RESIDUES AND POLES
Dr. Anil Kumar
CC205
Anil Kumar, BITS Goa Campus
2
Residues and Poles
The CauchyGoursat theorem states that if a function is
y
analytic at all points inside and on a simple closed contour
C, then the value of the integral of f(z) alo
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
K K BIRLA GOA CAMPUS
Second SEMESTER, 201415
Course no.: MATH F112
Course Title: Mathematics II
Problem for Section 2.3
1(a), 1(c), 2(a), 2(b),
3(a), 3(b), 4,
5(a),5(c),5(e), 6(a)
7. (a), (b), (c), (d),
8
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
K K BIRLA GOA CAMPUS
Second SEMESTER, 201415
Course no.: MATH F112
Course Title: Mathematics II
Problem for Section 2.4
1. (b),
2. (a), (c), (e),
3. (a), (c),
4(e), 5(c), 6,
7. (a), (c),
9(a), (b), (c),
1
Birla Institute of Technology and Science, PilaniK K Birla Goa Campus
Second Semester 20142015
MATHEMATICSII
(MATH F112)
Tutorial Sheet
CauchyRiemann (CR) Equations
1. Use CauchyRiemann (CR) equations to show that f (z) does not exist at any point if
CauchyRiemann (CR) Equations
MATHEMATICSII (MATH F112)
Dr. P. DHANUMJAYA
Department of Mathematics
BITSPilani K K Birla Goa Campus
Review of Complex Variables
Dr. P. Dhanumjaya
CauchyRiemann (CR) Equations
Supose that f (z) = u(x, y) + iv (x, y) and t
MATHEMATICSII (MATH F112)
Harmonic Functions
Dr. P. DHANUMJAYA
BITSPilani, Goa Campus
Dr. P. Dhanumjaya
Harmonic Functions
BITS PilaniK K Birla Goa Campus
Denition
A realvalued function H(x, y ) of real variables is said to be
harmonic in a domain D i
MATHEMATICSII (MATH F112)
Elementary Functions
Dr. P. DHANUMJAYA
BITSPilani, Goa Campus
Dr. P. Dhanumjaya
Elementary Functions
BITS PilaniK K Birla Goa Campus
Exponential Function
Dr. P. Dhanumjaya
Elementary Functions
BITS PilaniK K Birla Goa Campus
Lecture1
MATHEMATICSII (MATH F112)
Dr. P. DHANUMJAYA
Department of Mathematics
BITSPilani K K Birla Goa Campus
Review of Complex Variables
Dr. P. Dhanumjaya
Complex Variables
Complex numbers can be written in either Cartesian form of
polar form.
A comp
BIRLA INSTITUE OF TECHNOLOGY AND SCIENCE PILANI K.K. BIRLA GOA CAMPUS
Lab Week 7
Objectives
1. Arrays (random access lists) : One Dimensional and Multidimensional
2. Searching: Linear Search and Binary Search
3. Sorting: Selection and Bubble Sort
4. Exe
Birla Institute of Technology & Science, Pilani , K. K. BIRLA Goa Campus
Computer Programming (CS F111)
Second Semester 20142015
Lab2 (VI Editor)
Objectives:
1. Introduction
2. Creating your first file using VI Editor
3. Basic Operating Modes
4. Comman
Birla Institute of Technology & Science, Pilani
Computer Programming (CSF111)
Second Semester 20132014
Lab6
 Objectives
1. Iterative constructs
 Iterative Constructs (Loops)
There may be a situation when you need to execute a block of code several n
BIRLA INSTITUE OF TECHNOLOGY AND SCIENCE PILANI
K.K. BIRLA GOA CAMPUS
Second Semester 20142015
Lab3
Week #3 [January 27, 2015 TO January Feb 02, 2015]
Objectives:
1.
2.
3.
4.
I/O Redirection
Filters
Piping
Introduction to Unix Shell and Shell Script

BIRLA INSTITUE OF TECHNOLOGY AND SCIENCE PILANI
K.K. BIRLA GOA CAMPUS
Second Semester 20142015
Lab8
Lab Sheet on Functions
Objectives
1. Modular Programming: Functions in C
2. Passing Arguments to Functions
3. Passing Arrays to Functions
4. Exercises
BIRLA INSTITUE OF TECHNOLOGY AND SCIENCE PILANI
K.K. BIRLA GOA CAMPUS
Second Semester 20142015
Lab5
Lab5 (Introduction to C Programming Language)
Topics to be covered:
1. C Program structure and execution
2. Data type
3. Reading input and printing out
BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, K.K.BIRLAGOA CAMPUS
COMPUTER PROGRAMMING I
SEMESTER 20142015
STRINGS

Objectives:
1. Understanding what are strings.
2. Understanding two dimensional array of characters ( 2D Strings).
Strings
The way a group of
Birla Institute of Technology & Science  Pilani, K. K. Birla Goa Campus
Computer Programming (CS F111)
Second Semester 20132014
Lab11
Topics
Dynamic Memory Allocation and Structure
General Instructions:
o First create a directory with your full ID e.
