07_taylor's and laurent' s series

7/28/2019 07_taylor's and Laurent' s Series

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Mathematical Methods (2011-2012)7 - Taylors and Laurents series

Paolo Boieri

Dipartimento di Scienze Matematiche

10th May 2012

P. Boieri (Dip. Scienze Mat.) Math. Methods 2011/12 10th May 2012 1 / 16
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The fundamental Theorem

The following theorem is the central result in the theory of functions of a

complex variable.

Theorem

Let f be holomorphic in a domain . Fix z0 and let Br0 (z0) be aneighbourhood of z0 contained in . Then for all z Br0 (z0), we have

that

f(z) =n=0

f(n)(z0)

n!(z z0)

n

(i.e. the power series converges to f(z) if |z z0| < r0).

Remark. We can choose Br0 (z0) in an arbitrary way, with the conditionthat it is included in .The radius of convergence of Taylor series is thenat least equal to the distance of z0 from the boundary of .

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The Cauchys formula for the derivatives

The proof of the fundamental theorem uses the Cauchys integral formulafor the function f(z) at the point z0. As a result of this proof, we can givea generalization of the Cauchys integral formula.

Theorem (Cauchys integral formula for the derivatives)

If f is holomorphic in , then all its derivatives exist at z0; for each n 1and for each Jordan curve counter-clockwise oriented with trace includedin the neighbourhood Br0 (z0) where f is holomorphic we have that

f(n)

(z0) =

n!

2i

f(s)

(s z0)n+1 ds.

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Global results

An important application of the Cauchys integral formula for thederivatives is the following result, that states that there are no boundedfunctions in C (with the obvious exception of the constants). We remarkthat in R functions of these type exist (some examples are sine, cosine,arctangent).

Definition. A complex function w = f(z) is bounded in C if thereexists a real positive value M such that |f(z)| M, z .

Theorem (Liouvilles theorem)

If f is an entire function and it is bounded in C, then f(z) is constant.

Using Liouvilles theorem it is possible to prove the Fundamental Theoremof Algebra.

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Zeroes of an analytic function

If f is analytic at z0, then, in Br0 (z0), f is represented by its Taylors series

f(z) =

+n=0 cn(z z0)n , |z z0| < r0 .

If z0 is a zero of f, then c0 = 0; if, moreover, we have that

f(z0) = f(z0) = = f

(m1)(z0) = 0 e f(m)(z0) = 0 ,

then z0 is called a zero of order m and for |z z0| < r0 we have

f(z) =+n=m

cn(z z0)n =

= (z z0)m

(cm + cm+1(z z0) + cm+2(z z0)2 + . . .) == (z z0)

mg(z).


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Zeroes of an analytic function (continued)

Since g(z0) = 0 and g is continuous at z0, then g(z) = 0 in aneighbourhood of z0. Then we have the following result.

Theorem

The zero of an analytic function (non identically zero) are isolated points.In other words, if f is not identically zero, is analytic at z0 and f(z0) = 0,then there exists a neighbourhood of Br(z0), such that f(z) = 0 for allz Br(z0) \ {z0}.


L h
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Laurents theorem

Theorem

Let f be analytic in the annulus = {z C : r1 < |z z0| < r2} withz0 C and 0 r1 < r2. Then for all z , we have that

f(z) =+

n=

cn(z z0)n, where cn =

1

2i

C

f(s)

(s z0)n+1ds

and C is the curve, counter-clockwise oriented, whose trace is the circle

{s C

: |s z0| = r} con r1 < r < r2.


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Isolated singularities - 1

DefinitionA point z0 C is an isolated singularity for f if there exists aneighbourhood Br(z0) such that f is analytic in Br(z0) \ {z0}

In Br(z0) \ {z0} the function is represented by its Laurents expansion

f(z) = +c2

(z z0)2+

c1z z0

+ c0 + c1(z z0) + c2(z z0)2 +

Definition

The part of this series containing the negative powers of z z0 is calledprincipal part of f at z0.


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Definition

Let z = 0 be an isolated singularity of f . If the principal part of f at z0contains a finite number of terms, we say that z0 is a pole for f .More precisely, if there exists a non zero integer m such that cm = 0 and

cm1 = cm2 = = 0, i.e., if

f(z) =cm

(z z0)m+

cm+1(z z0)m1

+ +c1

z z0+ c0 + c1(z z0) +

we say that z0 is a pole of order m. In particular, if m = 1, it is a simplepole and if m = 2 is a double pole.


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If z0 is a pole of order m, then we can write f in the form

f(z) =1

(z z0)m

n=0

cm+n(zz0)n =

g(z)

(z z0)m, |zz0| < r, cm = 0

where g is an analytic function not zero in a neighbourhood of z0.

Definition

If the principal part of f at z0 contains an infinite number of terms, then

the point z0 is called an essential singularity.


Th f th sid s 1
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Theorem of the residues - 1

Definition

Let z0 be an isolated singularity for f ; suppose that for r > 0 we have that

f(z) =

+n=

cn(z z0)n , 0 < |z z0| < r.

Then the coefficient c1 is called residue of f at z0 and denoted byc1 = Resf(z0) or by c1 = Res(f; z0).


Theorem of the residues 2


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Theorem of the residues - 2

We know that

Resf(z0) = c1 = 12i

C

f(z) dz

where C is a Jordan curve with trace given (for instance) by the circle{z C : |z z0| = r}.

Theorem (Theorm of the residues)Let C be a Jordan arc and f be analytic on C and in its interior, with theexception of a finite number of points z1, z2, . . . , zn belonging to theinterior of C. Then

C

f(z) dz = 2in

k=1

Ref(zk) .


Computation of the residues Simple pole 1
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Computation of the residues - Simple pole - 1

If z0 is a simple pole for f, then for 0 < |z z0| < r we have

f(z) =c1

z z0+ c0 + c1(z z0) +

and(z z0)f(z) = c1 + c0(z z0) + c1(z z0)

2 + .

Thenc1 = Resf(z0) = lim

zz0

(z z0)f(z) .


Computation of the residues Simple pole 2
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Computation of the residues - Simple pole - 2

Moreover, suppose that f is of the form f(z) = n(z)d(z) , with n(z0) = 0 and

that z0 is a zero of order one for d(z) (this means that d(z0) = 0 andd(z0) = 0. Then we have that

Resf(z0) =n(z

0)

d(z0).

In fact

Resf

(z0) = limzz0(z z0)

n(z)

d(z) = limzz0

(z z0)

d(z) d(z0) n(z) =

n(z0)

d(z0) .


Computation of the residues Multiple pole
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Computation of the residues - Multiple pole

If z0

is a pole of order m for f, then

f(z) =cm

(z z0)m+

cm+1(z z0)m1

+ +c1

z z0+ c0 + c1(z z0) +

and

(z z0)mf(z) = cm + cm+1(z z0) + + c1(z z0)

m1 +

In order to compute the residue we differentiate m 1 times:

c1 = Resf(z0) = 1(m 1)! limzz0 d

m1

dzm1 (z z0)mf(z) .


Computation of the residues - Essential singularity
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Computation of the residues - Essential singularity

If z0 is an isolated essential singularity, the residue can be determined only

by the direct inspection of the Laurent expansion of the function.

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