2. Prepared By :
Mechanical – 2
Batch – C
Enrollment No. Name
150410119109 Shlok Desai
150410119110 Shreyansh Patel
150410119111 Siddharth Deshmukh
150410119112 Aakash Singh
150410119113 Harsh Sirsat
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3. Laurent’s Series
If a function fails tobe analytic at a point z0, 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.
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4. Laurent’s Series
Theorem
Suppose that a function f is analytic throughout an annular domain R1 < | z – z0 | < R2
centered at z0 and let C denote any positively oriented simple closed contour around
z0 and lying in that domain, Then,
Principal Part of Laurent’s
Series
(n = 0, 1, 2, … )
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(n = 1, 2, … )
5. 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.
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9. Singularity
In mathematics, a singularity is in general a point at which a given mathematical
object is not defined, or a point of an exceptional set where it fails to be well-
behaved in some particular way, such as differentiability.
For example, the function on the real line has a singularity at x = 0, where
it seems to "explode" to ±∞ and is not defined.
The function g(x) = |x| also has a singularity at x=0, since it is not differentiable there.
The various types of singularities are as follows :
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10. Isolated Singularity
A point z0 is called an Isolated Singularity of a function f(z) if f(z) has a singularity at
z0 but single valued and analytic in the annular region : 0 < | z-z0 | < R i.e., analytic in
neighborhood of z0.
Otherwise it is non-isolated singular points.
Example:
Here z=0, z = ± i and z = 2 are isolated singularities.
Has a singularity at z = 0
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11. Poles
If the principal part of Laurent’s series has finite number of terms, i.e.,
Then the singularity z = z0 is said to be the pole of order n, if b1 ≠ 0 andb2 = b3 = … = 0,
then
Then singularity z = z0 is said to be pole of order 1 or a Simple Pole.
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12. Removable Singularity
If in the Laurent’s series expansion, the principal part is zero, i.e.,
Then the singularity z = z0 is called Removable Singularity.
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13. Essential Singularity
If in the Laurent’s series expansion, the principal part contains an infinite number of
terms, then the singularity z = z0 is said to be an Essential Singularity.
Since the number of negative power terms of (z-2) is infinite, z = 2 is an essential
singularity.
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