MATEMATIK 4 KOMPLEKS FUNKTIONSTEORI MM 1'1

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MATEMATIK 4KOMPLEKS FUNKTIONSTEORIMM 1.1

• MM 1.1 Laurent rækker
• Emner Taylor rækker
• Laurent rækker
• Eksempler på udvikling af Laurent rækker
• Singulære punkter og nulpunkter
• Hævelig singularitet, pol, væsentlig
  singularitet
• Isoleret singularitet

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KURSUSPLANMATEMATIK 4

• 1. periode

Tirsdag, 1230-1615 Kompleks Funktionsteori Induk
tion Rekursion TKM
Torsdag, 1230-1615 Tidsdiskrete systemer og
sampling JoD
Torsdag, 1230-1615 Lineær Algebra HEb
2. periode
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KURSUSPLANMATEMATIK 4

• kom.aau.dk/tatiana/mat4
• Her findes alt materialet til funktionsteori
  samlet
• Opgaveløsninger, overheads, supplerende materiale
• Findes også på E4-hjemsiden.

Del B Induktion og Rekursion (3 mm)  
Del A Kompkeks Funktionsteori (2 mm)  
Kursuslitteratur
Kursuslitteratur Finn Jensen Sven
Skyum Induktion og Rekursion
kom.aau.dk/tatiana/mat4/IndukRekur.pdf
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What should we learn today?

• How to represent a function that is not
  analytical in singular points in form of a series
• We will call this series Laurent series
• How to classify singular points and zeros of a
  function and how singularities affect behavior of
  a function

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Reminder Taylor series

• Taylors theorem
• Let f(z) be analytic in a domain D, and let z0
  be any point in D. There exists precisely one
  Taylor series with center z0 that represents
  f(z)

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Radius of convergence

• Taylors theorem
• The representation as Taylor series is valid in
  the largest open disk with center z0 in which
  f(z) is analytic.
• Cauchy-Hadamard formula

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Laurent Series

• What to do if f(z) is not analytic in z0 ?
• If f(z) is singular at z0, we can not use a
  Taylor series. Instead, we will use a new kind of
  series that contains both positive integer powers
  and negative integer powers of z- z0 .
• Layrents theorem
• Let f(z) be analytic in a domain containing two
  circles C1 and C2 with center z0 and the ring
  between them. Then f(z) can be represented by the
  Laurent series

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Convergence region

• It is not enough to speak about radious of
  convergence.
• Laurents theorem
• The Laurent series converges and represets f(z)
  in the enlarged open ring obtained from the given
  ring by continuosly increasing the outer circle
  and decreasing the inner circle until each of the
  two circles reaches a point where f(z) is
  singular.
• Special case z0 is the only singular point of
  f(z) inside C2. Series is convergent in a disk
• Another way of determining region of convergence
  it is an intersection of convergence regions of
  two parts of the series

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Uniqueness of Laurent series

• The Laurent series of a given analytic function
  f(z) in its region of convergence is unique.
• However, f(z) may have different laurent series
  in different rings with the same center.

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Typeopgave

• Typical problem Find all Taylor and Laurent
  series of f(z) with center z0 and determine the
  precise regions of convergence.

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Typeopgave

• Typical problem Find all Taylor and Laurent
  series of f(z) with center z0 and determine the
  precise regions of convergence.
• To find coefficients, we dont calculate the
  integrals. Instead, we use already known series.

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Examples
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Singularities and Zeros

• Definition. Funktion f(z) is singular (has a
  singularity) at a point z0 if f(z) is not
  analytic at z0, but every neighbourhood of z0
  contains points at which f(z) is analytic.
• Definition. z0 is an isolated singularity if
  there exists a neighbourhood of z0 without
  further singularities of f(z).
• Example tan z and tan(1/z)

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Classification of isolated singularities

• Removable singularity. All bn 0. The function
  can be made analytic in z0 by assigning it a
  value .
  Example f(z)sin(z)/z, z0 0.
• Pole of m-th order. Only finitely many terms all
  bn 0, ngtm. Example 1 pole of the second order.
• Remark The first order pole simple pole.
• Essential singularity. Infinetely many terms.
  Example 2.

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Classification of isolated singularities

• The classification of singularotoes is not just a
  formal matter
• The behavior of an analytic function in a
  neighborhood of an essential singularity and a
  pole is different.
• Pole a function can be made analutic if we
  multiply it with (z- z0)m
• Essential singularity
• Picards theorem
• If f(z) is analytic and has an isolated
  essential singularity at point z0, it takes on
  every value, with at most one exeprional value,
  in an arbitrararily small neighborhood of z0 .

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Zeros of analytic function

• Definition. A zero has order m, if
• The zeros of an analytical function are isolated.
  
• Poles and zeros
• Let f(z) be analytic at z0 and have a zero of
  m-th order. Then 1/f(z) has a pole of m-th order
  at z0 .

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Analytic or singular at Infinity

• We work with extended complex plane and want to
  investigate the behavior of f(z) at infinity.
• Idea study behavior of g(w)f(1/w)f(z) in a
  neighborhood of w0. If g(w) has a pole at 0, the
  same has f(z) at infinity etc

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Typeopgave

• Typical problem Determine the location and kind
  of singularities and zeros in the extended
  complex plane.
• Examples

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Lhospital rule