(a) Locate the singularities of the function 23 sin z and classify each singularity as a...

 

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(a) Locate the singularities of the function 23 sin z and classify each singularity as a removable singularity, a pole (giving its order) or an essential singularity. (b) Find two Laurent series about 0 for the function f(z) = : one on {z z] <2}, giving four consecutive non-zero terms, and the other on {2:2>4}, giving two consecutive non-zero terms. f(z) = 24 (2+2)(z-4)' (+). Find the Laurent series about -i for f, giving four consecutive non-zero terms. (c) Let f(z) = z exp (i) State the annulus of convergence of the Laurent series. (ii) Determine whether the singularity of f at -i is a removable singularity, a pole or an essential singularity. (iii) Evaluate ( 2 Exp (+) dz. where C= {2:12 + i=2}. [8] [9] [2] [2] (a) Locate the singularities of the function 23 sin z and classify each singularity as a removable singularity, a pole (giving its order) or an essential singularity. (b) Find two Laurent series about 0 for the function f(z) = : one on {z z] <2}, giving four consecutive non-zero terms, and the other on {2:2>4}, giving two consecutive non-zero terms. f(z) = 24 (2+2)(z-4)' (+). Find the Laurent series about -i for f, giving four consecutive non-zero terms. (c) Let f(z) = z exp (i) State the annulus of convergence of the Laurent series. (ii) Determine whether the singularity of f at -i is a removable singularity, a pole or an essential singularity. (iii) Evaluate ( 2 Exp (+) dz. where C= {2:12 + i=2}. [8] [9] [2] [2]

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a To locate the singularities of the function fz sinz and classify each singularity we need to analyze the behavior of sinz for different values of z The singularities of sinz occur when the denominat... View the full answer