The region 0<|z-1|<1 lies completely within 0<|z-1|<3, right? So if you find the Laurent series that converges in 0<|z-1|<3, it will obviously converge when 0<|z-1|<1.nhrock3 said:[tex]f(x)=\frac{-2}{z-1}[/tex]+[tex]\frac{3}{z+2}[/tex]
our distance is from -2 till 1
we develop around 1
so our distances are 3 and zeo
so our areas are
0<|z-1|<3
0<|z-1|
3>|z-1|
but i was told to develop around
0<|z-1|<1
there is no such area
?
A Laurent series is a type of mathematical series that represents a complex function as a sum of infinite terms. It is used to approximate functions that have singularities or poles, and is often used in complex analysis.
Laurent series are used in various fields of science, including physics, engineering, and finance. They are also used in signal processing, image processing, and data compression. In physics, Laurent series are used to describe the behavior of physical systems, such as electric fields and fluid dynamics.
A Laurent series includes both positive and negative powers of the variable, while a Taylor series only includes positive powers. Additionally, a Laurent series can be used to represent functions with singularities, while a Taylor series is only applicable to functions that are infinitely differentiable.
The coefficients of a Laurent series can be found using the formula cn = 1/2πi ∫C f(z)(z-z0)-n-1 dz, where C is a contour that encircles the singularity z0 in a counterclockwise direction. This process is known as the Cauchy integral formula.
One limitation is that Laurent series can only be used to approximate functions that have isolated singularities. They cannot be used for functions with essential singularities, such as the function e1/z. Additionally, the convergence of a Laurent series may not be uniform, which can affect its accuracy in approximating the function.