Engineering Mathematics III

819 Pages·2021·8.57 MB·English

by  Dr. Manoj Kumar, Dr. S. B. Singh

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Engineering
Mathematics - III
for
B.E./B.Tech./B.Arch. students of first semester of all Engineering coleges affiliated
to U.P. Technical University, Lucknow
&
Uttrakhand Technical University, Dehradun
By
Dr. Manoj Kumar Dr. S. B. Singh
Assistant Professor Associate Professor
Department of Mathematics Deptt. of Maths, Stats. & Computer Science
R.K. College Shamli (Muzaffarnagar) G.B. Pant University of Agri. & Tech.
Pant Nagar (U.A.)
KRISHNA Prakashan Media (P) Ltd.
KRISHNA HOUSE, 11, Shivaji Road, Meerut-250 001 (U.P.), India
JAI SHRI RADHEY SHYAM
Dedicated
to
LORD
KRISHNA
Authors
and
Publishers
(v)
About the Authors
S.B. Singh
Received his Ph.D. in Mathematics from Institute of Advance Studies, C.C.S.
University, Meerut. Presently, he is Associate Professor in the Department of
Mathematics, Statistics and Computer Science, G. B. Pant University of Agriculture
and Technology, Pantnagar. He has around 15 years of teaching experience to
Undergraduate and Post Graduate students at different Engineering Colleges and
University. He is a member of Indian Mathematical Society, Operations Research
Society of India and National Society for Prevention of Blindness in India. He is a
regular reviewer of many books and some International Journals. He is Editor of the
Journal of Reliability and Statistical Studies. He has authored and coauthored eight
books on different courses of Applied/ Engineering Mathematics. He has been
conferred with four national awards. He has published his research works at
national and international journals of repute. His area of research is Reliability
Theory.
Manoj Kumar
received his Ph.D. in Cryptography from Dr. B. R. A. University Agra. Presently, he
is Assistant Professor in the Department of Mathematics, R. K. College Shamli-
Muzaffarnagar- U.P. He has around 10 years of teaching experience to
Undergraduate and Post Graduate students at different Engineering and degree
Colleges. He is a member of Indian Mathematical Society, Indian Society of
Mathematics and Mathematical Science, Ramanujan Mathematical society and
Cryptography Research Society of India. He is a regular reviewer of eight
International peer reviewed Journals. He is also working as a Technical Editor of
some International Journals: Asian Journal of Mathematics & Statistics, Asian
Journal of Algebra, Trends in Applied Sciences Research, Journal of Applied
Sciences. He has published his research works at national and international . Peer
review Journals. His area of research is cryptography.
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This book Engineering Mathematics - III has been written in conformity with the revised syllabus
for the third/ fourth semester students of the UPTU Lucknow. An ample part of the material in this book
has been rigorously class tested over the past several years. This book is basically the result of our
teaching experiences to the engineering and science students. The principal goal of this book is to
provide a study material to the student with a thorough knowledge of fundamental concepts and
methods of applied mathematics utilized in different engineering disciplines. A different aim of this text
book is to provide the students with the basic knowledge in the subject. We have given the definitions,
methods, theorems and observations, followed by typical problems and the step by step solution. Each
topic is covered in great detail, followed by several meticulously worked-out examples, and a problem
set containing a large number of additional related exercises including objective type problems. This
book aims at an exhaustive coverage of the syllabus and there is definitely an attempt to kindle the
student's creative ability. While preparing for the examination students should not restrict themselves
only to the questions / problems given in the self evaluation. They must be prepared to answer the
questions and problems from the entire text.
The authors feels rewarded for the labor and pains that they have been taken while writing this
book, if it is provides adequate study material covering all topics in details and enables the students to
secure better marks in the examination.
—Authors
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Authors feel great pleasure in presenting the revised edition of the book Engg. Maths-III. We would
like to thank all those who have used and adopted various editions of this book in past.
We are thankful to Prof. Sunder Lal, Pro V.C. of Dr. B. R. A. University, Agra. for his valuable
suggestions. Authors are particularly obliged to Dr. Jai Kishore, Mr. Vijayveer Singh, Dr. Anil Agarwal, Dr.
Ekata, Dr. Amit Awasthi, Dr. Atul Chaturvadi, Dr. Geetam Sharma, Dr. Vijayveer, Dr. Mukesh Rathor and
Miss Vandana Rathor and many more for their source of inspiration. This list is incomplete and we
