MATEMATIK 4
KOMPLEKS FUNKTIONSTEORI
MM 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
MAT 4 – Kompleks Funktionsteori
KURSUSPLAN
MATEMATIK 4
1. periode
Tirsdag, 12:30-16:15
Kompleks Funktionsteori
Induktion & Rekursion
TKM
Torsdag, 12:30-16:15
Tidsdiskrete systemer og
sampling
JoD
2. periode
Torsdag, 12:30-16:15
Lineær Algebra
HEb
MAT 4 – Kompleks Funktionsteori
KURSUSPLAN
MATEMATIK 4
kom.aau.dk/~tatiana/mat4
Her findes alt materialet til funktionsteori samlet:
Opgaveløsninger, overheads, supplerende materiale
Findes også på E4-hjemsiden.
Del A
Kompkeks Funktionsteori
(2 mm)
Kursuslitteratur:
Del B
Induktion og Rekursion (3 mm)
Kursuslitteratur:
Finn Jensen & Sven Skyum
Induktion og Rekursion
kom.aau.dk/~tatiana/mat4/IndukRekur.pdf
MAT 4 – Kompleks Funktionsteori
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
MAT 4 – Kompleks Funktionsteori
Reminder: Taylor series
• Taylor’s 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):
MAT 4 – Kompleks Funktionsteori
Radius of convergence
• Taylor’s 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:
MAT 4 – Kompleks Funktionsteori
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 .
• Layrent’s 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
MAT 4 – Kompleks Funktionsteori
Convergence region
• It is not enough to speak about radious of convergence.
• Laurent’s 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
MAT 4 – Kompleks Funktionsteori
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.
MAT 4 – Kompleks Funktionsteori
Typeopgave
• Typical problem: Find all Taylor and Laurent series of f(z) with center
z0 and determine the precise regions of convergence.
MAT 4 – Kompleks Funktionsteori
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.
MAT 4 – Kompleks Funktionsteori
Examples
MAT 4 – Kompleks Funktionsteori
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)
MAT 4 – Kompleks Funktionsteori
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, n>m. Example
1: pole of the second order.
Remark: The first order pole = simple pole.
• Essential singularity. Infinetely many terms. Example 2.
MAT 4 – Kompleks Funktionsteori
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:
• Picard’s 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 .
MAT 4 – Kompleks Funktionsteori
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 .
MAT 4 – Kompleks Funktionsteori
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 w=0. If
g(w) has a pole at 0, the same has f(z) at infinity etc
MAT 4 – Kompleks Funktionsteori
Typeopgave
• Typical problem: Determine the location and kind of singularities and
zeros in the extended complex plane.
• Examples:
MAT 4 – Kompleks Funktionsteori
L’hospital rule
MAT 4 – Kompleks Funktionsteori