1. Singularities ECE 6382 Notes are from D. R. Wilton, Dept. of ECE David R. Jackson 1

2. Singularity A point z s is a singularity of the function f(z) if the function is not analytic at z s. (The function does not necessarily have to be infinite there.) Recall from Liouville’s theorem that the only function that is analytic and bounded in the entire complex plane is a constant. Hence, the only function analytic everywhere (including the point at infinity) is a constant. Hence, all non-constant analytic functions have singularities somewhere (possibly at infinity). Note: If a function f (z) is unbounded at infinity, then f (1/z) is not analytic at the origin. 2

3. Taylor Series If f(z) is analytic in the region then for If a singularity exists at radius R, then the series diverges for Taylor’s theorem Note: R is called the radius of convergence of the Taylor series. 3

4. Laurent Series If f(z) is analytic in the region then for Laurent’s theorem Note: This theorem also applies to an isolated singularity ( a → 0 ). Isolated singularity: The function is singular at z 0 but is analytic for 4

5. Taylor Series Example Example: From the Taylor theorem we have: x y The point z = 1 is an isolated singularity (a first-order pole). 5

6. Example: Expand about z 0 = 1 : The series converges for The series diverges for Taylor Series Example x y 6

7. Example: Expand about z 0 = 1 : The series converges for The series diverges for Laurent Series Example x y Using the previous example, we have: Isolated singularity at z s = 1 (The coefficients are shifted by 1 from the previous example.) 7

8. Singularities Examples of singularities: (These will be discussed in more detail later.) pole of order p at z = z s ( if p = 1, pole is a simple pole) isolated essential singularity at z = z 0 (pole of infinite order) branch point (not an isolated singularity) removable singularity at z = 0 non-isolated essential singularity z = 0 T L L N N If expanded about the singularity, we can have: T = Taylor L = Laurent N = Neither 8

9. Isolated Singularity Isolated singularity: The function is singular at z s but is analytic for Examples: A Laurent series expansion about z s is always possible! 9

10. Singularities Isolated singularities Removable singularitiesPoles of finite order Isolated essential singularities (pole of infinite order) These are each discussed in more detail next. 10

11. Isolated Singularity: Removable Singularity The limit z → z 0 exists and f (z) is made analytic by defining Example: Removable singularity: Laurent series  Taylor series 11

12. Pole of finite order (order P ): The Laurent series expanded about the singularity terminates with a finite number of negative exponent terms. Examples: Isolated Singularity: Pole of Finite Order simple pole at z = 0 pole of order 3 at z = 3 12

13. Isolated Singularity: Isolated Essential Singularity Isolated Essential Singularity (pole of infinite order): The Laurent series expanded about the singularity has an infinite number of negative exponent terms Examples: 13

14. Classification of an Isolated Singularity at z s Analytic or removable Simple pole Pole of order p Isolated essential singularity Remember that in classifying a singularity, the series is expanded about the singular point, z 0 = z s ! Non-Isolated Essential Singularity => no Laurent series near the singularity exists! 14

15. Picard’s Theorem The behavior near an isolated essential singularity is pretty wild. Picard’s theorem: In any neighborhood of an isolated essential singularity, the function will assume every complex number (with possibly a single exception) an infinite number of times. For example: No matter how small  is, this function will assume all possible complex values (except possibly one). See the next slide. 15

16. Picard’s Theorem (cont.) Example: Set Hence The “exception” here is w 0 = 0 (r 0 = 0). 16

17. Picard’s Theorem (cont.) Example (cont.) This sketch shows that as n increases, the points where the function exp (1/z) equals the given value w 0 “spirals in” to the (essential) singularity. You can always find a solution now matter how small  (the “neighborhood”) is!  17

18. Essential Singularities A singularity that is not a removable singularity, not a pole of finite order, nor a branch point singularity is called an essential singularity. They are the singularities where the behavior is the “wildest”. Two types: isolated and non-isolated. Laurent series about the singularity has an infinite number of negative exponent terms. A Laurent series about the singularity is not possible, in general. 18

19. Non-Isolated Essential Singularity Non-Isolated Essential Singularity: By definition, this is an essential singularity that is not isolated. Example: simple poles at: Note: A Laurent series expansion in a neighborhood of z s = 0 is not possible! X X XX X X x y (Distance between successive poles decreases with m !) XX XXXXXX 19

20. Branch Point: This is another type of non-isolated essential singularity for which a Laurent expansion about the point is not possible Example: Not analytic on the branch cut x y Non-Isolated Essential Singularity (cont.) 20

21. Singularity at Infinity Example: pole of order 3 at w = 0 The function f (z) has a pole of order 3 at infinity. Note: When we say “finite plane” we mean everywhere except at infinity. The function f (z) in the example above is analytic in the finite plane. We classify the types of singularities at infinity by letting and analyzing the resulting function at 21

22. Other Definitions Meromorphic: The function is analytic everywhere in the finite plane except for isolated poles (of finite order). Examples: Entire: The function is analytic everywhere in the finite plane. Examples: Meromorphic functions can always be expressed as the ratio of two entire functions, with the zeros of the denominator function as the poles (proof omitted). 22