1. Chap 6 Residues and Poles
if f analytic.
Cauchy-Goursat Theorem:
What if f is not analytic at finite number of points interior to C Residues.
53. Residues
z0 is called a singular point of a function f if f fails to be analytic at z0 but is analytic at some point in every neighborhood of z0.
A singular point z0 is said to be isolated if, in addition, there is a deleted neighborhood of z0 throughout which f is analytic.
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2. The origin is a singular point of Log z, but is not isolated
Ex1.
Ex2.
The origin is a singular point of Log z, but is not isolated
Ex3.
not isolated
isolated
When z0 is an isolated singular point of a function f, there is a R2 such that f is analytic in
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3. Consequently, f(z) is represented by a Laurent series
and C is positively oriented simple closed contour
When n=1,
The complex number b1, which is the coefficient of in expansion (1) , is called the residue of f at the isolated singular point z0.
A powerful tool for evaluating certain integrals.
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4. Ex4.
湊出z-2在分母
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5. Ex5.
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6. 54. Residue Theorems
Thm1. Let C be a positively oriented simple closed contour. If f is analytic inside and on C except for a finite number of (isolated) singular points zk inside C, then
Cauchy’s residue theorem
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7. Ex1.
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8. Thm2:
If a function f is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour C, then
Pf:
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9. Ex2.
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10. 55. Three Types of Isolated Singular points
If f has an isolated singular point z0, then f(z) can be represented by a Laurent series
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11. (i) Type 1.
Ex1.
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12. is known as a removable singular point.
Ex2.
(ii) Type 2
bn=0, n=1, 2, 3,……
is known as a removable singular point.
* Residue at a removable singular point is always zero.
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13. * If we redefine f at z0 so that f(z0)=a0 define
Above expansion becomes valid throughout the entire disk
* Since a power series always represents an analytic function
Interior to its circle of convergence (sec. 49), f is analytic at z0 when it is assigned the value a0 there. The singularity at z0 is therefore removed.
Ex3.
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14. Infinite number of bn is nonzero.
(iii) Type 3:
Infinite number of bn is nonzero.
is said to be an essential singular point of f.
In each neighborhood of an essential singular point, a function assumes every finite value, with one possible exception, an infinite number of times. ~ Picard’s theorem.
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15. has an essential singular point at where the residue
Ex4.
has an essential singular point at
where the residue
an infinite number of these points clearly lie in any given neighborhood of the origin.
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16. an infinite number of these points clearly lie in any given neighborhood of the origin.
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17. 56. Residues at Poles
identify poles and find its corresponding residues.
Thm.
An isolated singular point z0 of a function f is a pole of order m iff f(z) can be written as
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18. Pf: “<=“
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19. “=>”
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20. Ex1.
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21. Need to write out the Laurent series for f(z) as in Ex 2. Sec. 55.
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22. Ex4.
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23. 57. Zeros and Poles of order m
Consider a function f that is analytic at a point z0.
(From Sec. 40).
Then f is said to have a zero of order m at z0.
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24. Thm. Functions p and q are analytic at z0, and
Ex1.
Thm. Functions p and q are analytic at z0, and
If q has a zero of order m at z0, then
has a pole of order m there.
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25. Corollary: Let two functions p and q be analytic at a point z0.
Ex2.
Corollary: Let two functions p and q be analytic at a point z0.
Pf:
Form Theorem in sec 56,
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26. The singularities of f(z) occur at zeros of q, or
Ex3.
The singularities of f(z) occur at zeros of q, or
try tan z
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27. Ex4
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28. 58. Conditions under which
Lemma : If f(z)=0 at each point z of a domain or arc
containing a point z0, then in any neighborhood N0 of z0 throughout which f is analytic. That is, f(z)=0 at each point z in N0.
Pf:
Under the stated condition, For some neighborhood N of z0
f(z)=0
Otherwise from (Ex13, sec. 57)
There would be a deleted neighborhood of z0 throughout which
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29. Since in N, an in the Taylor series for f(z) about z0 must be zero.
Thus in neighborhood N0 since that Taylor series also represents f(z) in N0. 圖解
Theorem.
If a function f is analytic throughout a domain D and f(z)=0 at each point z of a domain or arc contained in D, then in D.
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30. along real x-axis (an arc)
Corollary:
A function that is analytic in a domain D is uniquely determined over D by its values over a domain, or along an arc, contained in D.
Example:
along real x-axis (an arc)
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31. 59. Behavior of f near Removable and Essential Singular Points
Observation :
A function f is always analytic and bounded in some deleted neighborhood of a removable singularity z0.
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32. Assume f is not analytic at z0.
Thm 1:
Suppose that a function f is analytic and bounded in some deleted neighborhood of a point z If f is not analytic at z0, then it has a removable singularity there.
Pf:
Assume f is not analytic at z0.
The point z0 is an isolated singularity of f and f(z) is represented by a Laurent series
If C denotes a positively oriented circle
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33. is satisfied at some point z in each deleted neighborhood
Thm2.
Suppose that z0 is an essential singularity of a function f, and let w0 be any complex number. Then, for any positive number , the inequality
(a function assumes values arbitrarily
close to any given number)
(3)
is satisfied at some point z in each deleted neighborhood
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34. Since z0 is an isolated singularity of f. There is a
Pf:
Since z0 is an isolated singularity of f. There is a
throughout which f is analytic.
Suppose (3) is not satisfied for any z there. Thus
is bounded and analytic in
According to Thm 1, z0 is a removable singularity of g. We let g be defined at z0 so that it is analytic there,
becomes analytic at z0 if it is defined there as
But this means that z0 is a removable singularity of f, not an essential one, and we have a contradiction.
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35. tch-prob