Embed Size (px)
Citation preview
COURSE MATERIAL
MATHEMATICS-IV(15A54402)
LECTURE NOTES
B.TECH
II - YEAR & II - SEM
Prepared by:
Dr. B. Nagabusanam, Assistant Professor
Department of H&S
VEMU INSTITUTE OF TECHNOLOGY (Approved By AICTE, New Delhi and Affiliated to JNTUA, Ananthapuramu)
Accredited By NAAC & ISO: 9001-2015 Certified Institution Near Pakala, P. Kothakota, Chittoor- Tirupathi Highway
Chittoor, Andhra Pradesh - 517 112 Web Site: www.vemu.org
COURSE MATERIAL
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY ANANTAPURII B.Tech II-Sem (E.C.E) T Tu C
3 1 3(15A54402) MATHEMATICS -IV
(Common to ECE, EEE)OBJECTIVES: To enable the students to understand the mathematical concepts of special functions & complex variables and their applications in science and engineering. OUTCOMES: The student achieves the knowledge to analyse the problems using the methods of special functions and complex variables UNIT – I: Special Functions: Gamma and Beta Functions – their properties – Evaluation of improper integrals. SeriesSolutions of ordinary differential equations (Power series and Frobenius Method). UNIT – II: Bessel functions – Properties – Recurrence relations – Orthogonality. Legendre polynomials – Properties –Rodrigue’s formula – Recurrence relations – Orthogonality. UNIT – III Functions of a complex variable – Continuity – Differentiability – Analyticity – Properties – Cauchy-Riemann equations in Cartesian and polar coordinates.Harmonic and conjugate harmonic functions – Milne– Thomson method. Conformal mapping: Transformation of ez, Inz, z2, Sin z, cos z, Bilinear transformation- Translation, rotation, magnification and inversion – Fixed point – Cross ratio – Determination of bilineartransformation. UNIT – IV Complex integration: Line integral – Evaluation along a path and by indefinite integration – Cauchy’sintegral theorem – Cauchy’s integral formula – Generalized integral formula.Complex power series: Radiusof convergence – Expansion in Taylor’s series, Maclaurin’s series and Laurent series. Singular point –Isolated singular point – Pole of order m – Essential singularity. UNIT – V Residue – Evaluation of residue by formula and by Laurent’s series – Residue theorem. Evaluation ofintegrals of the type (a) Improper real integrals
TEXT BOOKS: 1.Higher Engineering Mathematics, B.S.Grewal, Khanna publishers. 2.Engineering Mathematics, Volume - III, E. Rukmangadachari & E. Keshava Reddy, Pearson Publisher
REFERENCES: 1. Mathematics III by T.K.V. Iyengar, B.Krishna Gandhi, S.Ranganatham and M.V.S.S.N.Prasad, S.Chandpublications. 2. Advanced Engineering Mathematics, Peter V.O’Neil, CENGAGE publisher. 3. Advanced Engineering Mathematics by M.C. Potter, J.L. Goldberg, Edward F.Aboufadel, Oxford.
1
UNIT-I
SPECIAL FUNCTIONS-I
Beta Function
The definite integral ( )1
11
0
1nmx x dx−− − is called the Beta function and is
denoted by B (m, n) and read as “Beta m, n”. The above integral converges for m > 0,
n > 0.
( ) ( )1
11
0
, 1nmB m n x x dx−− = − , where m > 0, n > 0.
Beta function is also called Eulerian integral of the properties of Beta functions.
(i) Symmetry of Beta function i.e. B (m, n) = B (n, m)
(ii) Trigonometric form B (m, n) =
22 1 2 1
0
2 sin .cosm n d
− −
(iii) B (m, n) = B (m + 1, n) + B (m, n + 1)
(iv) If m and n are positive integers then B (m, n) = ( ) ( )( )
1 ! 1 !
1 !
