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Laurent's Series & Types of Singularities

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Laurent's Series & Types of Singularities

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Detailed explanation of Laurent's series and various types of singularities like Essential Singularity, Removable Singularity, Poles, Isolated Singularity, etc.

Detailed explanation of Laurent's series and various types of singularities like Essential Singularity, Removable Singularity, Poles, Isolated Singularity, etc.

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Laurent's Series & Types of Singularities

  1. 1. Laurent’s Series & Types of Singularities
  2. 2. Prepared By : Mechanical – 2 Batch – C 2 Enrollment No. Name 150410119109 Shlok Desai 150410119110 Shreyansh Patel 150410119111 Siddharth Deshmukh 150410119112 Aakash Singh 150410119113 Harsh Sirsat
  3. 3. Laurent’s Series 3 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.
  4. 4. Laurent’s Series 4 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, … ) (n = 1, 2, … )
  5. 5. Calculating Laurent’s Series Expansion To calculate the Laurent series we use the standard and modified geometric series which are : 5 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.
  6. 6. Example Determine the Laurent series for : that are valid in the regions 6 Solution :
  7. 7. Example 7
  8. 8. “ 88 Types of Singularities
  9. 9. Singularity 9 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 :
  10. 10. Isolated Singularity 10 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
  11. 11. Poles 11 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.
  12. 12. Removable Singularity 12 If in the Laurent’s series expansion, the principal part is zero, i.e., Then the singularity z = z0 is called Removable Singularity.
  13. 13. Essential Singularity 13 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.
  14. 14. 14 THANKS! Any questions? You can find me at ak_singh@gmail.com

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