6 complex variables
TRANSCRIPT
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
P aul Lim Complex Variables – 2 / 69
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
P aul Lim Complex Variables – 3 / 69
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
P aul Lim Complex Variables – 4 / 69
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
P aul Lim Complex Variables – 5 / 69
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
❖ Cauchy-Riemann 3
Singularity-Zero
Complex Integrals
Cauchy’s Theorem
T-L Series
Residue Theorem
Applications
Inverse LT
P aul Lim Complex Variables – 6 / 69
The Cauchy-Riemann Relations
Complex Variables
Cauchy-Riemann
❖ Cauchy-Riemann
❖ Cauchy-Riemann 2
❖ Cauchy-Riemann 3
Singularity-Zero
Complex Integrals
Cauchy’s Theorem
T-L Series
Residue Theorem
Applications
Inverse LT
P aul Lim Complex Variables – 7 / 69
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
❖ Cauchy-Riemann 2
❖ Cauchy-Riemann 3
Singularity-Zero
Complex Integrals
Cauchy’s Theorem
T-L Series
Residue Theorem
Applications
Inverse LT
P aul Lim Complex Variables – 8 / 69
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
❖ Cauchy-Riemann 2
❖ Cauchy-Riemann 3
Singularity-Zero
Complex Integrals
Cauchy’s Theorem
T-L Series
Residue Theorem
Applications
Inverse LT
P aul Lim Complex Variables – 9 / 69
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
P aul Lim Complex Variables – 10 / 69
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
P aul Lim Complex Variables – 11 / 69
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
❖ 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
P aul Lim Complex Variables – 12 / 69
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
❖ 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
P aul Lim Complex Variables – 13 / 69
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
❖ 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
P aul Lim Complex Variables – 14 / 69
(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
❖ 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
P aul Lim Complex Variables – 15 / 69
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
P aul Lim Complex Variables – 16 / 69
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
P aul Lim Complex Variables – 17 / 69
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
❖ 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
P aul Lim Complex Variables – 18 / 69
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
❖ 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
P aul Lim Complex Variables – 19 / 69
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
❖ 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
P aul Lim Complex Variables – 20 / 69
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
❖ 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
P aul Lim Complex Variables – 21 / 69
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
❖ 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
P aul Lim Complex Variables – 22 / 69
(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
❖ 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
P aul Lim Complex Variables – 23 / 69
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
❖ 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
P aul Lim Complex Variables – 24 / 69
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
❖ 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
P aul Lim Complex Variables – 25 / 69
(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
P aul Lim Complex Variables – 26 / 69
Cauchy’s Theorem
P aul Lim Complex Variables – 27 / 69
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
❖ 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
P aul Lim Complex Variables – 28 / 69
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
❖ 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
P aul Lim Complex Variables – 29 / 69
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
❖ 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
P aul Lim Complex Variables – 30 / 69
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
❖ 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
P aul Lim Complex Variables – 31 / 69
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
❖ 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
P aul Lim Complex Variables – 32 / 69
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
❖ 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
P aul Lim Complex Variables – 33 / 69
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
❖ 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
P aul Lim Complex Variables – 34 / 69
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
❖ Example 11 contd
❖ Example 11 contd
Residue Theorem
Applications
Inverse LT
P aul Lim Complex Variables – 35 / 69
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
❖ Example 11 contd
❖ Example 11 contd
Residue Theorem
Applications
Inverse LT
P aul Lim Complex Variables – 36 / 69
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 10 contd
❖ Example 11
❖ Example 11 contd
❖ Example 11 contd
Residue Theorem
Applications
Inverse LT
P aul Lim Complex Variables – 37 / 69
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.
Taylor and Laurent Series 3
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
❖ Example 11 contd
❖ Example 11 contd
Residue Theorem
Applications
Inverse LT
P aul Lim Complex Variables – 38 / 69
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.