m n
m n
− −
+ −
Other forms of Beta functions:
Form: Form-I:
( )( ) ( ) ( )
1 1 1
0 0 0
,1 1 1
m n n
m n m n m n
x x yB m n dx dx dy
x x y
− − −
+ + += = =
+ + +
(or)
( )( )
1
0
,1
q
p q
yB p q dy
y
−
+=
+
Hint: ( ) ( )1
11
0
, 1nmB m n x x dx−−= −
2
Put 1
1x
y=
+
( )2
1
dydx
y= −
+
Form-II:
( )( )
1 1 1
0
,1
m n
m n
x xB m n dx
x
− −
+
+=
+ (or)
( )( )
1 1 1
0
,1
p q
p q
x xB p q dx
x
− −
+
+=
+
Hint: ( )( ) ( ) ( )
11 1 1
0 0 1
,1 1 1
m m m
m n m n m n
x x xB m n dx dx dx
x x x
− − −
+ + += = +
+ + +
(I) (II)
II. ( )
1
0 1
m
m n
xdx
x
−
++
Put 2
1 1x dx dy
y y
−= =
Form-III:
( )( )
1
0
,m
m n
m n
xB m n a b dx
ax b
−
+=
+
Hint: R.H.S ( )
1
0
mm n
m n
xa b dx
ax b
−
++
=
1
0
mm n
m n
xa b dx
ax bb
b b
−
+
+
=
1
0 1
mm n
m n
m n
xa b dx
axb
b
−
+
+ +
3
Put ax by
y xb a= =
bdx dy
a=
Form-IV: ( )
( )
( )
( )
111
0
1 ,
1
nm
m n mn
x x B m ndx
x a a a
−−
+
−=
+ +
Hint: By def ( ) ( )1
11
0
, 1nmB m n x x dx−−= + … (1)
Put ( )
( )1
1a y y
x ay a y a
+ = = +
+ + ( )1
y a aa
y a
+ −= +
+
( )1y a a
ay a y a
+= + −
+ +
( )1 1a
ay a
= + −
+
( )( )
2
11 0dx a a dy
y a
− = + −
+
( )
( )2
1a adx dy
y a
+=
+
Form-V: ( ) ( ) ( ) ( )1 1 1
, , 0, 0
am n m n
b
x b a x dx a b B m n m n− − + −
− − = −
Hint: By def ( ) ( )1
11
0
, 1nmB m n x x dx−−= −
Put t b
xa b
−=
−
dtdx
a b=
−
4
1. 2
0
1 1 1sin cos ,
2 2 2
m n m nd B
+ +
=
2. ( ) ( ) ( ), 1 1, ,
, 0, 0B p q B p q B p q
p qq p p q
+ += =
+
Gamma Function: The definite integral 1
0
x ne x dx
− −
is called the Gamma function
and is denoted by ( )n and read as “Gamma n”.
The integral converges only for n > 0
( ) 1
0
, 0x nn e x dx where n
− − =
Gamma function is also called Eulerian Integral of the second kind. The integral
1
0
x ne x dx
− −
does not converge if 0n .
Properties of Gamma Function:
1. 1 1=
2. ( ) ( )1 1 , 1n n n n= − −
3. 1n n n+ =
4. ( )( ) ( ) ( ) ( )1 2 ..... , 0n n n n r n r n r= − − − − −
5. 1 !n n n n+ = =
6. 1 !n n+ = , n is a non-negative integer
7. ( ), , 0, 0m n
B m n m nm n
= +
8. ( )1sin
n nn
− =
9. 1 1
, 22 2
= − = −
5
10. ( )1
0
. 0x nn e x dx n
− −=
11. ( )1
0, 1, 2, 3n
n nn
+= − − −
12. ( ) ( ) ( )0, 1 , 2 , 3− − − ….. are all undefined..
Note: 1. ( )n is defined when n > 0.
2. ( )n is defined when ‘n’ is a negative fraction
3. But n is not defined when n = 0 and n is a negative integer.
13. 2
02
xe dx
− =
14. 2
0
2
xe dx−
−
=
15. 2xe dx
−
−
=
6
Series Solutions of Ordinary Differential Equations
✓ Power Series:
An infinite series of the form
( ) ( ) ( )2
0 0 1 0 2 0
0
...n
n
n
a x x a a x x a x x
=
− = + − + − +
where 0 1 2, , ,...a a a are real constants, is called a power series in powers of
( )0x x− .
✓ Analytic Function:
A function f(x) defined on an interval containing the point x = x0 is said to be
analytic at x0 if the Taylor series of f(x) given by
( )
( )0
0
0 !
n
n
n
x xf x
n
=
− ….. (1)
Exists and converges to f(x) for all x in the interval of convergence of eqn. (1)
✓ Ordinary Point:
A point x = a is called an ordinary point of the equation
2
20
d y dyP Qy
dx dx+ + = ….. (2)
where P and Q are polynomials in x, if both the functions P and Q are analytic
at x = a.
That is, x = a is an ordinary point o the differential equation (1) if the
denominators of P and Q do not vanish for x = a i.e., ,P Q .
✓ Singular Point:
If the point x = a is not an ordinary point of the differential equation
( ) ( )2
20
d y dyP x Q x y
dx dx+ + = ….. (1)
then it is called a singular point of eqn. (1)
There are two types of singular points named Regular Singular Point, Irregular
Singular Point.
i. Regular Singular Point:
A singular point x = a of differential equation is called regular if both
( )x a P− and ( )2
x a Q− are analytic at x = a.
7
i.e., ( )x a P− and ( )2
x a Q− are not infinite at x = a.
ii. Irregular Singular Point:
A singular point, which is not regular is called an irregular singular
point. Thus if ( )x a P− and ( )2
x a Q− are infinite at x = a, then x = a is an
irregular singular point.
