Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Winter 2013
SUBSPACES
Definition: Let S be a nonempty subset of
a vector space V. S is said to be a
subspace of V, if S is a vector space
under the same operations of addition
and scalar multiplication as in V.
Theorem A nonempty subset S of a vector
space V is a sub
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Fall 2013
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, Pilani
Pilani Campus
Instruction Division
Course Handout (Part II)
Second Semester 201314
Dated: 13/01/2013
In addition to Part I (General Handout for all courses appended to the time table) this portion gives
f
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Winter 2013
Chapter 7: Evaluation of
Improper Integrals
Advice 1:
Page No. 257:
Q. Nos.: 1  5
(1) Let f(x) is continuous for all
x 0, then
R
f ( x)dx lim
f ( x) dx
0
R
0
provided the limit on RHS
exists.
2 Let f(x) is continuous for all x.
then
f ( x) dx

0
li
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Winter 2013
Eigen Values & Eigen Vectors
Let A be n x n matrix. is eigen value of A if there exists a nonzero vector n x R such that Ax = x x is called an eigen vector of A associated with
Eigen value / proper value / characteristic value / latent value eigen vector
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Winter 2013
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
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Winter 2013
Taylors Theorem: Suppose that
a function f(z) is analytic
throughout a disk z z0 R0
centered at z0 and with radius R0.
Then f(z) has the power series
representation
n
f z an z z0 ,
z z0 R0
n 0
where
an
f
n
z0
n!
(n 0,1, 2.)
Maclaurin Series
Taylor S
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Winter 2013
SpanofanysetS
Definition : Span of set S is set of all
finite linear combinations of elements of
S.
Thm:[S]isasubspaceofV
foranynonemptysetSofV.
RajivKumarMathII
Triviallinearcombination
Ifallscalars 1 , 2 ,., n arezeros,
thenlinearcombination
u u u
,.
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Winter 2013
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
wt u t i vt
dt
provided each of the derivatives
u & v exists at t
2 If z 0 is a complex const
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Winter 2013
Taylors Theorem: Suppose that
a function f(z) is analytic
throughout a disk z z0 R0
centered at z0 and with radius R0.
Then f(z) has the power series
representation
n
f z an z z0 ,
z z0 R0
n 0
where
an
f
n
z0
n!
(n 0,1, 2.)
Maclaurin Series
Taylor S
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Winter 2013
Section 25
Harmonic Function :
A real valued function u(x, y) is said to be
harmonic in a given domain D if
(i) u x , u xx , u y & u yy exist & they are
continuous in D,
(ii) u satisfies Laplace eqution
2
u u xx u yy 0
Theorem 1 :
If f(z) u(x, y) i v(x,
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Winter 2013
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
1
y = imaginary part of z = Im z
We usually write
z= (x, y) = x + i y,
where i =
1
= (0, 1)
i2 = i. i = (0, 1) . (0, 1) = ( 
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Winter 2013
Linear Transformations
Rajiv Kumar Math II
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
u , v U
(ii)T(u)= T(u)
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Winter 2013
System of Linear Equations
Rajiv Kumar Math II
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
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Winter 2013
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
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 112

Winter 2013
Department of Mathematics, UMIST
MATHEMATICAL FORMULA TABLES
Version 2.0
September 1999
CONTENTS
page
Greek Alphabet
3
Indices and Logarithms
3
Trigonometric Identities
4
Complex Numbers
6
Hyperbolic Identities
6
Series
7
Derivatives
9
Integrals
11
Laplac