6 Complex Variables

Complex Variables

Complex Variables

❖ F.z/

❖ F.z/ 2

❖ Example 1

❖ Example 2

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT

P aul Lim Complex Variables – 1 / 69

Functions of a Complex Variable

Complex Variables

❖ F.z/

❖ F.z/ 2

❖ Example 1

❖ Example 2

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


The quantity f .z/ is said to be a function of the complex variable z

if to every value of z in a certain domain R (a region of the Arganddiagram) there corresponds one or more values of f .z/.

Thus,f .z/ D u.x; y/ C iv.x; y/

We are only interested with functions that are single-valued, so thateach value of z there corresponds just one value of f .z/.

Functions of a Complex Variable 2

Complex Variables

❖ F.z/

❖ F.z/ 2

❖ Example 1

❖ Example 2

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


A function f .z/ that is single-valued in some domain R isdifferentiable at the point z in R if the derivative

f 0.z/ D lim�z!0

�

f .z C �z/ � f .z/

�z

�

(1)

exists and is unique, in that its value does not depend upon thedirection in the Argand diagram from which �z tends to zero.

Example 1

Complex Variables

❖ F.z/

❖ F.z/ 2

❖ Example 1

❖ Example 2

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


ExampleShow that the function f .z/ D x2 � y2 C i2xy is differentiable forall values of z

SolutionTaking �z D �x C i�y,

f .z C �z/ � f .z/

�z

D.x C �x/2 � .y C �y/2 C 2i.x C �x/.y C �y/ � x2 C y2 � 2ixy

�x C i�y

D2x�x C .�x/2 � 2y�y � .�y/2 C 2i.x�y C y�x C �x�y/

�x C i�y

D 2x C i2y C.�x/2 � .�y/2 C 2i�x�y

�x C i�y

The last term on the right will tend to zero and the unique limit2x C i2y obtained. Since z was arbitrary, f .z/ is differentiable atall points in the complex plane. Note that f .z/ D z2.

Example 2

Complex Variables

❖ F.z/

❖ F.z/ 2

❖ Example 1

❖ Example 2

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


ExampleShow that the function f .z/ D 2y C ix is not differentiableanywhere in the complex plane.

Solution

f .z C �z/ � f .z/

�zD

2y C 2�y C ix C i�x � 2y � ix

�x C i�y

D2�y C i�x

�x C i�y

Let �z ! 0 along a line through z of slope m, so that �y D m�x,

lim�z!0

�

f .z C �z/ � f .z/

�z

�

D lim�x;�y!0

�

2�y C i�x

�x C i�y

�

D2m C i

1 C im

This limit is dependent on m and hence the direction from which�z ! 0.

The Cauchy-Riemann Relations

Complex Variables

Cauchy-Riemann

❖ Cauchy-Riemann

❖ Cauchy-Riemann 2


Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


The Cauchy-Riemann Relations

Complex Variables

Cauchy-Riemann

❖ Cauchy-Riemann



Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


If the limit

L D lim�z!0

�

f .z C �z/ � f .z/

�z

�

(2)

is to exits and be unique, then two particular ways of letting�z ! 0, by moving parallel to the real axis or by moving parallel tothe imaginary axis, must produce the same limit (necessarycondition).

If we let f .z/ D u.x; y/ C iv.x; y/, �z D �x C i�y, then

f .z C �z/ D u.x C �x; y C �y/ C iv.x C �x; y C �y/;

and the limit is given by

L D lim�x;�y!0

�

u.x C �x; y C �y/ C iv.x C �x; y C �y/ � u.x; y/ � iv.x; y/

�x C i�y

�

The Cauchy-Riemann Relations 2

Complex Variables

Cauchy-Riemann

❖ Cauchy-Riemann



Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


If we first suppose that �z is purely real so that �y D 0, we obtain

L D lim�x!0

�

u.x C �x; y/ � u.x; y/

�xC i

v.x C �x; y/ � v.x; y/

�x

�

D@u

@xC i

@v

@x(3)

provided each limit exists at the point z.

Similarly, if �z is taken as purely imaginary, so that �x D 0,

L D lim�y!0

�

u.x; y C �y/ � u.x; y/

i�yC i

v.x; y C �y/ � v.x; y/

i�y

�

D1

i

@u

@yC

@v

@y(4)

The Cauchy-Riemann Relations 3

Complex Variables

Cauchy-Riemann

❖ Cauchy-Riemann



Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


For f to be differentiable, Eqs. 3 and 4 must be identical, and thusequating real and imaginary parts (necessary condition),

@u

@xD

@v

@yand

@v

@xD �

@u

@y(5)

(Cauchy-Riemann relations)

In our earlier examples,(i) f .z/ D x2 � y2 C i2xy:

@u

@xD 2x D

@v

@yand

@v

@xD 2y D �

@u

@y

(ii) f .z/ D 2y C ix:

@u

@xD 0 D

@v

@ybut

@v

@xD 1 6D �2 D

@u

@y

Singularities and Zeroes of Complex Function

Complex Variables

Cauchy-Riemann

Singularity-Zero

❖ Singularity-Zero

❖ Singularity-Zero 2

❖ Example 3

❖ Example 3 contd

❖ Example 3 contd

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


Singularities and Zeroes of Complex Function

Complex Variables

Cauchy-Riemann

Singularity-Zero



❖ Example 3

❖ Example 3 contd

❖ Example 3 contd

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


A singular point of a complex function f .z/ is any point in theArgand diagram at which f .z/ fails to be analytic.