✓ Power Series Solution about the Ordinary Point x = 0:
Working rule to solve the equation
( ) ( )2
20
d y dyP x Q x y
dx dx+ + = ….. (1)
When x = 0 is an ordinary point of the equation.
S1. Assume the solution of eqn. (1) to be of the form.
2
0 1 2
0
..... ....n n
n r
r
y a a x a x a x a x
=
= + + + + + = …… (2)
Where 0 1 2, , ,...,a a a are constants to be found.
S2. Find
2
2,
dy d y
dx dx from eqn (2) and substitute the values of ,
dyy
dx and
2
2
d y
dx
in eqn (1). The result of this substitution is an identity.
S3. Equate to zero the coefficients of the various powers of x. Now we will
get a number of equations involving 0 1 2, , ,...a a a
The result obtained by equating the coefficient of xn to zero is
called recurrence relation and it can be used to compute additional
constants.
S4. Determine the values of 2 3 4, , ,...a a a in terms of a0 and a1.
S5. Finally substitute the values of 2 3 4, , ,...a a a in eqn. (2) to get the desired
series solution involving two arbitrary constants a0 and a1.
✓ Frobenius Methods:
When x = a is a regularity singularity of the differential equation
( ) ( ) ( )2
0 1 220
d y dyP x P x P x y
dx dx+ + = ..... (1)
atleast one of the solutions can be expressed in the form
8
( ) ( )0
m r
r
r
y x a a x a
=
= − −
= ( ) ( ) ( )2
0 1 2 ....m
x a a a x a a x a − + − + − +
where the exponent ‘m’ may be real or complex number.
The method for solving the eqn. (1) when x = 0 is a regular singularity is
based on this theorem and is called the Frobenius method.
9
UNIT-II
SPECIAL FUNCTIONS-II
(Bessel Functions)
✓ Legendres’s Differential Equation
The differential equation
( ) ( )21 2 1 0x y xy n n y − − + + = …. (1)
is known as the Legendre’s differential equation. Hence n is a real number.
The Legendre’s equation (1) can also be written as
( ) ( )21 1 0d dy
x n n ydx dx
− + + =
✓ General Solution of Legendre’s Equation:
The Legendre’s eqn. (1) can be solved in series of ascending or descending
powers of x.
The solution of eqn. (1) in series is
0
k r
r
r
y a x
−
=
=
CASE-1: If k = n,
( )( )
( )( )( )( )( )
2 4
0
1 1 2 3...
2 2 1 2.4. 2 1 2 3
n n nn n n n n ny a x x x
n n n
− − − − − −
= − + − − − −
Which is one solution of Legendre’s equation.
If ( )
0
1.3.5... 2 1
!
na
n
−= then the above solution is called the Legendre’s
function of first kind and it is denoted by ( )nP x .
CASE-2: If ( )1k n= − +
10
( )( )( )
( )( )( )( )( )( )
1 3 5
0
1 2 1 2 3 4...
2 2 3 2.4. 2 3 2 5
n n nn n n n n ny a x x x
n n n
− − − − − − + + + + + +
= + + + + + +
Which is the other solution of Legendre’s equation.
If ( )
0
!
1.3.5... 2 1
na
n=
+ then the above solution is called the Legendre’s
function of second kind and it is denoted by ( )nQ x
Thus most general solution of Legendre’s equation is given by
( ) ( )n ny aP x bQ x= +
where a and b are arbitrary constants.
→ ( )nP x is a terminating series and gives what are known as Legendre’s
polynomials for different values of n.
We can write
( )( ) ( )( ) ( )
2
0
1 2 2 !
2 ! 2 ! !
rN
n r
n nr
n rP x x
r n r n r
−
=
− −=
− −
Where
( )
/ 2,
1 / 2,
n if nis evenN
n if nisodd
=
−
✓ Rodrigue’s Formula:
( ) ( )211
2 !
