 times

1. Consider the function f(z) z²³ 2²-2iz+3 (a) Find the poles of this function and state the order of each. (b) Let C₁ be a circle of radius 1 centred at i, traversed anticlockwise; compute the integral of f(z) over this contour. (c) Let C₂ be a circle of radius 3 centred on i, traversed anticlockwise; compute f(z) dz. (d) C3 is a square with corners at 2 +2i, 2 - 2i, −2+ 2i and −2 − 2i, traversed anticlockwise. Integrate f(z) over this contour. (In all cases, it may be very useful to plot the poles and sketch the contours in the complex plane.)
2. Consider the function g(z) = sin(z) (z+)4 (a) Find the first three nonzero terms of the Taylor series expansion of sin(z) (not zo = 0). around the point zo = - (b) Using your result from (a), find the principal part of the Laurent series expansion of g(z) around zo = -. (c) Let C be a circle of radius 2 centred on the point -2, traversed anticlock- wise. Compute the contour integral of g(z) over C.
3. We have the function iz³ h(z) (3z + 1)²(z²+9)* (a) Find the poles of this function and state what order each one is. (b) Find the residue of h(z) at each pole you found in (a). (c) Consider the circle centred at -i and of radius 3, traversed anticlockwise. Sketch this curve in the complex plane, plot each of the singularities from (a) and and determine which, if any, lie in the circle's interior. (d) Integrate h(z) over the circle in (c).