If f .z/ has a singular point at z D z0 but is analytic at all points insome neighbourhood containing z0 but no other singularities, thenz D z0 is called an isolated singularity. The most important type ofisolated singularity is the pole.

If f .z/ has the form

f .z/ Dg.z/

.z � z0/n(6)

where n is a positive integer, g.z/ is analytic at all points in someneighbourhood containing z D z0 and g.z0/ 6D 0, the f .z/ has apole of order n at z D z0.

Singularities and Zeroes of Complex Function 2

Complex Variables

Cauchy-Riemann

Singularity-Zero



❖ Example 3

❖ Example 3 contd

❖ Example 3 contd

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


An alternative definition is that

limz!z0

Œ.z � z0/nf .z/� D a (7)

where a is a finite, non-zero complex number. If no finite value of n

can be found such that Eq. 7 is satisfied then z D z0 is called anessential singularity.

Example 3

Complex Variables

Cauchy-Riemann

Singularity-Zero



❖ Example 3

❖ Example 3 contd

❖ Example 3 contd

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


ExampleFind the singularities of the functions

.i/ f .z/ D1

1 � z�

1

1 C z; .i i/ f .z/ D tanh z

Solution(i) If we write f .z/ as

f .z/ D1

1 � z�

1

1 C zD

2z

.1 � z/.1 C z/

f .z/ has poles of order 1 (or simple poles) at z D 1 and z D �1

Example 3 contd

Complex Variables

Cauchy-Riemann

Singularity-Zero



❖ Example 3

❖ Example 3 contd

❖ Example 3 contd

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


(ii) In this case we write

f .z/ D tanh z Dsinh z

cosh zD

exp z � exp.�z/

exp z C exp.�z/

Thus f .z/ has a singularity when exp z D � exp.�z/, orequivalently when

exp z D expŒi.2n C 1/�� exp.�z/

where n is any integer. Equating the arguments of theexponentials, we find z D .n C 1

2/�i , for integer n.

Example 3 contd

Complex Variables

Cauchy-Riemann

Singularity-Zero



❖ Example 3

❖ Example 3 contd

❖ Example 3 contd

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


Furthermore, we have

limz!.nC

12

/�i

�

Œz � .n C 1=2/�i� sinh z

cosh z

�

D limz!.nC

12

/�i

�

Œz � .n C 1=2/�i� cosh z C sinh z

sinh z

�

D 1

Therefore each singularity is a simple pole.

Complex Integrals

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

❖ Complex Integrals

❖ Complex Integrals 2

❖ Example 4

❖ Example 4 contd

❖ Example 5

❖ Example 5 contd

❖ Example 5 contd

❖ Example 6

❖ Example 6 contd

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


Complex Integrals

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals



❖ Example 4

❖ Example 4 contd

❖ Example 5

❖ Example 5 contd

❖ Example 5 contd

❖ Example 6

❖ Example 6 contd

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


If a complex function f .z/ is single-valued and continuous in someregion R in the complex plane, then we can define the complexintegral of f .z/ between two points A and B along some curve inR; its value will depend, in general, upon the path taken betweenA and B .

Let a particular path C be described by a continuous (real)parameter t (˛ � t � ˇ) that gives successive positions on C bymeans of the equations

x D x.t/; y D y.t/ (8)

with t D ˛ and t D ˇ corresponding to the points A and B

respectively.

Complex Integrals 2

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals



❖ Example 4

❖ Example 4 contd

❖ Example 5

❖ Example 5 contd

❖ Example 5 contd

❖ Example 6

❖ Example 6 contd

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


Then the integral along the path C of a continuous function f .z/ iswritten

Z

C

f .z/ dz (9)

and is given more explicitly as the sum of the real integralsobtained as follows:Z

C

f .z/ dz D

Z

C

.u C iv/.dx C i dy/

D

Z

C

u dx �

Z

C

v dy C i

Z

C

u dy C i

Z

C

v dx

D

Z ˇ

˛

udx

dtdt �

Z ˇ

˛

vdy

dtdt

C i

Z ˇ

˛

udy

dtdt C i

Z ˇ

˛

vdx

dtdt (10)

Example 4

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals



❖ Example 4

❖ Example 4 contd

❖ Example 5

❖ Example 5 contd

❖ Example 5 contd

❖ Example 6

❖ Example 6 contd

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


ExampleEvaluate the complex integral of f .z/ D z�1 along the circlejzj D R, starting and finishing at z D R.

SolutionThe path C1 is parametrised as follows:

z.t/ D R cos t C iR sin t; 0 � t � 2�

FIG. 1: Different paths for an integral of f .z/ D z�1.