nn
n n n
dP x x
n dx= −
✓ Legendre polynomials:
→ ( )0 1P x =
→ ( )1P x x=
→ ( ) ( )2
2
13 1
2P x x= −
→ ( ) ( )3
3
15 3
2P x x x= −
→ ( ) ( )4 2
4
135 30 3
8P x x x= − +
11
→ ( ) ( )5 3
5
163 70 15
8P x x x x= − +
Generating Function for ( )nP x :
( ) ( )1
2 2
0
1 2 n
n
n
xt t P x t−
=
− + = +
( )1 1nP = and ( ) ( )1 1n
nP − = −
✓ Orthogonality of Legendre polynomials:
( ) ( )1
1
0,
2,
2 1
m n
if m n
P x P x dxif m n
n−
= = +
(or)
( ) ( )1
1
2
2 1m n mnP x P x dx
n−
= +
Where mn is called ‘Kronecker delta’ and is defined by
0,
1,mn
if m n
if m n
=
=
✓ Laplace’s First integral for ( )nP x :
If n is a positive integer, then
( ) 2
0
11cos
n
nP x x x d = −
✓ Laplace’s Second integral for ( )nP x :
If n is a positive integer, then
( )( )1
2
0
11cos
n
nP x x x d
− +
= −
✓ Recurrence Relations:
→ ( ) ( ) ( ) ( ) ( )1 12 1 1n n nn xP x n P x nP x+ −+ = + +
→ ( ) ( ) ( )1n n nnP x xP x P x− = −
→ ( ) ( ) ( ) ( )1 12 1 n n nn P x P x P x+ − + = −
→ ( ) ( ) ( ) ( )11 n n nn P x P x xP x+ + = −
12
→ ( ) ( ) ( ) ( )2
11 n n nx P x n P x xP x−− = −
→ ( ) ( ) ( ) ( ) ( )2
11 1n n nx P x n xP x P x+− = + −
✓ Beltrami’s Result:
( )( ) ( ) ( ) ( ) ( )2
1 12 1 1 1n n nn x P x n n P x P x+ −+ − = + −
✓ Chiristoffel’s Expansion:
( ) ( ) ( ) ( ) ( ) ( ) ( )1 3 52 1 2 5 2 9 ...n n n nP x n P x n P x n P x− − − = − + − + − +
the last term of the series being 3P1 or P0 according as n is even or odd.
✓ Bessel’s Equation:
The differential equation
( )2
2 2 2
20
d y dyx x x n y
dx dx+ + − =
where n is a positive constant, is known as Bessel’s Equation. Its particular
solutions are called Bessel’s functions of order n.
✓ Solution of Bessel’s Differential Equation:
( )( )
( )
2
0
12
! 1
n rr
n
r
x
J xr n r
+
=
−
=+ +
In particular, Bessel’s functions of orders 0 and 1 are given by,
( )2 4 6
0 2 2 2 2 2 21 ...
2 2 .4 2 .4 .6
x x xJ x = − + − +
( )3 5
1 2 2 2...
2 2 .4 2 .4 .6
x x xJ x = − + +
✓ Recurrence Formulae for ( )nJ x :
→ ( ) ( ) ( )1n n nxJ x nJ x xJ x+ = −
→ ( ) ( ) ( )1n n nxJ x nJ x xJ x− = − +
→ ( ) ( ) ( )1 1
1
2n n nJ x J x J x− + = −
→ ( ) ( ) ( )1 12
n n n
xJ x J x J x
n− += +
→ ( ) ( )1
n n
n n
dx J x x J x
dx−
=
13
→ ( ) ( )1
n n
n n
dx J x x J x
dx
− −
+ = −
✓ Generating Function for ( )nJ x :
( )1
2
xt
nt
n
n
e t J x
−
=−
=
✓ Orthogonality of Bessel’s Functions:
( ) ( )( )
1
2
10
0,
1,
2
n n
n
if
xJ x J x dxJ if
+
= =
where and are the roots of ( )nJ x .
14
UNIT-III
FUNCTIONS OF COMPLEX VARIABLE
Functions of complex variable:
Suppose “D” is set of a complex numbers. A rule ‘f’ defined on D which
assign to every Z in D a complex number “W” is called a function f and we write
( )w f Z= .
Here Z is a complex variable and can be written as Z x iy= + x, y are real
and 1i = − .
The set D is called domain of definition of ‘f’. The set of all ( )w f Z=
where Z D is called the range of f.
Since Z x iy= + and Z depends on x and y, we notice that u and v also depend
on x and y
i.e. ( ),u u x y= and ( ),v v x y=
( ) ( ) ( ), ,w f Z u x y iv x y = = + . These u and v are called the real and
imaginary parts of ( )W f Z= .
-Disc around W = W0 in the complex W-plane:
Let us represent all the complex numbers w = u + iv where u, v are real on a
rectangular system of Cartesian (u, v) co-ordinate plane. This is called the w-plane or
(u, v) plane.
Let W0 be a point represented on this plane. Then the set of all points w for
which 0 0. . :w w i e w w w − − is called the -disc around w0. This is
also called as an -neighbouring w0. 0/ 0w w w − called the deleted -disc
around W0.
15
Similarly, we represent the complex number z x iy= + on the rectangular
system of Cartesian (x, y) plane, and this plane is referred to as Complex Z-plane (or)
(x, y) plane.
Limit and continuity:
Def: Limit of f(Z): A function ( )w f z= is said to tend to limit ‘l´ as Z approaches a
point Z0, if for every real ‘’ we can find a positive “” such that ( )f z l − for
00 z z s − i.e. for every 0z z in the -disc in Z-plane, f(z) has a value lying in
the -disc of w-plane.