Taylor and Laurent Series 4
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
❖ Example 11 contd
❖ Example 11 contd
Residue Theorem
Applications
Inverse LT
P aul Lim Complex Variables – 39 / 69
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
P aul Lim Complex Variables – 40 / 69
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 10 contd
❖ Example 11
❖ Example 11 contd
❖ Example 11 contd
Residue Theorem
Applications
Inverse LT
P aul Lim Complex Variables – 41 / 69
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
P aul Lim Complex Variables – 42 / 69
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 10 contd
❖ Example 11
❖ Example 11 contd
❖ Example 11 contd
Residue Theorem
Applications
Inverse LT
P aul Lim Complex Variables – 43 / 69
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 10 contd
❖ Example 11
❖ Example 11 contd
❖ Example 11 contd
Residue Theorem
Applications
Inverse LT
P aul Lim Complex Variables – 44 / 69
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
❖ Residue Theorem 3
❖ Arc
❖ Arc 2
Applications
Inverse LT
P aul Lim Complex Variables – 45 / 69
Residue Theorem
Complex Variables
Cauchy-Riemann
Singularity-Zero
Complex Integrals
Cauchy’s Theorem
T-L Series
Residue Theorem
❖ Residue Theorem
❖ Residue Theorem 2
❖ Residue Theorem 3
❖ Arc
❖ Arc 2
Applications
Inverse LT
P aul Lim Complex Variables – 46 / 69
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
❖ Residue Theorem 2
❖ Residue Theorem 3
❖ Arc
❖ Arc 2
Applications
Inverse LT
P aul Lim Complex Variables – 47 / 69
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
❖ Residue Theorem 2
❖ Residue Theorem 3
❖ Arc
❖ Arc 2
Applications
Inverse LT
P aul Lim Complex Variables – 48 / 69
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
❖ Residue Theorem 2
❖ Residue Theorem 3
❖ Arc
❖ Arc 2
Applications
Inverse LT
P aul Lim Complex Variables – 49 / 69
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
❖ Residue Theorem 2
❖ Residue Theorem 3
❖ Arc
❖ Arc 2
Applications
Inverse LT
P aul Lim Complex Variables – 50 / 69
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
❖ Example 12 contd
❖ Infinite Integrals
❖ Infinite Integrals 2
❖ Example 13
❖ Example 13 contd
❖ Pole on Axis
❖ Jordan’s lemma
❖ Jordan’s lemma 2
❖ Example 14
❖ Example 14 contd
Inverse LT
P aul Lim Complex Variables – 51 / 69
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 12 contd
❖ Infinite Integrals
❖ Infinite Integrals 2
❖ Example 13
❖ Example 13 contd
❖ Pole on Axis
❖ Jordan’s lemma
❖ Jordan’s lemma 2
❖ Example 14
❖ Example 14 contd
Inverse LT
P aul Lim Complex Variables – 52 / 69
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 12 contd
❖ Infinite Integrals
❖ Infinite Integrals 2
❖ Example 13
❖ Example 13 contd
❖ Pole on Axis
❖ Jordan’s lemma
❖ Jordan’s lemma 2
❖ Example 14
❖ Example 14 contd
Inverse LT
P aul Lim Complex Variables – 53 / 69
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
P aul Lim Complex Variables – 54 / 69
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 12 contd
❖ Infinite Integrals
❖ Infinite Integrals 2
❖ Example 13
❖ Example 13 contd
❖ Pole on Axis
❖ Jordan’s lemma
❖ Jordan’s lemma 2
❖ Example 14
❖ Example 14 contd
Inverse LT
P aul Lim Complex Variables – 55 / 69
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 12 contd
❖ Infinite Integrals
❖ Infinite Integrals 2
❖ Example 13
❖ Example 13 contd
❖ Pole on Axis
❖ Jordan’s lemma
❖ Jordan’s lemma 2
❖ Example 14
❖ Example 14 contd
Inverse LT
P aul Lim Complex Variables – 56 / 69
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 12 contd
❖ Infinite Integrals
❖ Infinite Integrals 2
❖ Example 13
❖ Example 13 contd
❖ Pole on Axis
❖ Jordan’s lemma
❖ Jordan’s lemma 2
❖ Example 14
❖ Example 14 contd
Inverse LT
P aul Lim Complex Variables – 57 / 69
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 12 contd
❖ Infinite Integrals
❖ Infinite Integrals 2
❖ Example 13
❖ Example 13 contd
❖ Pole on Axis
❖ Jordan’s lemma
❖ Jordan’s lemma 2
❖ Example 14
❖ Example 14 contd
Inverse LT
P aul Lim Complex Variables – 58 / 69
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
P aul Lim Complex Variables – 59 / 69
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 12 contd
❖ Infinite Integrals
❖ Infinite Integrals 2
❖ Example 13
❖ Example 13 contd
❖ Pole on Axis
❖ Jordan’s lemma
❖ Jordan’s lemma 2
❖ Example 14
❖ Example 14 contd
Inverse LT
P aul Lim Complex Variables – 60 / 69
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 12 contd
❖ Infinite Integrals
❖ Infinite Integrals 2
❖ Example 13
❖ Example 13 contd
❖ Pole on Axis
❖ Jordan’s lemma
❖ Jordan’s lemma 2
❖ Example 14
❖ Example 14 contd
Inverse LT
P aul Lim Complex Variables – 61 / 69
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 12 contd
❖ Infinite Integrals
❖ Infinite Integrals 2
❖ Example 13
❖ Example 13 contd
❖ Pole on Axis
❖ Jordan’s lemma
❖ Jordan’s lemma 2
❖ Example 14
❖ Example 14 contd
Inverse LT
P aul Lim Complex Variables – 62 / 69
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 12 contd
❖ Infinite Integrals
❖ Infinite Integrals 2
❖ Example 13
❖ Example 13 contd
❖ Pole on Axis
❖ Jordan’s lemma
❖ Jordan’s lemma 2
❖ Example 14
❖ Example 14 contd
Inverse LT
P aul Lim Complex Variables – 63 / 69
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
❖ Example 15 contd
P aul Lim Complex Variables – 64 / 69
Inverse Laplace Transform
P aul Lim Complex Variables – 65 / 69
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
❖ Example 15 contd
P aul Lim Complex Variables – 66 / 69
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
❖ Example 15 contd
P aul Lim Complex Variables – 67 / 69
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
❖ Example 15 contd
P aul Lim Complex Variables – 68 / 69
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
❖ Example 15 contd
P aul Lim Complex Variables – 69 / 69
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