Example 4 contd

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals



❖ Example 4

❖ Example 4 contd

❖ Example 5

❖ Example 5 contd

❖ Example 5 contd

❖ Example 6

❖ Example 6 contd

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


whilst f .z/ is given by

f .z/ D1

x C iyD

x � iy

x2 C y2

) u Dx

x2 C y2D

R cos t

R2and v D

�y

x2 C y2D �

R sin t

R2

)

Z

C1

1

zdz D

Z 2�

0

cos t

R.�R sin t/ dt �

Z 2�

0

�

� sin t

R

�

R cos t dt

C i

Z 2�

0

cos t

RR cos t dt C i

Z 2�

0

�

� sin t

R

�

.�R sin t/ dt

D 0 C 0 C i� C i� D 2�i

(R

C1

dzz

DR 2�

0�R sin tCiR cos t

R cos tCi sin tdt D

R 2�0 i dt D 2�i )

Example 5

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals



❖ Example 4

❖ Example 4 contd

❖ Example 5

❖ Example 5 contd

❖ Example 5 contd

❖ Example 6

❖ Example 6 contd

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


ExampleEvaluate the complex integral of f .z/ D z�1 along

(i) the contour C2 consisitng of the semicircle jzj D R in thehalfplane y � 0.

(ii) the contour C3 made up of the two straight lines C3a andC3b .

Solution(i) This is just as in the above example, except that now 0 � t � � .With this change we have that

Z

C2

dz

zD �i (11)

Example 5 contd

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals



❖ Example 4

❖ Example 4 contd

❖ Example 5

❖ Example 5 contd

❖ Example 5 contd

❖ Example 6

❖ Example 6 contd

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


(ii) The straight lines that make up the contour C3 may beparametrised as follows:

C3a; z D .1 � t/R C i tR for 0 � t � 1

C3b; z D �sR C i.1 � s/R for 0 � s � 1

With these parametrizations, the required integrals may be written

Z

C3

dz

zD

Z 1

0

�R C iR

R C t.�R C iR/dt

C

Z 1

0

�R � iR

iR C s.�R � iR/ds (12)

Example 5 contd

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals



❖ Example 4

❖ Example 4 contd

❖ Example 5

❖ Example 5 contd

❖ Example 5 contd

❖ Example 6

❖ Example 6 contd

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


The first integral is

Z 1

0

�R C iR

R.1 � t / C i tRdt D

Z 1

0

.�1 C i/.1 � t � i t/

.1 � t /2 C t2dt

D

Z 1

0

2t � 1

1 � 2t C 2t2dt C i

Z 1

0

1

1 � 2t C 2t2dt

D1

2

�

ln.1 � 2t C 2t2/�1

0C

i

2

�

2 tan�1

�

t � 1=2

1=2

��1

0

D 0 Ci

2

h�

2��

��

2

�i

D1

2�i

The second integral on the right of Eq. 12 can also be shown to have thevalue �i=2. Thus

Z

C3

dz

zD �i

Example 6

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals



❖ Example 4

❖ Example 4 contd

❖ Example 5

❖ Example 5 contd

❖ Example 5 contd

❖ Example 6

❖ Example 6 contd

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


ExampleEvaluate the complex integral of f .z/ D Re z along the paths C1

and C2 and C3.

Solution(i) If we take f .z/ D Re z and the contour C1, then

Z

C1

Re z dz D

Z 2�

0

R cos t .�R sin t C iR cos t / dt D i�R2

(ii) Using C2 as the contourZ

C2

Re z dz D

Z �

0

R cos t .�R sin t C iR cos t / dt D i�R2=2

Example 6 contd

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals



❖ Example 4

❖ Example 4 contd

❖ Example 5

❖ Example 5 contd

❖ Example 5 contd

❖ Example 6

❖ Example 6 contd

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT


(iii) Finally the integral along C3 D C3a C C3b is given by

Z

C3

Re z dz D

Z 1

0

.1 � t/R.�R C iR/ dt

C

Z 1

0

.�sR/.�R � iR/ ds

D1

2R2.�1 C i/ C

1

2R2.1 C i/ D iR2

Cauchy’s Theorem

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

❖ Cauchy’s Theorem

❖ Example 7

❖ Example 7 contd

❖ Example 8

❖ Example 8 contd

❖ Integral Formula

❖ Integral Formula 2

❖ Example 9

T-L Series

Residue Theorem

Applications

Inverse LT


Cauchy’s Theorem


Cauchy’s theorem states that if f .z/ is an analytic function, and f 0.z/ iscontinuous at each point within and on a closed contour C , then

I

C

f .z/ dz D 0 (13)

Proof: Green’s theorem in a plane for a closed contour C bounding a domain R

gives

Z Z

R

�

@p

@xC

@q

@y

�

dxdy D

I

C

.p dy � q dx/:

With f .z/ D u C iv and dz D dx C i dy, this can be applied to

I D

I

C

f .z/ dz D

I

C

.u dx � v dy/ C i

I

C

.v dx C u dy/ to give

I D

Z Z

R

�

@.�u/

@yC

@.�v/

@x

�

dxdy C i

Z Z

R

�

@.�v/

@yC

@u

@x

�

dxdy (14)

f .z/ is analytic and each integrand is zero (Cauchy-Riemann relations). ) I D 0.

Example 7

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem


❖ Example 7

❖ Example 7 contd

❖ Example 8

❖ Example 8 contd



❖ Example 9

T-L Series

Residue Theorem

Applications

Inverse LT


ExampleSuppose two points A and B in the complex plane are joined bywto different paths C1 and C2. Show that if f .z/ is an analyticfunction on each path and in the region enclosed by the two pathsthen the integral of f .z/ is the same along C1 and C2.

