Final answer:
The function sin(z)/z³ has a pole of second order at z = 0, as determined by the presence of a 1/z² term in the Laurent series expansion of the sine function divided by z³.
Step-by-step explanation:
The question is about determining the type of singularity of the complex function sin(z)/z³ at z = 0. A singularity in complex analysis is a point at which a mathematical object (in this case, a function) is not well-behaved in some sense, such as not being differentiable or not being defined.
In the case of sin(z)/z³, we can determine the type of singularity by examining the behavior of the function near z = 0. An essential tool in this situation is to use a Laurent series expansion of the sine function around 0. The sine function can be expanded as sin(z) = z - z³/3! + zµ/5! - ..., which when divided by z³ would result in 1/z² - 1/3! + z²/5! - .... The presence of the 1/z² term suggests that the function has a pole of order 2 at z = 0. This is because a pole of a given order is characterized by a term of the form 1/zn in the Laurent series where n is a positive integer, which indicates the order of the pole. In this case, n=2, which demonstrates that the singularity is a pole of second order.