( )0z z
Lt f z l→
=
Continuity: A function f(z) is said to be continuous at z = z0 if f (z0) is defined and
( ) ( )0
0z zLt f z f z→
=
(or)
A function ‘f’ is continuous at a point z0, if corresponding to each positive number , a
number exists such that ( ) ( )0f z f z − whenever 0z z −
Def: f(z) is said to be continuous in a domain if it is continuous at every point of that
domain.
Derivative of f(z):
Let ( )w f z= be a given function defined for all z in a neighbourhood of z0.
If ( ) ( )0 0
0z
f z z f zLt
z →
+ −
exists, the function f(z) is said to be derivable at z0 and
the limit is denoted by ( )0f z . ( )0f z if exists is called the derivative of f(z) at z0.
Taking 0z z z− = we noticed that ( )( ) ( )
0
0
0
0z z
f z f zf z Lt
z z→
− =
− if f is
differentiable at each point of a set, then we say that f differentiable on that set.
16
Analytic functions:
Def: Let a function f(z) be derivable at every point z in an neighbourhood of z0. i.e.
( )f z exists for all z such that 0z z − whenever 0 .
Then, f(z) is analytic at z0.
Note: If f(z) is analytic at z0.
(i) ( )0f z exists and (ii) ( )f z exists at every point z in a neighbourhood of z0.
Definitions:
Let D be the domain of complex numbers. If ( )f z is analytic at every ( ),Z D f z
is said to be analytic in the domain D.
❖ If ( )f z is analytic at every point “z” on the complex plane, f(z) is said to be
entire function (or) integral function.
❖ A point at which an analytic function ceases to have a derivative is called a
singular point.
❖ An analytic function is also known as regular or holomorphic.
❖ If ( )0f z does not exist then 0z z= is called a singular point of ( )f z .
❖ If ( )f z exists at every point in a neighbourhood of 0z but ( )0f z does not
exist, then 0z is said to be an isolated singular point of ( )f z .
Eg: ( )1
f zz
= is analytic at every point 0z
( ) 2
1f z
z
− = if 0z
at ( )0,Z f z= does not exists.
Z = 0 is an isolated singular point of ..
Cauchy-Riemann (C – R) Equations
17
( ) ( ) ( ), ,f z w u x y iv x y z= = + in domain R
(i) , , ,u u v v
x y x y
are continuous functions of x and y in R.
(ii) ,u v u v
x y y x
= = −
The above relations are known as Cauchy-Riemann equations.
❖ Laplace Equation: If ( ) ( ) ( ), ,f z u x y iv x y= + is analytic in a domain D,
then u and v satisfy Laplace equation
2 22
2 20
u uu
x y
= + =
and
2 22
2 20
v vv
x y
= + =
❖ Harmonic Functions: Solutions of Laplace equations having continuous
second order partial derivatives are called Harmonic functions. Their theory is
called potential theory. Hence the real and imaginary parts of an analytic
function are harmonic functions.
Thus the functions satisfying the Laplace equations
2 2
2 20
x y
+ =
are known
as Harmonic functions.
Conjugate Harmonic function:
❖ If two harmonic functions u and v satisfy the Cauchy-Riemann in a domain D
and there are the real and imaginary parts of an analytic function ‘f’ in D then
v is said to be a conjugate Harmonic function of u in D.
(or)
❖ Two harmonic functions u & v which are such that u + iv is an harmonic
functions are called conjugate harmonic functions.
(or)
❖ If ( )f z u iv= + is analytic and if u & v satisfy Laplace equation, the u and v
are called conjugate harmonic functions.
❖
18
Polar Form of Cauchy-Riemann Equations:
❖ If ( ) ( ) ( ) ( ), ,if z f re u r iv r = = + and ( )f z is derivable at 0
0 0
iz r e
=
then
1u V
r r
=
1v u
r r
= −
Orthogonal Trajectories:
❖ Every analytic function of ( )f z u iv= + defines two families of curves
( ) 1,u x y k= , and ( ) 2,v x y k= forming an orthogonal system.
(or)
❖ If ( )w f z= is an analytic function, then the family of curves defined by
( ),u x y = constant cuts orthogonally the family of curves ( ),v x y = constant.
Conformal Mapping
✓ Mapping or Transformation:
The correspondence defined by the equation ( )w f z= or ( ),u u x y= and
( ),v v x y= between the points in the z-plane and w-plane is called a
‘mapping’ or transformation from z-plane to the w-plane.
✓ Conformal Mapping:
Suppose the mapping takes c1 and c2 are any two curves into the curves 1c and
2c which are intersecting from ( )0 0,P x y to ( )0 0,P u v .