Solution

FIG. 2: Two paths C1 and C2 enclosing a region R.

Example 7 contd

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem


❖ Example 7

❖ Example 7 contd

❖ Example 8

❖ Example 8 contd



❖ Example 9

T-L Series

Residue Theorem

Applications

Inverse LT


Since f .z/ is analytic in R, from Cauchy’s theorem we haveZ

C1

f .z/ dz �

Z

C2

f .z/ dz D

I

C1�C2

f .z/ dz D 0

since C1 � C2 forms a closed contour enclosing R. Thus we obtainZ

C1

f .z/ dz D

Z

C2

f .z/ dz

and so the values of the integrals along C1 and C2 are equal.

Example 8

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem


❖ Example 7

❖ Example 7 contd

❖ Example 8

❖ Example 8 contd



❖ Example 9

T-L Series

Residue Theorem

Applications

Inverse LT


ExampleConsider two closed contoursC and in the Argand diagram,with small enough that it liescompletely within C . Show thatif the function f .z/ is analytic inthe region between the two con-tours, thenI

C

f .z/ dz D

I

f .z/ dz

(15)

Solution

FIG. 3: The contour used to prove

the result of Eq. 15.

Example 8 contd

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem


❖ Example 7

❖ Example 7 contd

❖ Example 8

❖ Example 8 contd



❖ Example 9

T-L Series

Residue Theorem

Applications

Inverse LT


The two close parallel lines C1 and C2 join and C . The newcontour � consists of C , C1, and C2.

Within the area bounded by � , f .z/ is analytic

)

I

�

f .z/ dz D 0: (16)

C1 and C2 of � are traversed in opposite directions and lie (in thelimit) on top of each other, and so their contributions to Eq. 16cancel. Thus

I

C

f .z/ dz C

I

f .z/ dz D 0 (17)

The sense of the integral round is opposite to the conventional(anticlockwise) one, and so by traversing in the usual sense, weestablish the result (Eq. 15).

Cauchy’s Integral Formula

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem


❖ Example 7

❖ Example 7 contd

❖ Example 8

❖ Example 8 contd



❖ Example 9

T-L Series

Residue Theorem

Applications

Inverse LT


If f .z/ is analytic within and on a closed contour C and z0 is apoint within C then

f .z0/ D1

2�i

I

f .z/

z � z0dz (18)

Proof:

Take to be a circle centered on the point z D z0, of small enoughradius � that it all lies inside C . Since f .z/ is analytic inside C , theintegrand f .z/=.z � z0/ is analytic in the space between C and .

Thus, from Eq. 15, the integrand around has the same value asthat around C .

Cauchy’s Integral Formula 2

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem


❖ Example 7

❖ Example 7 contd

❖ Example 8

❖ Example 8 contd



❖ Example 9

T-L Series

Residue Theorem

Applications

Inverse LT


Any point z on is given by z D z0 C � exp i�

) I D

I

f .z/

z � z0dz D

I 2�

0

f .z0 C � exp i�/

� exp i�i� exp i� d�

D i

I 2�

0

f .z0 C � exp i�/ d�

If � ! 0, I ! 2�if .z0/, thus establishing Eq. 18.

An extension to Cauchy’s integral formula:

f 0.z0/ D1

2�i

I

C

f .z/

.z � z0/2dz (19)

More generally,

f .n/.z0/ DnŠ

2�i

I

C

f .z/

.z � z0/nC1dz (20)

Example 9

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem


❖ Example 7

❖ Example 7 contd

❖ Example 8

❖ Example 8 contd



❖ Example 9

T-L Series

Residue Theorem

Applications

Inverse LT


ExampleSuppose that f .z/ is analytic inside and on a circle C of radius R

centred at the point z D z0. If jf .z/j � M on the circle, where M

is some constant, show that

jf .n/.z0/j �MnŠ

Rn(21)

Solution

From Eq. 20, we have jf .n/.z0/j DnŠ

2�i

ˇ

ˇ

ˇ

ˇ

I

C

f .z/

.z � z0/nC1dz

ˇ

ˇ

ˇ

ˇ

But

ˇ

ˇ

ˇ

ˇ

I

C

f .z/ dz

ˇ

ˇ

ˇ

ˇ

�

I

C

jf .z/jjdzj � M

I

C

dl D ML where L is

the length of the path C (circumference).

) jf .n/.z0/j �nŠ

2�

M

RnC12�R D

MnŠ

Rn

Taylor and Laurent Series

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

❖ T-L Series

❖ T-L Series 2

❖ T-L Series 3

❖ T-L Series 4

❖ Example 10

❖ Example 10 contd

❖ Example 11



Residue Theorem

Applications

Inverse LT


Taylor and Laurent Series

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

❖ T-L Series

❖ T-L Series 2

❖ T-L Series 3

❖ T-L Series 4

❖ Example 10


❖ Example 11



Residue Theorem

Applications

Inverse LT


If f .z/ is analytic inside and on a circle C of radius R centred onthe point z D z0, and z is a point inside C , then

f .z/ D

1X

nD0

an.z � z0/n (22)

where an is given by f .n/.z0/=nŠ. The Taylor expansion is validinside the region of analyticity.

