Classify the singularity of the function f(z) = z − 2/z2 sin( 1/z−

1.	Classify the singularity of the function f(z) = z − 2/z2 sin( 1/z−1)
Answer» Poles of f(z) are z = 0, 0. That is z = 0 is a pole of order 2. Zeros of f(z) are obtained by, (z − 2) sin(1/z−1 = 0 ⇒ z − 2 = 0 and sin( 1 z−1 ) = 0 ⇒ z = 2 and z = 1 nπ + 1, n = 0, 1, 2, . . . The limits of zeros is 1. Therefore z = 1 is isolated and essential singularity.

Classify the singularity of the function f(z) = z − 2/z2 sin( 1/z−1)

Answer»

Poles of f(z) are z = 0, 0.

That is z = 0 is a pole of order 2.

Zeros of f(z) are obtained by, (z − 2) sin(1/z−1 = 0

⇒ z − 2 = 0 and sin( 1 z−1 ) = 0

⇒ z = 2 and z = 1 nπ + 1, n = 0, 1, 2, . . .

The limits of zeros is 1.

Therefore z = 1 is isolated and essential singularity.

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Classify the singularity of the function f(z) = z − 2/z2 sin( 1/z−1)

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