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(1) Negative Powers (2) Convergence within an Annulus (3) Laurent’s Theorem (4) Singular Points

Section 7. SECTION 7 Power Series II - Laurent Series. (1) Negative Powers (2) Convergence within an Annulus (3) Laurent’s Theorem (4) Singular Points. We saw that we can expand a function in a Taylor series about a centre. For example,. Section 7.

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(1) Negative Powers (2) Convergence within an Annulus (3) Laurent’s Theorem (4) Singular Points

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  1. Section 7 SECTION 7 Power Series II - Laurent Series (1) Negative Powers (2) Convergence within an Annulus (3) Laurent’s Theorem (4) Singular Points

  2. We saw that we can expand a function in a Taylor series about a centre. For example, Section 7 Also, we can expand functions about different centres. For example Notice that (a) we always have positive powers of (zz0) (b) the series converges inside a disk

  3. Section 7 But there is another type of series we can have which (a) includes negative powers of (zz0) (b) converges within an annulus Such series are called Laurent Series Example converges for 1z 2 centre singular points at z1, 2

  4. Section 7 But there is another type of series we can have which (a) includes negative powers of (zz0) (b) converges within an annulus Such series are called Laurent Series Example converges for 1z 2 centre singular points at z1, 2

  5. Laurent Series always converge within an annulus Section 7 If we take a function and plot its singular points, we’ll be able to separate the complex plane into different regions of convergence. Example centre centre inside a disk z 1 - ordinary Taylor series with positive powers in an annulus 1z  - Laurent series

  6. Laurent Series always converge within an annulus Section 7 If we take a function and plot its singular points, we’ll be able to separate the complex plane into different regions of convergence. Example centre centre inside a disk z 1 - ordinary Taylor series with positive powers in an annulus 1z  - Laurent series

  7. Laurent Series always converge within an annulus Section 7 If we take a function and plot its singular points, we’ll be able to separate the complex plane into different regions of convergence. Example centre centre inside a disk z 1 - ordinary Taylor series with positive powers in an annulus 1z  - Laurent series

  8. Section 7 Of course we could have different centres ... centre centre in an annulus 2z1 - Laurent series inside a disk z12 - Taylor series

  9. Section 7 We could even have the centre at the singular point ... In this case the series is only be valid for 0z1 - a disk with the singular point / centre punched out of it centre In fact the series in this case is simply the single term !

  10. Section 7 We could even have the centre at the singular point ... In this case the series is only be valid for 0z1 - a disk with the singular point / centre punched out of it centre In fact the series in this case is simply the single term !

  11. Section 7 Example (1) How many series does the function have about the centre z14 ? The function has two singularities (simple poles), at 1, 2. 7/4z14 z1454 5/4z147/4

  12. The annulus is always between two singular points Section 7 Example (1) How many series does the function have about the centre z14 ? The function has two singularities (simple poles), at 1, 2. 7/4z14 z1454 5/4z147/4

  13. Section 7 Example (2) How many series does the function have about the centre z0 ? The function has one singularity (second order pole). z2 2z

  14. Example (3) Section 7 How many series does the function have about the centre z2 ? The function has three singularities (simple poles). z21 1z22 2z25 5z2

  15. Question: Section 7 How many series does the function have about the centre z1 ?

  16. How do we find these Laurent Series ? Laurent’s Theorem Section 7 Suppose that the function f(z) is analytic in an annulus with centre z0. Then the Laurent series is where C Pierre Alphonse Laurent (1843) compare with Section 6, slide 15

  17. Since converges for z 1, the series converges for 1z 1, or z 1 Finding Laurent Series Section 7 As with the Taylor series, there are many ways to find the Laurent series of a function. We don’t actually use the complicated formulae on the previous slide. One method is to use the geometric series, as we did with Taylor series. Example (1) Expand the function 1(1z) in negative powers of z

  18. Since converges for z 1, the series converges for (z 2)(i2)1, or (z 2)5 Example (2) Section 7 Expand the function 1(iz) in powers of z2

  19. Since converges for z 1, the series converges for (i2) (z2)1, or (z2)(i2) 1, or z 25 Example (2) cont. Section 7 But there is another possibility - expand the function 1(iz) in negative powers of z2

  20. Example (3) Section 7 Expand the function about the centre z1 converges for 0z 1

  21. Example (3) Section 7 Expand the function about the centre z1 converges for 0z 1 Here, the centre is the actual singular point !