Suppose f .z/ has a pole of order p at z D z0 but is analytic atevery other point inside C and on C itself. Then the functiong.z/ D .z � z0/pf .z/ is analytic at z D z0, and so may beexpanded as a Taylor series about z D z0,

g.z/ D

1X

nD0

bn.z � z0/n (23)

Taylor and Laurent Series 2

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

❖ T-L Series

❖ T-L Series 2

❖ T-L Series 3

❖ T-L Series 4

❖ Example 10


❖ Example 11



Residue Theorem

Applications

Inverse LT


Thus, for all z inside C , f .z/ will have a power seriesrepresentation of the form

f .z/ Da�p

.z � z0/pC� � �C

a�1

z � z0

Ca0Ca1.z�z0/Ca2.z�z0/2C: : : (24)

with a�p 6D 0. Such a series is called a Laurent series. Bycomparing the coefficients in Eq. 23 and 24, we see thatan D bnCp. Coefficients bn in the Taylor expansion of g.z/ aregiven by

bn Dg.n/.z0/

nŠD

1

2�i

I

g.z/

.z � z0/nC1dz

and so for the coefficients an in Eq. 24 we have

an D1

2�i

I

g.z/

.z � z0/nC1Cpdz D

1

2�i

I

f .z/

.z � z0/nC1dz

an expression that is valid for both positive and negative n.


Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

❖ T-L Series

❖ T-L Series 2

❖ T-L Series 3

❖ T-L Series 4

❖ Example 10


❖ Example 11



Residue Theorem

Applications

Inverse LT


The terms in the Laurent series with n � 0 are collectively calledthe analytic part, whilst the remainder of the series, consisting ofterms in inverse powers of z, is called the principal part, which maycontain an infinite number of terms:

f .z/ D

C1X

nD�1

an.z � z0/n (25)

If f .z/ is analytic at z D z0 then in Eq. 25 all an for n < 0 must bezero. It may happen that not only are all an zero for n < 0 but a0,a1, : : : , am�1, are all zero as well. In this case, the firstnon-vanishing term in Eq. 25 is am.z � z0/m with m > 0, and f .z/

is then said to have a zero of order m at z D z0.


Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

❖ T-L Series

❖ T-L Series 2

❖ T-L Series 3

❖ T-L Series 4

❖ Example 10


❖ Example 11



Residue Theorem

Applications

Inverse LT


If f .z/ is not analytic at z D z0 then two cases arise

(i) It is possible to find an integer p such that a�p 6D 0 buta�p�k D 0 for all integers k > 0.

i.e. f .z/ is of the form of Eq. 24 and is described as having apole of order p at z D z0: the value of a�1 (not a�p) is calledthe residue of f .z/ at the pole z D z0.

(ii) It is not possible to find such a lowest value of �p.

f .z/ is said to have an essential singularity.

Example 10


ExampleFind the Laurent series of

f .z/ D1

z.z � 2/3

about the singularities z D 0 and z D 2 (separately). Hence verify that z D 0 is a poleof order 1 and z D 2 is a pole of order 3, and find the residue of f .z/ at each pole.

SolutionTo obtain the Laurent series about z D 0, we simply write f .z/ as

f .z/ D �1

8z.1 � z=2/3

D �1

8z

�

1 C .�3/.�z

2/ C

.�3/.�4/

2Š.�

z

2/2 C

.�3/.�4/.�5/

3Š.�

z

2/3 C : : :

�

D �1

8z�

3

16�

3z

16�

5z2

4� : : :

Example 10 contd

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

❖ T-L Series

❖ T-L Series 2

❖ T-L Series 3

❖ T-L Series 4

❖ Example 10


❖ Example 11



Residue Theorem

Applications

Inverse LT


Since the lowest power of z is �1, the point z D 0 is a pole oforder 1. The residue of f .z/ at z D 0 is the coefficient of z�1

about that point, and is equal to �1=8.

The Laurent series about z D 2 is found by setting z D 2 C � andsubstituting into the expression for f .z/:

f .z/ D1

.2 C �/�2D

1

2�2.1 C �=2/

D1

2�3

�

1 � .�

2/ C .

�

2/2 � .

�

2/3 C .

�

2/4 � : : :

�

D1

2�3�

1

4�2C

1

8��

1

16C

�

32� : : :

D1

2.z � 2/3�

1

4.z � 2/2C

1

8.z � 2/�

1

16C

z � 2

32� : : :

) z D 2 is a pole of order 3, and residue of f .z/ at z D 2 is 1/8.

