Page 1 Laurent’s Series If a function fails tobe analytic at a point z 0 , one cannot apply Taylors’s Theorem at that point. Laurent’s series may be used to express complex functions in cases where Taylor’s series of expansion cannot be apllied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Unlike the Taylor series which expresses f(z) as a series of terms with non- negative powers of z, a Laurent series includes terms with negative powers. Page 2 Laurent’s Series If a function fails tobe analytic at a point z 0 , one cannot apply Taylors’s Theorem at that point. Laurent’s series may be used to express complex functions in cases where Taylor’s series of expansion cannot be apllied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Unlike the Taylor series which expresses f(z) as a series of terms with non- negative powers of z, a Laurent series includes terms with negative powers. Laurent’s Series Theorem Suppose that a function f is analytic throughout an annular domain R 1 < | z – z 0 | < R 2 centered at z 0 and let C denote any positively oriented simple closed contour around z 0 and lying in that domain, Then, Principal Part of Laurent’s Series (n = 0, 1, 2, … ) (n = 1, 2, … ) Page 3 Laurent’s Series If a function fails tobe analytic at a point z 0 , one cannot apply Taylors’s Theorem at that point. Laurent’s series may be used to express complex functions in cases where Taylor’s series of expansion cannot be apllied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Unlike the Taylor series which expresses f(z) as a series of terms with non- negative powers of z, a Laurent series includes terms with negative powers. Laurent’s Series Theorem Suppose that a function f is analytic throughout an annular domain R 1 < | z – z 0 | < R 2 centered at z 0 and let C denote any positively oriented simple closed contour around z 0 and lying in that domain, Then, Principal Part of Laurent’s Series (n = 0, 1, 2, … ) (n = 1, 2, … ) Calculating Laurent’s Series Expansion To calculate the Laurent series we use the standard and modified geometric series which are : Here f(z) = 1/1-z is analytic everywhere apart from the singularity at z = 1. Above are the expansions for f in the regions inside and outside the circle of radius 1, centered on z = 0, where |z| < 1 is the region inside the circle and |z| > 1 is the region outside the circle. Page 4 Laurent’s Series If a function fails tobe analytic at a point z 0 , one cannot apply Taylors’s Theorem at that point. Laurent’s series may be used to express complex functions in cases where Taylor’s series of expansion cannot be apllied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Unlike the Taylor series which expresses f(z) as a series of terms with non- negative powers of z, a Laurent series includes terms with negative powers. Laurent’s Series Theorem Suppose that a function f is analytic throughout an annular domain R 1 < | z – z 0 | < R 2 centered at z 0 and let C denote any positively oriented simple closed contour around z 0 and lying in that domain, Then, Principal Part of Laurent’s Series (n = 0, 1, 2, … ) (n = 1, 2, … ) Calculating Laurent’s Series Expansion To calculate the Laurent series we use the standard and modified geometric series which are : Here f(z) = 1/1-z is analytic everywhere apart from the singularity at z = 1. Above are the expansions for f in the regions inside and outside the circle of radius 1, centered on z = 0, where |z| < 1 is the region inside the circle and |z| > 1 is the region outside the circle. Example Determine the Laurent series for : that are valid in the regions Solution : Page 5 Laurent’s Series If a function fails tobe analytic at a point z 0 , one cannot apply Taylors’s Theorem at that point. Laurent’s series may be used to express complex functions in cases where Taylor’s series of expansion cannot be apllied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Unlike the Taylor series which expresses f(z) as a series of terms with non- negative powers of z, a Laurent series includes terms with negative powers. Laurent’s Series Theorem Suppose that a function f is analytic throughout an annular domain R 1 < | z – z 0 | < R 2 centered at z 0 and let C denote any positively oriented simple closed contour around z 0 and lying in that domain, Then, Principal Part of Laurent’s Series (n = 0, 1, 2, … ) (n = 1, 2, … ) Calculating Laurent’s Series Expansion To calculate the Laurent series we use the standard and modified geometric series which are : Here f(z) = 1/1-z is analytic everywhere apart from the singularity at z = 1. Above are the expansions for f in the regions inside and outside the circle of radius 1, centered on z = 0, where |z| < 1 is the region inside the circle and |z| > 1 is the region outside the circle. Example Determine the Laurent series for : that are valid in the regions Solution : ExampleRead More

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