Notes 8 Singularities

Transcription

1 ECE 6382 Fall 27 David R. Jackso Notes 8 Sigularities Notes are from D. R. Wilto, Dept. of ECE

2 Sigularity A poit s is a sigularity of the fuctio f () if the fuctio is ot aalytic at s. (The fuctio does ot ecessarily have to be ifiite there.) Recall from Liouville s theorem that the oly fuctio that is aalytic ad bouded i the etire complex plae is a costat. Hece, all o-costat aalytic fuctios have sigularities somewhere (possibly at ifiity). 2

3 Taylor Series The radius of covergece R c is the distace to the closest sigularity. Sice f () is aalytic i the regio the < Rc for < Rc f a ( ) ( ) a ( ) ( ) ( )! 2π i C ( ) f f + d R c s The series coverges for - < R c. The series diverges for - > R c (proof omitted). 3

4 Lauret Series If f () is aalytic i the regio a< < b a b the s for f a ( ) ( ) f ( ) ( ) a < < b a d,,, 2 2π j ± ± + C s The series coverges iside the aulus The series diverges outside the aulus (proof omitted). 4

5 Taylor Series Example Example: f ( ) y 2 3 f ( ) s x The poit is a sigularity (a first-order pole). From the property of Taylor series we have:, < diverges, > 5

6 Taylor Series Example Example: f ( ) y Expad about : f a ( ) ( ) x a a a 3/2 2 etc.! ! /2 R c π < θ < π + ( ) 3 ( ) The series coverges for < The series diverges for > 6

7 Lauret Series Example Example: f ( ) π < θ < π y Expad about : f a ( ) ( ) x Usig the previous example, we have: a a 2 3 a 8 etc. (The coefficiets are shifted by from the previous example.) 3 ( ) The series coverges for < < The series diverges for > 7

8 Isolated Sigularity Isolated sigularity: The fuctio is sigular at s but is aalytic for < s < δ δ > s / Examples: e si,,, at si A Lauret series expasio about s is always possible! This is a special case of a Lauret series with a 8

9 No-Isolated Sigularity No-Isolated Sigularity: By defiitio, this is a sigularity that is ot isolated. Example: f ( ) Simple poles at: si mπ X X X X X X X X X X X X π (Distace betwee successive poles decreases with m!) y s π x Note: The fuctio is ot aalytic i ay regio < < δ. Note: A Lauret series expasio i a eighborhood of s is ot possible! 9

10 No-Isolated Sigularity (cot.) Brach Poit: This is a type of o-isolated sigularity. Example: ( ) /2 f Not aalytic at the brach poit. y s x Note: The fuctio is ot aalytic i ay regio < < δ. Note: A Lauret series expasio i a eighborhood of s is ot possible!

11 Examples: Examples of Sigularities (These will be discussed i more detail later.) T L si ( ) ( ) s p removable sigularity at If expaded about the sigularity, we ca have: T Taylor L Lauret N Neither pole of order p at s ( if p, pole is a simple pole) L N N / e si /2 essetial sigularity at (pole of ifiite order) o-isolated sigularity brach poit (ot a isolated sigularity)

12 Classificatio of Isolated Sigularities Isolated sigularities Removable sigularities Poles of fiite order Essetial sigularities (poles of ifiite order) ( ) ( ) si cos, 2,,, 2 ( ) ( ) ( + 2) m si, e / These are each discussed i more detail ext. 2

13 Isolated Sigularity: Removable Sigularity Removable sigularity: The limit exists ad f () is made aalytic by defiig ( ) f( ) lim f si Example: ( ) s δ L'Hospital's ( Rule ) cos( ) si lim lim Lauret series Taylor series 3

14 Isolated Sigularity: Pole of Fiite Order Pole of fiite order (order P): f a ( ) ( ) P s s δ The Lauret series expaded about the sigularity termiates with a fiite umber of egative expoet terms. Examples: f ( ), ( P ) simple pole at f , ( P 3) ( ) 3 2 ( 3) ( 3) ( 3) ( ) pole of order 3 at 3 4

15 Isolated Sigularity: Essetial Sigularity Essetial Sigularity (pole of ifiite order): δ ( ) ( ) f a s s The Lauret series expaded about the sigularity has a ifiite umber of egative expoet terms Examples: f ! 6 2 ( ) si ( ) odd / f ( ) e ! 2 6 5

16 Graphical Classificatio of a Isolated Sigularity at s Lauret series: 2 s p s s s 2 s p ( ) ( ) + ( ) + + ( ) + + ( ) + ( ) f a a a a a a Isolated sigularities Simple pole Pole of order p Essetial sigularity Aalytic or removable 6

17 Picard s Theorem The behavior ear a essetial sigularity is pretty wild. Picard s theorem: I ay eighborhood of a essetial sigularity, the fuctio will assume every complex umber (with possibly a sigle exceptio) a ifiite umber of times. s δ For example: / f ( ) e No matter how small δ is, this fuctio will assume all possible complex values (except possibly oe). (See the ext slide.) Picard 7

18 Picard s Theorem (cot.) / Example: f ( ) e δ Set e / w Re iθ ( a give arbitrary complex umber) s Hece e i θ cosθ isi iθ r r θ r i + re e e e w Re ( Θ 2π ) ( Θ ) cosθ rl R, siθ r + 2π ( Θ ) cos θ + si θ r l R π ( π ) 2 l R 2 ( Θ 2π ) Take the l of both sides. + r 2 + Θ + l R, θ ta π / 2 2 The exceptio here is w (R ). 8

19 Example (cot.) Picard s Theorem (cot.) y / f ( ) e s x δ r, θ π / 2 This sketch shows that as icreases, the poits where the fuctio exp (/) equals the give value w spirals i to the (essetial) sigularity. You ca always fid a solutio for ow matter how small δ (the eighborhood ) is! 9

20 Picard s Theorem (cot.) Example (cot.) / f ( ) e i re θ cosθ r / i e e e siθ r Plot of the fuctio exp(/), cetered o the essetial sigularity at. The hue represets the complex argumet, the lumiace represets the absolute value. This plot shows how approachig the essetial sigularity from differet directios yields differet behaviors (as opposed to a pole, which, approached from ay directio, would be uiformly white). 2

21 Sigularity at Ifiity We classify the types of sigularities at ifiity by lettig w / ad aalyig the resultig fuctio at w. Example: 3 f ( ) f ( ) g( w) 3 pole of order 3 at w w The fuctio f () has a pole of order 3 at ifiity. Note: Whe we say fiite plae we mea everywhere except at ifiity. The fuctio f () i the example above is aalytic i the fiite plae. 2

22 Other Defiitios Etire: The fuctio is aalytic everywhere i the fiite plae. Examples: 2 ( ) f e, si, Meromorphic: The fuctio is aalytic everywhere i the fiite plae except for isolated poles of fiite order. Examples: si f( ), g ( ) si ( )( + ) 3 Meromorphic fuctios ca always be expressed as the ratio of two etire fuctios, with the eros of the deomiator fuctio as the poles (proof omitted). 22