Example 11


ExampleSuppose f .z/ has a pole of order m at the point z D z0. By considering theLaurent series of f .z/ about z0, derive a general expression for the residue off .z/ at z D z0. Hence evaluate the residue of the function, at the point z D i ,

f .z/ Dexp iz

.z2 C 1/2

Solutionf .z/ has a pole of order m at z D z0, its Laurent series:

f .z/ Da�m

.z � z0/mC � � � C

a�1

.z � z0/C a0 C a1.z � z0/ C a2.z � z0/2 C : : :

which on multiplying both sides of the equation by .z � z0/m, gives

.z � z0/mf .z/ D a�m C a�mC1.z � z0/ C � � � C a�1.z � z0/m�1 C : : :

Example 11 contd

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

❖ T-L Series

❖ T-L Series 2

❖ T-L Series 3

❖ T-L Series 4

❖ Example 10


❖ Example 11



Residue Theorem

Applications

Inverse LT


Differentiating both sides m � 1 times, we obtain

dm�1

dzm�1Œ.z � z0/mf .z/� D .m � 1/Ša�1 C

1X

nD1

bn.z � z0/n

for some coefficients bn. In the limit z ! z0, however, the terms inthe sum disappear, and after rearranging we obtain the formula

R.z0/ D a�1 D limz!z0

�

1

.m � 1/Š

dm�1

dzm�1Œ.z � z0/mf .z/�

�

(26)

which gives the value of the residue of f .z/ at the point z D z0.

If we now consider the function

f .z/ Dexp iz

.z2 C 1/2D

exp iz

.z C i/2.z � i/2

we see that it has poles of order 2 at z D i and z D �i .

Example 11 contd

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

❖ T-L Series

❖ T-L Series 2

❖ T-L Series 3

❖ T-L Series 4

❖ Example 10


❖ Example 11



Residue Theorem

Applications

Inverse LT


To calculate the residue at z D i , we may apply the formula(Eq. 26) with m D 2. Performing the required differentiation weobtain

d

dzŒ.z � i/2f .z/� D

d

dz

�

exp iz

.z C i/2

�

D1

.z C i/4Œ.z C i/2i exp iz � 2.exp iz/.z C i/�

Setting z D i we find the residue is given by

R.i/ D1

1Š

1

16.�4ie�1 � 4ie�1/ D �

i

2e

An important special case of Eq. 26 occurs when f .z/ has asimple pole at z D z0. Then the residue at z0 is given by

R.z0/ D limz!z0

Œ.z � z0/f .z/� (27)

Residue Theorem

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

❖ Residue Theorem

❖ Residue Theorem 2


❖ Arc

❖ Arc 2

Applications

Inverse LT


Residue Theorem

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

❖ Residue Theorem



❖ Arc

❖ Arc 2

Applications

Inverse LT


Suppose f .z/ has a pole of order m at the point z D z0:

f .z/ D

1X

nD�m

an.z � z0/n (28)

ConsiderH

C f .z/ dz around a closed contour C that enclosesz D z0, but no other singular points. Using Cauchy’s theorem thisintegral has the same value as the integral around a circle ofradius � centred on z D z0:

I D

I

f .z/ dz D

1X

nD�m

an

I

.z � z0/n dz

D

1X

nD�m

an

Z 2�

0

i�nC1 expŒi.n C 1/�� d�

(dz D i� exp i� d� )

Residue Theorem 2

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

❖ Residue Theorem



❖ Arc

❖ Arc 2

Applications

Inverse LT


For every term in the series with n 6D �1, we have

Z 2�

0

i�nC1 expŒi.n C 1/�� d� D

�

i�nC1 expŒi.n C 1/��

i.n C 1/

�2�

0

D 0

but for the n D �1 term we obtainZ 2�

0

i d� D 2�i

Therefore only the term in .z � z0/�1 contributes to the value ofthe integral around (and therefore C ), and I takes the value

I D

I

C

f .z/ dz D 2�ia�1 (29)

Thus the integral around any closed contour containing a singlepole of order m is equal to 2�i times the residue of f .z/ at z D z0.

Residue Theorem 3

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

❖ Residue Theorem



❖ Arc

❖ Arc 2

Applications

Inverse LT


In general, if f .z/ is continuous within and on a closed contour C

and analytic, except for a finite number of poles, within C , thenI

C

f .z/ dz D 2�iX

j

Rj (residue theorem) (30)

whereP

j Rj is the sum of the residues of f .z/ at its poles withinC .

Arc of a circle

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

❖ Residue Theorem



❖ Arc

❖ Arc 2

Applications

Inverse LT


Suppose f .z/ has a simple pole at z D z0, then

f .z/ D �.z/ C a�1.z � z0/�1

where �.z/ is analytic within some neighbourhood surrounding z0.

Consider the integral I of f .z/ along the open contour C , which isthe arc of a circle of radius � centred on z D z0 given by

jz � z0j D �; �1 � arg.z � z0/ � �2 (31)

where � is chosen small enough that no singularity of f , otherthan z D z0, lies within the circle.

) I D

Z

C

f .z/ dz D

Z

C

�.z/ dz C a�1

Z

C

.z � z0/�1 dz

Arc of a circle 2

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

❖ Residue Theorem



❖ Arc

❖ Arc 2

Applications

Inverse LT


As � ! 0 the first integral tends to zero, since the path becomes ofzero length and � is analytic and therefore continuous along it.

On C , z D �ei� ,

) I D lim�!0

Z

C

f .z/ dz D lim�!0

a�1

Z �2

�1

1

�ei�i�ei� d�

!