Birla Institute of Technology & Science, Pilani
First Semester 20142015, Computer Programming [CS F111]
Lab #1
Week #1
OBJECTIVES
Introduction (Computing Machine, Operating System, UNIX)
Getting started with UNIX
UNIX treats everything as Files!
Print
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
K K BIRLA GOA CAMPUS
Second SEMESTER, 201415
Course no.: MATH F112
Course Title: Mathematics II
Problem for Section 2.2
1. (a), (b), (c), (d), (e), (f),
2. (a), (b), (c), (e),
3,4(a),4(c),5(c),6(a),7(a),
Span of Vector Space
Tarkeshar Singh
Department of Mathematics
BITS Pilani KK Birla Goa Campus, Goa
12th Feb. 2015
Linear Combination
Denition
A vector w V is called a linear combination of vectors
v1 , v2 , . . . , vr if it can be written as
r
w=
ki vi .
MathematicsII (Math F 112)
Dr. Amit Setia, (cc211)
Assistant Professor
Department of Mathematics
1
Lecture (23.01.2015)
2
Solving a System Using the Inverse of the Coefficient Matrix
Theorem 2.15
Let AX = B represent a system
where the coefficient matri
APPLICATIONS OF RESIDUES
Dr. Anil Kumar
CC205
Anil Kumar, BITS Goa Campus
2
Evaluation of improper integrals
Some important applications of the theory of residues will be
p
pp
y
discussed here.
The applications include evaluation of certain types of de
Basis of Vector Space
Tarkeshar Singh
Department of Mathematics
BITS Pilani KK Birla Goa Campus, Goa
18th Feb. 2015
Basis
Denition
Let V be a vector space and B be a subset of V . Then B is called
a basis of V if
B is linearly independent.
span(B) = V .
T
MathematicsII (Math F 112)
Dr. Amit Setia, (cc211)
Assistant Professor
Department of Mathematics
1
Text Books:
1) Elementary Linear Algebra
by S. Andrilli & D. Hecker
2) Complex Variables and applications
by R.V. Churchill & J.W. Brown
2
Instructors:
Th
MathematicsII (Math F 112)
Dr. Amit Setia, (cc211)
Assistant Professor
Department of Mathematics
1
Lecture (17.01.2015)
2
Example(Type III operation is also required)
Using Gaussian elimination method,
solve the following linear system of equations
3 x
MathematicsII (Math F 112)
Dr. Amit Setia, (cc211)
Assistant Professor
Department of Mathematics
1
Lecture 19.01.2015
2
Definition. Equivalent Systems
Two systems of m linear equations in n variables
are equivalent
they have exactly the same solution s
MathematicsII (Math F 112)
Dr. Amit Setia, (cc211)
Assistant Professor
Department of Mathematics
1
Theorem 2.8:
If A and B are two m n row equivalent matrices,
then the row spaces of A and B are equal.
proof:
If A and B are row equivalent,
then the rows
SERIES
Dr. Anil Kumar
CC205
Anil Kumar, BITS Goa Campus
2
Power Series
Given a sequence cfw_an of complex numbers, the series
a (z z )
n
0
n
= a0 + a1 ( z z0 ) + a2 ( z z0 ) 2 + an ( z z0 ) n +
n =0
is called the power series about the point z0.
In par
MathematicsII (Math F 112)
Dr. Amit Setia, (cc211)
Assistant Professor
Department of Mathematics
1
Exercise:
2
Let V=R
where addition and scalar multiplications are
defined as
( x1 , y1 ) ( x2 , y2 ) = ( x1 + x2 , y1 + y2 ) &
a ( x, y ) = ( ax, 0 )
Does
MathematicsII (Math F 112)
Dr. Amit Setia, (cc211)
Assistant Professor
Department of Mathematics
1
4.1 Finite dimensional
vector space
2
Definition:
A vector space is a set V together with 2
operations called vector addition &
scalar multiplication on w