If the transformation is such that the angles between c1 and c2 at ( )0 0,x y is
equal both in magnitude and sense to the angle between 1c and 2c at ( )0 0,u v
then it is said to be a conformal transformation at ..
✓ Sufficient conditions for ( )w f z= to represent a conformal Mapping:
Let ( )f z be an analytic function of z in a domain of the Z-plane and let
( ) 0f z in D. Then ( )w f z= is a conformal mapping at all points of D.
19
✓ The points where 0dw
dz= or are called critical and the points where
0dw
dz are called ordinary points.
✓ Standard Transformations:
1) Translation: Consider the transformation w z C= + , where C is any
complex constant is a translation.
2) Expansion or contraction & Rotation (Magnification): Consider the
transformation w Cz= where C is any complex constant is an expansion.
3) Inversion: The mapping 1
wz
= is called inversion.
✓ Bilinear Transformation (Mobius Transformation):
The transformation az b
wcz d
+=
+ where a, b, c, d are complex constants
and 0ad bc− is known as ‘Bilinear Transformation’.
❖ A bilinear transformation is conformal.
❖ A bilinear transformation preserves cross ratio property of four points.
( )( )( )( )
( )( )( )( )
1 2 3 4 1 2 3 4
1 4 3 2 1 4 3 2
w w w w z z z z
w w w w z z z z
− − − −=
− − − −
❖ Three given distinct points 1 2 3, ,z z z can always be mapped onto three
prescribed points 1 2 3, ,w w w by one and only linear fractional transformation
( )w f z= .
1 2 3 1 2 3
3 2 1 3 2 1
. .w w w w z z z z
w w w w z z z z
− − − −=
− − − −
20
UNIT-IV
COMPLEX INTEGRATION
Definition: A set of points (x, y) such that
( ) ( ) ( ),x x t y y t a t b= =
Where x (t), y (t) are continuous functions of the real variable ‘t’ is called a
continuous arc. If two distinct values of ‘t’ correspond to the same point (x, y) the arc
is called a Jordan arc.
❖ If ( ) ( ) ( ) ( ),x a x b y a y b= = and if no other two values of ‘t’ correspond to
the same point (x, y) the continuous arc is a simple closed curve.
❖ A simple closed curve is also called a Jordan curve.
Line integral: Let ( )f z be a function of complex variable defined in a
domain D.
Let C be an arc in the domain joining from z = to z = .
Let C be defined by ( ) ( ) ( ),x x t y y t a t b= =
Where ( ) ( )x a iy a = + and ( ) ( )x b iy b = +
❖ Cauchys (integral) theorem
❖ Let ( ) ( ) ( ), ,f z u x y iv x y= + , be analytic on and within a simple closed
contour ‘C’ and let ( )f z be continuous. Then
( ) 0C
f z dz =
❖ Cauchy-Goursat theorem for a multiply connected region
Let C denote a closed contour and 1 2 3, , ... kC C C C be a finite number of closed
contours interior to ‘C’ such that the interiors of the .. s do not have any points
in common.
Let R be the region consisting of points on and within “C” except the interior
points of Cj. If B denotes the positively oriented boundary of the region R,
then ( ) 0B
f z dz = where ( )f z is analytic in the region “R”.
21
❖ Indefinite integral:
If ( )f z is analytic in a region R and P and Q are two points in ‘R’ then prove
that ( )Q
P
f z dz is independent of the path joining P and Q.
❖ Cauchys integral formula:
Let ( )f z be an analytic function everywhere on and within a closed contour
C. If Z = a is any point within C, then
( )( )1
2C
f zf a dZ
i Z a=
−
Where the integral is taken in the positive sense around “C”.
Generalization of Cauchys integral formula
❖ If ( )f z is analytic on and within a simple closed curve “C” and if a is any
point within C, then
( )( )
( )1
!
2
n
n
C
f znf a dZ
i Z a +=
−
Complex Power Series
❖ A series of the form n
na Z is called a power series. If n
na Z converges at
Z = Z, then it converges absolutely Z such that 1Z Z .
❖ If n
na Z converges for Z R and diverges for Z R , then R is called
the radius of convergence of the power series and Z R= is called the circle
of convergence of the power series.
❖ If R is the radius of convergence of the power series n
na Z , then the power
series converges uniformly for 1Z R R .
Taylors theorem:
❖ Let ( )f z be analytic at all points within a circle “C0” with centre at a and
radius “r”. Then at each point Z within C0
22
( ) ( )( )
( )( )
( )( )
( )2
.....1! 2! !
nnf a f a f a
f z f a z a z a z an
= + − + − + + − ...
(1)
i.e. the series on the right hand side in Eq. (1) converges to ( )f z whenever
0z a r− .
The expansion in Eq.(1) on the R.H.S. is called the Taylors series expansion of
( )f z in powers of ( )z a− (or) Taylors Series expansion of ( )f z about
z=a.
❖ Maclaurins Series expansions:
i) 2 3
0
1 .... ... , . .2! 3! ! !
n nz
n
z z z ze z z i e z
n n
=
= + + + + + =
ii) ( )( )
( )
2 13 5
0
sin ... 13! 5! 2 1 !
nn
n
z z zz z for z
n
+
=
= − + = − +
iii) ( )( )
2 4 2
0
cos 1 ... 1 .2! 4! 2 !
nn
n
z z zz for z
n
=
= − + = −
iv) ( )
2 1
0
sin h2 1 !
n
n
zz for z
n
+
=
= +
v) ( )
2
0
cosh2 !
n
n
zz for z
n
=
=
vi) ( )0
11 . , 1
1
n n
n
z for zz
=
= − +
vii) 0
1, 1
1
n
n
z for zz
=
= −
Laurents theorem:
❖ Let C1 and C2 be two circles given by 1z a r − = and 2z a r −
respectively where 2 1r r , and z is any point on C1 or C2.
❖ Let ( )f z be analytic on C1 and C2 and throughout the region between the two
circles.
23
❖ Let Z be any point in the ring shaped region between the two circles C1 and C2
then
( ) ( )( )0 1
n nn n
n n
bf z a z a
z a
= =
= − +−
.... (1)
Where ( )
( )1
1
1
2n n
C
f za dz
i z a +
=
− .... (2)
( )
( )2
1
1
2n n
C
f zb dz
i z a − +
=
− .... (3)
Where the integrals are taken around C1 and C2 in the anti-clockwise.
The series in Eq.(1) is called the Laurents Series Expansion of ( )f z around
0z z= .
24
UNIT-V
CALCULUS OF RESIDUES
Zero of an Analytic Function
A zero of an analytic function ( )f z is a value of Z such that ( ) 0f z =
Particularly a point ‘a’ is called a zero of an analytic function ( )f z if ( ) 0f a = .
Zero of mth order:
If an analytic function ( )f z can be expressed in the form
( ) ( ) ( ).m
f z z a z= −
Where ( )z is analytic and ( ) 0a , then Z = a is called zero of mth order
of the function ( )f z .
Simple zero: Simple zero is a zero of order one
Eg:
i) ( ) ( )3
1 ,f z z= − then 1z = is an zero of order ‘3’.
ii) ( )1
1f z
z=
−, then z = is a simple zero of ( )f z .
iii) ( ) sinf z z= , then 0, , 2 .....z = are simple zeros of ( )f z
iv) ( ) tan zf z e= , has no zeros since 0ze z
❖ Singular points: A singular point (or singularity) of a function ( )f z is the point
at which the function ( )f z ceases to be analytic.
❖ Isolated singularity: A point z = a is called an isolated singularity of an analytic
function ( )f z if
a) ( )f z is not analytic at the point z = a
b) ( )f z is analytic in the deleted neighbourhood of z = a i.e. there exists a
neighbourhood of the point z = a which contains no other singularity.
25
Eg: If ( ) 2 1
zef z
z=
+ then z i= are two isolated singular points of ( )f z .
If ( )2
sinf z
z= then , 2 , 3 ....z = are infinite number of isolated
singular points of ( )f z .
❖ Poles of an analytic function: If z = a is an isolated singular point of an analytic
function ( )f z then ( )f z can be expanded in Laurents series about the point
z = a.
i.e. ( ) ( )( )0 1
n nn n
n n
bf z a z a
z a
= =
= − +−
….. (1)
series of negative integral powers of ( )z a− namely ( )1
n
nn
b
z a
= − is known as the
“Principal part” of the Laurents series of ( )f z .
If the principal part contains a finite number of terms say “m”, (i.e. 0nb n=
such that n > m) then the singular point z = a is called a poly of order m of ( )f z .
“Simple pole” is a pole of order one.
Eg: ( )( )( )
2
21 2
zf z
z z=
− + then z = 1 is a simple pole and z = – 2 is a pole of
order 2.
❖ Essential singularity:
If the Principal part of ( )f z contains an infinite number of terms i.e. the series
( )1
n
nn
b
z a
= − contains an infinite number of terms, then the point z = a is called
essential singularity of ( )f z . In this case, ( )z aLt f z→
does not exist.
e.g: z = 0 is an essential singularity of
1
ze , since the principal part of
1
ze contains
infinite number of terms containing negative powers of (z – 0).
❖ Removable singularity:
If the principal part of ( )f z contains no term i.e. if 0nb n= , then the
singularity z = a is called removable singularity of ( )f z .
26
In this case ( ) ( )0
n
n
n
f z a z a
=
= −
Here the singularity can be removed by defining ( )f z at z = a in such a way that
it becomes analytic at z = a. Such a singularity is called removable singularity.
Thus if ( )z aLt f z→
exists finitely, then z = a is removable singularity.
Eg:- ( )1 cos z
f zz
−= , then z = 0 is removable singularity.
❖ Singularities of Infinity:
Taking 1
zt
= in ( )t z we obtain ( )1
f F tt
=
. Then the nature of the singularity
at z = is defined to be the same as that of ( )F t at t = 0
Eg: i) ( ) 3f z z= has a pole of order ‘3’ at z =, since 3
1 1f
t t
=
has a pole of
order ‘3’ at t = 0.
ii) ( ) zf z e= has an essential singularity at z =, since
11
tf et
=
has an
essential singularity at t = 0.
Residues: The coefficient of 1
z a− in the Laurent series expansion of ( )f z
about the isolated singularity z = a is called the residue of ( )f z at that point.
Thus the residue of ( )f z at z = a is b1.
From Laurent series, we know that the coefficient b1 is given by
( )1
1
2C
b f z dzi
=
( ) ( )2 Rec
f z dz i sidue of f z at z a= =
( )2 Rez a
i sf z=
=
Where ‘C’ is a closed curve containing the point z = a.
27
❖ Residue at a pole: The residue of f at z = z0 is defined as the coefficient of
( )1
0z z−
− which is 1a− in the Laurent’s series expansion of ( )f z and is denoted
by ( )0
Resz z
f z=
( ) ( )0
1Res
2z zC
f z f z dzi→
=
❖ Cauchys Residue theorem: If ( )f z is analytic within and on a closed curve C,
except at a finite number of poles 1 2 3, , ..... nz z z z within ‘C’ and 1 2, ,.... nR R R be
the residues of ( )f z at these poles, then
( ) ( )1 22 .... n
C
f z dz i R R R= + + + (or)
( ) ( )2 " "C
f z dz i Sumof theresidues at the poles within C=
Calculations of Residues
Type-I: If ( )f z has a simple pole at z = a,
then ( ) ( ) ( )Rez az a
s f z Lt z a f z→=
= −
Type-II: If ( )f z has a simple pole at z = a and ( )( )( )
zf z
z
=
where ( ) ( ) ( )z z a f z = −
Where ( ) 0f a then ( )( )( )
Rez a
as f z
a
==
Type-III: If ( )f z is analytic within a curve ‘C’ and has a pole of order ‘m’ at z = a
then the residue at z = a is ( )
( ) ( )1
1
1
1 !
mm
mz a
dLt z a f z
m dz
−
−→
− −
Evaluation of Real Definite integrals by Contour integration
The process of evaluating a definite integral by making the path of integration about a
suitable contour (curve) in the complex plane is called contour integration.
28
INTEGRATION ROUND THE UNIT CIRCLE
❖ Integrals of the type-I ( )2
0
cos ,sinF d
We consider the evaluation of the integrals of the type
( )2
0
cos ,sinF d
… (1)
where F is real, rational function of sin and cos .
1i iz e ez
−= =
idz ie d =
dz iz d=
dz
diz
=
But w.k.t.
1
cos2 2
i i ze e z
− +
+= = and
1
sin2 2
i i ze e z
i i
− −
−= =
and also 0 2 , we have travels on the entire unit circle and 1iz e = =
Eg. 1) ( )2
0
cos ,sinF d
= 1 1 1 1
,2 2
C
dzF z z
z i z iz
+ −
=
2 21 1,
2 2C
z z dzF
z iz iz
+ −
= ( )C
f z dz (say)
where C is the unit circle observe that ( )f z is a rational function
Now by Residue theorem
( ) ( )2 ' 'C
f z dz i Sumof theresiduesof f z at its poles inside C=
29
Integrals of the type-II ( )f x dx
−
Integration around semi-circles:
Here all the singularities of ( )f z are in the upper half of the z-plane.
( ) 2C
f z dz i R +=
( ) ( )R
R
f z dz f z dz−
+ …. (1)
Now as ( ) 0z f z → as z →
( ) 0f z dz
= …. (2)
As z → , R→ and along real axis z = x
From (1) & (2)
( ) 2f x dx i R
+
−
=
( ) ( )2 intf x dx i sumof residues at erior poles
−
=
Integrals of the Type-III ( )imxe f x dx
−
(Jordans Lemma)
If ( )f z is analytic except at finite number of singularities and if ( ) 0f z →
uniformly as z → , then
( ) ( )0 0
R
imz
RC
Lt e f z dz m→
=
where CR denotes the semi-circle z R=
The above theorem is useful in evaluating integrals of the form
( )( )
sinP x
mx dxQ x
−
( )( )
( )cos 0P x
mx dx mQ x
−
Where (i) ( )P x and ( )Q x are polynomials
(ii) degree of ( )Q x degree of ( )P x
(iii) ( ) 0Q x = has no real roots.