D ia�1.�2 � �1/ (32)

Applications

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

❖R

of Sinuoidal

❖ Example 12


❖ Infinite Integrals

❖ Infinite Integrals 2

❖ Example 13


❖ Pole on Axis

❖ Jordan’s lemma

❖ Jordan’s lemma 2

❖ Example 14


Inverse LT


Integrals of Sinusoidal Functions

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

❖R

of Sinuoidal

❖ Example 12




❖ Example 13


❖ Pole on Axis



❖ Example 14


Inverse LT


Suppose that an integral of the form

Z 2�

0

F.cos �; sin �/ d� (33)

is to be evaluated. It can be made into a contour integral aroundthe unit circle C by writing z D exp i� and hence

cos � D1

2.z C z�1/; sin � D �

1

2i.z � z�1/; d� D �iz�1 dz (34)

This contour integral can then be evaluated using the residuetheorem, provided the transformed integrand has only a finitenumber of poles inside the unit circle and none on it.

Example 12

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

❖R

of Sinuoidal

❖ Example 12




❖ Example 13


❖ Pole on Axis



❖ Example 14


Inverse LT


ExampleEvaluate

I D

Z 2�

0

cos 2�

a2 C b2 � 2ab cos �d�; b > a > 0 (35)

Solution

By de Moivre’s theorem, cos n� D1

2.zn C z�n/,

) I Di

2ab

I

C

z4 C 1

z2.z � a=b/.z � b=a/dz (n D 2)

) a double pole at z D 0 and a simple pole z D a=b (Note: b > a)

Example 12 contd


Using Eq. 26 with m D 2,

d

dzŒz2f .z/� D

d

dz

�

z4 C 1

.z � a=b/.z � b=a/

�

D.z � a=b/.z � b=a/4z3 � .z4 C 1/Œ.z � a=b/ C .z � b=a/�

.z � a=b/2.z � b=a/2

Setting z D 0 and applying (Eq. 26), we find, R.0/ Da

bC

b

a. For the simple

pole at z D a=b, using Eq. 27 the residue is given by

R.a=b/ D limz!a=b

Œ.z � a=b/f .z/� D.a=b/4 C 1

.a=b/2.a=b � b=a/D �

a4 C b4

ab.b2 � a2/

Therefore by residue theorem

I D 2�i �i

2ab

�

a2 C b2

ab�

a4 C b4

ab.b2 � a2/

�

D2�a2

b2.b2 � a2/

Some Infinite Integrals

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

❖R

of Sinuoidal

❖ Example 12




❖ Example 13


❖ Pole on Axis



❖ Example 14


Inverse LT


Suppose we wish to evaluate an integral of the form

Z

1

�1

f .x/ dx

where f .z/ has the following properties.

(i) f .z/ is analytic in the upper half-plane, Im z � 0, except fora finite number of poles, but with none on the real axis.

(ii) On the semicircle � of radius R, R times the maximum ofjf j on � tends to zero as R ! 1 (a sufficient condition isthat zf .z/ ! 0 as jzj ! 1).

(iii)R 0

�1f .x/ dx and

R

1

0 f .x/ dx both exist.

Some Infinite Integrals 2

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

❖R

of Sinuoidal

❖ Example 12




❖ Example 13


❖ Pole on Axis



❖ Example 14


Inverse LT


The required integral is then given by

Z

1

�1

f .x/ dx D 2�i � .sum of the residues at poles with Im z � 0/ (36)

Condition (ii) ensures thatˇ

ˇ

ˇ

ˇ

Z

�

f .z/ dz

ˇ

ˇ

ˇ

ˇ

� 2�R � .maximum of jf j on � /

which tends to zero as R ! 1.

FIG. 4: A semicircular contour in the upper half-plane.

Example 13

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

❖R

of Sinuoidal

❖ Example 12




❖ Example 13


❖ Pole on Axis



❖ Example 14


Inverse LT


ExampleEvaluate

I D

Z

1

0

dx

.x2 C a2/4

where a is real.

SolutionThe complex function .z2 C a2/�4 has poles of order 4 atz D ˙ai of which only z D ai is in the upper half-plane.Conditions (ii) and (iii) are clearly satisfied.

Let z D ai C � and expand for small �,

1

.z2 C a2/4D

1

.2ai� C �2/4D

1

.2ai�/4

�

1 �i�

2a

��4

Example 13 contd

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

❖R

of Sinuoidal

❖ Example 12




❖ Example 13


❖ Pole on Axis



❖ Example 14


Inverse LT


The coefficient of ��1 is

1

.2a/4

.�4/.�5/.�6/

3Š

�

�i

2a

�3

D�5i

32a7

and hence by the residue theorem

Z

1

�1

dx

.x2 C a2/4D

10�

32a7

and so I D 5�=32a7.

Simple Pole on Axis


FIG. 5: An indented contour used when

the integrand has a simple pole on the

real axis.

If simple pole is on axis, the contouris indented at the pole in the form ofa semicircle of radius � in the up-per half-plane, thus excluding the polefrom the interior of the contour.

P

Z R

�R

f .x/ dx �

Z z0��

�R

f .x/ dx

C

Z R

z0C�

f .x/ dx

for � ! 0. Eq. 32 shows that sinceonly a simple pole is involved its con-tribution is

�ia�1� (37)

where a�1 is the residue at the poleand the minus sign arises because

is traversed in the clockwise (nega-tive) sense.

