Question 3.26: (a) Locate and name all the singularities of f(z) = z^8 + z^...

Schaum’s Outline of Complex Variables

(a) Locate and name all the singularities of $f(z)=\frac{z^8+z^4+2}{(z-1)^3(3 z+2)^2}$ .

(b) Determine where f(z) is analytic.

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(a) The singularities in the finite z plane are located at z = 1 and z = -2/3; z = 1 is a pole of order 3 and z = -2/3 is a pole of order 2.
To determine whether there is a singularity at z = ∞ (the point at infinity), let z = 1/w. Then

f(1 / w)=\frac{(1 / w)^8+(1 / w)^4+2}{(1 / w-1)^3(3 / w+2)^2}=\frac{1+w^4+2 w^8}{w^3(1-w)^3(3+2 w)^2}

Thus, since w = 0 is a pole of order 3 for the function f(1/w), it follows that z = ∞ is a pole of order 3 for the function f(z).
Then the given function has three singularities: a pole of order 3 at z = 1, a pole of order 2 at z = -2/3, and a pole of order 3 at z =∞.

(b) From (a) it follows that f(z) is analytic everywhere in the finite z plane except at the points z = 1 and -2/3.

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