Question

Find the lament series expansion for:

a) z² sin z1/2, 0<|z|b) z-7/z²+z−2 on i) 1<|z|<2;ii)|z|>2
c) 1−cosz/(z−2π)³ ,|z−2π|>0 iii) 0<|z−1|<1
d) 2+3z/z²+z⁴ ,0<|z|<1

Answer 1

Laurent series expansions are not directly addressed in the reference material, and one would typically use standard techniques like the binomial theorem or Taylor series to find these for the specified functions and domains.

The student's question is seeking Laurent series expansions for various complex functions in specific domains. However, the provided reference material does not seem directly relevant to constructing these expansions. In the case of series expansions, a binomial theorem or Taylor series might be used to express a function as a sum of terms derived from the derivatives of the function at a specific point.

The domain information given (e.g., 0<|z|<1 or |z|>2) indicates the annular region in the complex plane where the expansion is valid. Unfortunately, the reference material does not provide enough context or formulas for a comprehensive answer to the question regarding the Laurent series. So it's necessary to use standard series expansion techniques tailored to each specific function and domain in question.