The Laurent Expansion of sin z/(z-1) is a mathematical series that represents the function in terms of powers of (z-1) and 1/(z-1). It is given by the formula: ∑n=0 (-1)^n z^n / (n+1)! + ∑n=1 (-1)^n z^(-n) / (n+1)!. This expansion is valid for all complex numbers z except z=1.
The Laurent Expansion is important in mathematics because it allows us to represent complex functions in a simpler form, making them easier to work with and analyze. It also helps us understand the behavior of a function around singularities, which are points where the function is not defined or behaves in a special way.
The Laurent Expansion is a generalization of the Taylor Series. While the Taylor Series only considers powers of (z-a), the Laurent Expansion includes both positive and negative powers of (z-a), making it more versatile in representing complex functions. The Taylor Series is a special case of the Laurent Expansion when all the negative powers are equal to 0.
Yes, the Laurent Expansion can be used to find the poles and residues of a function. The poles of a function are the values of z for which the Laurent Expansion has infinite terms in the negative powers of (z-a). The residue at a pole is the coefficient of the 1/(z-a) term in the Laurent Expansion. These values are important in complex analysis and have many applications in physics and engineering.
Yes, there is a shortcut to finding the Laurent Expansion of a function. If the function is analytic at the point z=a, meaning it can be represented by a Taylor Series around that point, then the Laurent Expansion can be obtained by replacing z with (z-a) in the Taylor Series. This gives us the Laurent Expansion centered at z=a. However, this shortcut only works if the function is analytic at the point of expansion.