  22. Section 7 Note - this will help make sense of Laurent Series Each Laurent series consists of two parts: INSIDE positive powers (Taylor series) OUTSIDE negative powers (the “Principal Part”)

  23. Example (4) Section 7 Expand the function about the centre z0 How many ways can we do this ? centre (a) z 1 (b) 1z 3 (c) 3z 

  24. Example (4) Section 7 Expand the function about the centre z0 How many ways can we do this ? centre (a) z 1 (b) 1z 3 (c) 3z 

  25. Example (4) Section 7 Expand the function about the centre z0 How many ways can we do this ? centre (a) z 1 (b) 1z 3 (c) 3z 

  26. Example (4) Section 7 Expand the function about the centre z0 How many ways can we do this ? centre (a) z 1 (b) 1z 3 (c) 3z 

  27. Section 7 (a) z 1 inside disk - positive terms

  28. Section 7 (a) z 1 inside disk - positive terms

  29. Section 7 (b) 1z 3 negative powers 1z  positive powers z 3

  30. On the previous slide, how did we know which term to expand in negative powers and which, if any, to expand in positive powers ? Section 7 The term is “outside” - negative terms The term is “inside” - positive terms The final annulus is where they overlap

  31. Section 7 (c) 3z  negative powers 3z  positive powers z 

  32. Section 7 Singular Points A Singular Point z0 of a function f (z) is where f (z) is not analytic. There are two main different types of singular point. Isolated Singularity Non-isolated Singularity isolated not isolated

  33. Section 7 Isolated Singularities There are two types of isolated singularity We can form the Laurent series with centre z0, valid or 0z z0R Pole of order m The Laurent series “stops” (at the mth negative power) Essential Singularity The Laurent series is infinite (in negative powers) here, centre is singular point

  34. Section 7 Isolated Singularities There are two types of isolated singularity We can form the Laurent series with centre z0, valid or 0z z0R Pole of order m The Laurent series “stops” (at the mth negative power) Essential Singularity The Laurent series is infinite (in negative powers) here, centre is singular point

  35. Section 7 Isolated Singularities There are two types of isolated singularity We can form the Laurent series with centre z0, valid or 0z z0R Pole of order m The Laurent series “stops” (at the mth negative power) Essential Singularity The Laurent series is infinite (in negative powers) here, centre is singular point

  36. Section 7 Example (1) Classify the singularity of the function The Laurent series with z00 as centre is simply the one term , valid for 0z . This is a simple pole 0z Example (2) Classify the singularity of the function The Laurent series with z01 as centre is simply the two terms valid for 0z1. This is a pole of order 3 0z1

  37. Section 7 Example (1) Classify the singularity of the function The Laurent series with z00 as centre is simply the one term , valid for 0z . This is a simple pole 0z Example (2) Classify the singularity of the function The Laurent series with z01 as centre is simply the two terms valid for 0z1. This is a pole of order 3 0z1

  38. Section 7 Example (1) Classify the singularity of the function The Laurent series with z00 as centre is simply the one term , valid for 0z . This is a simple pole 0z Example (2) Classify the singularity of the function The Laurent series with z01 as centre is simply the two terms valid for 0z1. This is a pole of order 3 0z1

  39. Example (3) Section 7 Classify the singularity of the function This is a pole of order 3 0z Example (4) Classify the singularity of the function This is an essential singularity 0zi

  40. Example (3) Section 7 Classify the singularity of the function This is a pole of order 3 0z Example (4) Classify the singularity of the function This is an essential singularity 0zi

  41. Example (3) Section 7 Classify the singularity of the function This is a pole of order 3 0z Example (4) Classify the singularity of the function This is an essential singularity 0zi

  42. Section 7 Note: there are a couple of good reasons for classifying singularities into poles and essential singularities. (1) When we have poles we have lots of formulae for evaluating integrals (see next section) (2) Functions with poles   as z0 is approached (from any direction) - those with essential singularities take on many different values depending on the direction of approach

  43. Section 7 Topics not Covered (1) Proof of Laurent’s Theorem (formulae for Laurent’s series) - slide 8 (2) Removable Singularities removable singularity at z00 (3) Singularities “at infinity” has a 2nd order pole at 0, so has a 2nd order pole at  (4) Zeros has 2nd order zeros at 2

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