Final answer:
Isolated singularities for tan(z) are simple poles at π/2 + kπ for integers k. The singularity for cos(1-1/z) is essential, and the function (sin(3z)/z²) - (3/z) has a removable singularity at z=0.
Step-by-step explanation:
The student is seeking assistance in identifying and classifying isolated singularities of given complex functions. These singular points can be poles, essential singularities, or removable singularities, and their classification requires analysis of the behavior of each function near the points of singularity.
(a) ταν(z)
For the function tan(z), the singularities occur at points where cos(z) is zero. These are points of the form π/2 + kπ, for k an integer. Each of these points is a simple pole because tan(z) behaves like 1/(z - z0) near each z0, which is where cos(z0) is zero.
(b) τος(1-1/z)
The singularities for cos(1-1/z) are present when 1-1/z= π/2 + kπ, solving for z provides a singularity at z=1/(1-π/2 - kπ), for k an integer. However, this is more complicated because the function itself doesn't have a standard limit as z approaches this point. Hence, the singularities here could be classified as essential.
(c) (sin(3z)/z²) - (3/z)
In the function (sin(3z)/z²) - (3/z), z=0 is an isolated singularity. Here, we have cancellation of the highest order pole (since the Laurent series expansion for sin(3z) around z=0 gives us a z³ term). Therefore, the singularity at z=0 is classified as a removable singularity.