PPT: Taylor Series & Laurent Series | Engineering Mathematics - Civil Engineering (CE) PDF Download

 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 :
Example