D. R. Wilton Adapted from notes of Prof. David R. Jackson ECE Dept.

1. ECE 6382 Fall 2008 Singularities D. R. Wilton Adapted from notes of Prof. David R. Jackson ECE Dept.

2. Singularity A point zs is a singularity of the function f(z) if the function is not analytic at zs. (The function does not necessarily have to go to infinity there.)

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

4. Laurent Series Laurent’s theorem If f(z) is analytic in the region then for Note: this theorem applies for an isolated singularity (a→ 0).

5. y x Taylor Series Example Example: From the Taylor theorem we have:

6. y x Taylor Series Example Example: Expand about z0 =1: The series converges for The series diverges for

7. Singularities Examples of singularities: removable singularity at z = 0 pole of order p at z = z0( if p= 1, pole is a simple pole) isolated essential singularity at z = z0(pole of infinite order) non-isolated essential singularity z = 0 branch point not isolated

8. Isolated Singularity Isolated singularity: The function is singular at z0 but is analytic for Examples:

9. Singularities Isolated singularities removable singularities poles of finite order isolated essential singularities

10. Isolated Singularity: Removable Singularity The limit z→z0 exists and f(z) is made analytic by defining Example:

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

12. Isolated Singularity: Isolated Essential Singularity Isolated Essential Singularity: The Laurent series expanded about the singularityhas an infinite number of negative exponent terms Examples:

13. Isolated Essential Singularity: 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 come arbitrarily close to every complex number.

14. Essential Singularities Essential Singularities A singularity that is not a removable singularity, a pole of finite order, or 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.

15. Analytic Simple Pole Pole of Order p Essential Singularity Classification of an Isolated Singularity at zs

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

17. y x not analytic on the branch cut Branch Point Branch Point: This is another type of singularity for which a Laurent expansion about the point is not possible. Example:

18. 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) above is analytic in the finite plane.

19. Other Definitions Entire: The function is analytic everywhere in the finite plane. Examples: Meromorphic: The function is analytic everywhere in the finite plane except for a finite number of poles of finite order. Examples: