Complex Number: A complex
number z is an ordered pair (x, y),
where x & y are real nos. i.e.
z = (x, y)
x = real part of z = Re z
y = imaginary part of z = Im z
1
We usually write
z= (x, y) = x + i y,
where i =
i2
= (0, 1)
1
= i. i = (0, 1) . (0, 1) = ( -
Derivatives of functions w(t)
1 Let w t u t i v t
be a complex - valued function of a
real variable t, where u and v are
real - valued functions of t.
dw
Then
w t u t i v t
dt
provided each of the derivatives
u & v exists at t
2 If z 0 is a complex con
Sec 20 : Cauchy - Riemann Equations
Suppose that
f(z) u(x, y) iv(x, y)
and that f (z) exists at a point z 0 x0 iy 0
Then the first - order partial derivatives
u x , u y , v x and v y must exist at (x 0 , y0 )
and they satisfy the CR - equations
u x v y ,
BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, PILANI
II Semester 2005-2006, Math C 192 Mathematics II
Assignment sheet for Quiz II
Page 93 Q.1(a), (c)
Q 2 (b)
Page 94
Q.3 (a) , (d) Q.4 (e)
Page 104 Q.1 (a) , (d)
Q.2 (d), (f)
Q.1 (i) If U x1 , x2 , x3 / x1 x2 x
Maclaurin Series
Taylor Series about the point
z 0 0 is called Maclaurin
series, i. e.
f ( z )
n 0
f
(n)
( 0) n
z , ( z R0 )
n!
Examples:
n
1.
z
e , ( z )
n 0 n!
2.
z
sin z ( 1)
,
(2n 1)!
n 0
z
n
2 n 1
( z )
2n
z
3. cos z ( 1)
,
(2n)!
n 0
n
( z )
4.
2 n
Derivatives: Let f(z) be a fn defined on a
set S and S contains a nbd of z0. Then
derivative of f(z) at z0, written as
f ( z0 ),
is defined by the equation
lim f ( z ) f ( z0 )
f ( z0 )
,
z z0
z z0
provided the limit on RHS exists.
The function f(z) is s
Linear Transformations
Rajiv Kumar Math I
Definition : Let U and V be real
vector spaces. A map T: U V from
U to V is called a linear map, or
Linear transformation, if T satisfies
the following conditions
(i )T u v T u T v
for all
Rajiv Kumar Math I
u ,
Complex Number: A complex
number z is an ordered pair (x, y),
where x & y are real nos. i.e.
z = (x, y)
x = real part of z = Re z
y = imaginary part of z = Im z
1
We usually write
z= (x, y) = x + i y,
where i =
i2
= (0, 1)
1
= i. i = (0, 1) . (0, 1) = ( -
Function of a complex variable:
Let S be a set of complex numbers.
Then function f defined on S is a
rule that assigns to each z S a
unique complex number w, and we
write
f (z) = w
The set S is called domain of
definition of f.
Let z = x+ i y &
w = u (x,y
x
Q. If u ( x, y ) 2
,
find
a
2
x y
harmonic conjugate v of u.
Soln : Observe the following :
1
(i ) If f ( z ) , then u Re f ( z ).
z
(ii ) f ( z ) is analytic in a domain
D C - cfw_(0, 0).
y
(iii) Im f(z) v 2
.
2
x y
Conclude that v is a H.C. of u.
Chap
CHAPTER 11
Conic Sections and Polar
Coordinates
Section 11.3
Polar Coordinates
What are cartesian coordinates of a point in the
plane?
What are cartesian coordinates of a point in the
plane?
What are polar coordinates of a point in the plane?
Cartesian Co
BIRLA INSTITUTE OF TECHNOLGY & SCIENCE, PILANI
SECOND-SEMSETER 2012-2013
MATH F112: Mathematics-II
List of problems from the text book (Complex Variables and Applications; R.V Churchill and J.W
Brown, 8th edition) to be attempted by the students: Not for
LECTURE NO.-4
3.1 VECTOR SPACES
A (real) vector space
is a set V of objects
called vectors, together with a rule for adding
any two vectors u and v of V to produce a
vector u +v in V and a rule for multiplying
any vector u in V by any scalar in R to
produ
Derivatives of functions w(t)
1 Let w t u t i v t
be a complex - valued function of a
real variable t, where u and v are
real - valued functions of t.
dw
Then
w t u t i v t
dt
provided each of the derivatives
u & v exists at t
2 If z 0 is a complex con
Function of a complex variable:
Let S be a set of complex numbers.
Then function f defined on S is a
rule that assigns to each z S a
unique complex number w, and we
write
f (z) = w
The set S is called domain of
definition of f.
Let z = x+ i y &
w = u (x,y
BIRLA INSTITUTE OF TECHNOLGY & SCIENCE, PILANI
SECOND-SEMSETER 2013-2014
MATH F112: Mathematics-II
List of problems and theoretical exercises from the text
book: Elementary Linear Algebra by Andrilli and Hecker
Chapter 2:
Section Number
2.1
2.2
2.3
2.4
Pr
Derivatives: Let f(z) be a fn defined on a
set S and S contains a nbd of z0. Then
derivative of f(z) at z0, written as
f ( z0 ),
is defined by the equation
lim f ( z ) f ( z0 )
f ( z0 )
,
z z0
z z0
provided the limit on RHS exists.
The function f(z) is s
Maclaurin Series
Taylor Series about the point
z 0 0 is called Maclaurin
series, i. e.
f ( z )
n 0
f
(n)
( 0) n
z , ( z R0 )
n!
Examples:
1.
n
z
e , ( z )
n
!
n 0
z
2.
2 n 1
z
sin z ( 1)
,
(
2
n
1
)!
n 0
n
( z )
2n
z
3. cos z ( 1)
,
(
2
n
)!
n 0
n
( z )
Sec 20 : Cauchy - Riemann Equations
Suppose that
f(z) u(x, y) iv(x, y)
and that f (z) exists at a point z 0 x0 iy 0
Then the first - order partial derivatives
u x , u y , v x and v y must exist at (x 0 , y0 )
and they satisfy the CR - equations
u x v y ,
Rajiv Kumar
Determinant and their properties
(i) det (AT) = det (A)
(ii) det (C)= -det (A) if C is determinant of
matrix where two rows of A are
interchanged.
(iii) If two rows or two columns of A are
identical then det (A)=0
(iv) If D is obtained by mult
Sec 20 : Cauchy - Riemann Equations
Suppose that
f(z) u(x, y) iv(x, y)
and that f (z) exists at a point z 0 x0 iy 0
Then the first - order partial derivatives
u x , u y , v x and v y must exist at (x 0 , y0 )
and they satisfy the CR - equations
u x v y ,
Complex Number: A complex
number z is an ordered pair (x, y),
where x & y are real nos. i.e.
z = (x, y)
x = real part of z = Re z
y = imaginary part of z = Im z
1
We usually write
z= (x, y) = x + i y,
where i =
i2
= (0, 1)
1
= i. i = (0, 1) . (0, 1) = ( -
Analytic function
A function f(z) is said to be
analytic at a point z 0 if
(i) f(z) is differentiable at z 0
(ii ) f(z) is differentiable in some
neighbourhood of z 0 .
A function f(z) is analytic in an
open set S if f is differentiable at
each point of t
Derivatives of functions w(t)
1 Let w t u t i v t
be a complex - valued function of a
real variable t, where u and v are
real - valued functions of t.
dw
Then
w t u t i v t
dt
provided each of the derivatives
u & v exists at t
2 If z 0 is a complex con
Function of a complex variable:
Let S be a set of complex numbers.
Then function f defined on S is a
rule that assigns to each z S a
unique complex number w, and we
write
f (z) = w
The set S is called domain of
definition of f.
Let z = x+ i y &
w = u (x,y
Complex Number: A complex
number z is an ordered pair (x, y),
where x & y are real nos. i.e.
z = (x, y)
x = real part of z = Re z
y = imaginary part of z = Im z
1
We usually write
z= (x, y) = x + i y,
where i =
i2
= (0, 1)
1
= i. i = (0, 1) . (0, 1) = ( -
Matrices related to Linear
Transformation.
Definition: Let B = cfw_u1 ,u2,. un, be
an ordered basis for V.
Then a vector v V can be written as:
v = 1u1+ 2u2 +.+ nun.
The vector ( 1, 2,., n) is called the
coordinate vector of v relative to the
ordered basi