Jordan’s lemma

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

❖R

of Sinuoidal

❖ Example 12




❖ Example 13


❖ Pole on Axis



❖ Example 14


Inverse LT


Jordan’s lemma. For a function f .z/ of a complex variable z, if

(i) f .z/ is analytic in the upper half-plane except for a finitenumber of poles in Im z > 0,

(ii) the maximum of jf .z/j ! 0 as jzj ! 1 in the upperhalf-plane,

(iii) m > 0

then

I� D

Z

�

eimzf .z/ dz ! 0 as R ! 1 (38)

where � is the semicircular contour.

Jordan’s lemma 2

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

❖R

of Sinuoidal

❖ Example 12




❖ Example 13


❖ Pole on Axis



❖ Example 14


Inverse LT


For 0 � � � �=2

1 �sin �

��

2

�(39)

Then, since on � we have j exp.imz/j D j exp.�mR sin �/j

I� �

Z

�

jeimzf .z/jjdzj � MR

Z �

0

e�mR sin � d�

D 2MR

Z �=2

0

e�mR sin � d�

Thus, using Eq. 39

I� < 2MR

Z �=2

0

e�mR.2�=�/ d� D�M

m.1 � e�mR/ <

�M

m

and hence tends to zero since M does, as R ! 1.

Example 14

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

❖R

of Sinuoidal

❖ Example 12




❖ Example 13


❖ Pole on Axis



❖ Example 14


Inverse LT


ExampleFind the principal value of

Z

1

�1

cos mx

x � adx

for a real, m > 0.

SolutionConsider the function .z � a/�1 exp.imz/; it has a simple pole atz D a, and further j.z � a/�1j ! 0 as jzj ! 1. Applying theresidue theorem,

)

Z a��

�R

C

Z

C

Z R

aC�

C

Z

�

D 0 (40)

Example 14 contd

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

❖R

of Sinuoidal

❖ Example 12




❖ Example 13


❖ Pole on Axis



❖ Example 14


Inverse LT


Now as R ! 1 and � ! 0,R

� ! 0 by Jordan’s lemma, and fromEqs. 36 and 37 we obtain

P

Z

1

�1

eimx

x � adx � i�a�1 D 0 (41)

where a�1 is the residue of .z � a/�1 exp.imz/ at z D a, which isexp.ima/.

Then taking the real and imaginary parts of Eq. 41 gives

P

Z

1

�1

cos mx

x � adx D �� sin ma

P

Z

1

�1

sin mx

x � adx D � cos ma

Inverse Laplace Transform

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT

❖ Inverse LT

❖ Inverse LT 2

❖ Inverse LT 3

❖ Example 15



Inverse Laplace Transform


FIG. 6: The integration path of the inverse

Laplace transform is along the infinite line

L. The quantity � must be positive and

large enough for all poles of the integrand

to lie to the left of L.

Laplace transform Nf .s/ of a functionf .x/, x � 0, is given by

Nf .s/ D

Z

1

0

e�sxf .x/ dx; Re s > 0

Inverse Laplace transform is given bythe Bromwich integral:

f .x/ D1

2�i

Z �Ci�

��i1

esx Nf .s/ ds; � > 0

(42)where s is treated as a complex vari-able and integration is along line L.Position of line is dictated by the re-quirements that � > 0 and that all sin-gularities of Nf .s/ lie to left of line.

Inverse Laplace Transform 2

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT

❖ Inverse LT

❖ Inverse LT 2

❖ Inverse LT 3

❖ Example 15



The path L must be made into a closed contour in such a way thatthe contribution from the completion either vanishes or is simplycalculable.

A typical completion path is shown below.

FIG. 7: Some contour completions for the integration path L of the

inverse Laplace transform.

Inverse Laplace Transform 3

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT

❖ Inverse LT

❖ Inverse LT 2

❖ Inverse LT 3

❖ Example 15



We consider only the simple case (Fig. 7(a)). If there exist

constants M > 0 and ˛ > 0 such that on � j Nf .s/j �M

R˛then

R

� ! 0 as R ! 1. This condition always holds when Nf .s/ hasthe form

Nf .s/ DP.s/

Q.s/

where P.s/ and Q.s/ are polynomials and the degree of Q.s/ isgreater than that of P.s/.

The inverse Laplace transform (Eq. 42) is then given by

f .t/ DX

.residues of Nf .s/esx at all poles/ (43)

Example 15

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT

❖ Inverse LT

❖ Inverse LT 2

❖ Inverse LT 3

❖ Example 15



ExampleFind the function f .x/ whose Laplace transform is

Nf .s/ Ds

s2 � k2

where k is a constant

SolutionIt is clear that Nf .s/ is of the form required for the integral over thecircular arc � to tend to zero as R ! 1, and so we may use theresult (Eq. 43). Now

Nf .s/esx Dsesx

.s � k/.s C k/

and thus has simple poles at s D k and s D �k.

Example 15 contd

Complex Variables

Cauchy-Riemann

Singularity-Zero

Complex Integrals

Cauchy’s Theorem

T-L Series

Residue Theorem

Applications

Inverse LT

❖ Inverse LT

❖ Inverse LT 2

❖ Inverse LT 3

❖ Example 15



Using Eq. 27, the residues at each pole can be easily calculated as

R.k/ Dkekx

2kand R.�k/ D

ke�kx

2k

Thus the inverse Laplace transform is given by

f .x/ D1

2.ekx C e�kx/ D cosh kx

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6 Complex Variables