apologize to those faculty members whose names have not been included here, as the list is very long.
Without mentioning the names, a particular gratitude is to the all family members who have
supported us at home. SB Singh expresses his appreciation to his wife Mrs. Neelam Singh for her
constant support and encouragement through out the time of revision of the book. Manoj Kumar
especially appreciate the ongoing interest and support of his wife Mrs. Chhaya, father in law Dr. Subey
Singh; mother in law Smt Sureshvati, brother in law Mr. Anurag Verma and his wife Mrs. Meenu.
Finally, authors owe a debt of appreciation to MD Mr. Satyendra Rastogi 'Mitra' for his constant
motivation and SMM Mr. M. R. Sharma of Krishna Prakashan Media Pvt. Ltd. Meerut for his enthusiasm
in the project. We are also thankful to the printing, publishing and editorial staff of Krishna Prakashan
Media (P) Ltd., for their patience with us.
At last, but not the least, we thank to everyone who has lent a helping hand to provide a shape to
our bricks, cement etc. in the preparation of this book.
We welcome valuable remarks and suggestions from students, teachers and academicians so that
this book may further be improved upon. To make this new edition as complete and user friendly as
possible, the reader's remarks will be appreciated always. Please contact us via electronic mail at the
address listed below for remarks, suggestions, questions, intimation of errors, misprints and feedback
regarding this book.
S.B. Singh
[email protected]
Manoj Kumar
[email protected]
(vii)
CONTENTS
Unit-1 Function of Complex Variables…………………….(1-176)
1. Analytical Functions……………………………………………………….3
2. Complex Integration……………………………………………………..43
3. Power Series and Calculus of Residues …………………………………75
Unit-2 Statistical Techniques …………………………...(177-298)
4. Statistics ………………………………………………………………...179
5. Curve Fitting ……………………………………………………………217
6. Correlation and Regression…………………………………………….239
7. Probability Theory ……………………………………………………..275
Unit-3 Statistical Techniques -II ………………………..(299-558)
8. Binomial Poisson and Normal Distribution…………………………….301
9. Sampling theory ………………………………………………………..347
10. Tests of Significations and its Applications …………………………….447
11. Time Series and Forecasting …………………………………………...471
12. Statistical Quality Control ……………………………………………...499
Unit-4 Number Techniques-I……………………………..(559-674)
13. Solutions of Algebraic and Transcendental Equation………………….561
14. Final Differences and Interpolation…………………………………….593
Unit-5 Numerical Techniques-II…………………………(675-774)
15. Solution of system of Linear Equation………………………………….677
16. Numerical differentiation and Integration………………………………701
17. Solution of differential Equation……………………………………….733
Appendix A Errors in the Numerical Computations………………………...(777-806)
Appendix B Question Bank……………………………………………………(807-820)
(viii)
Unit-1
1. Functions of Complex Variable Analytic Functions
2. Complex Integration
3. Power Series and Calculus of Residues
Unit-1
Chapter
F f C
unctions o omplex
1
A F
Variable– nalytic unctions
1.1 Introduction
Complex analysis is the branch of mathematics investigating functions of complex
numbers. It is enormously useful in many branches of mathematics, including
number theory and applied mathematics. The method of complex analysis are
widely applicable in treating and solving many engineering problems. Many of the
physical problems like heat conduction, fluid dynamics, electrostatics etc. are solved
with the help of complex analysis. The present chapter deals with the functions of a
complex variable and their properties.
1.2 Some Basic Definitions
Before starting the study of complex variable, let us introduce some basic terms
which are important in complex analysis :
(i) Complex plane : The xy-plane in which complex numbers z = x + iy are
represented are complex plane. Here x and y are called real and imaginary axis
respectively.
(ii) Point set : Any collection of points in the complex plane is called a point set
and each point is called an element of the set.
(iii) Circle : z -z =r represents a circle with centre at the point z and of radius r.
0 0
Open and closed disk : The set of points which satisfies the equation
z -z <r defines an open disk while the set of point which satisfies the
0
equation z -z £r is called a closed disk.
0
Neighborhood : The neighborhood of a point z consists of all points z lying
0
inside but not on the circle of radius d with centre at z , i.e. z -z <d.
0 0
A deleted neighorhood of a point z in the set of points in z -z <d except
0 0
the point z itself.
0
Annulus of z is given by r < z -z <r .
0 1 0 2
(iv) Interior point : A point z is said to be an interior point of a set S if there exist
0
some neighborhood of z which contains only points of S.
0
(v) Boundary point : If every neighborhood of z contains points not belonging
0
to set S then z is called boundary points of S.
0
4 Engineering Mathematics-III
(vi) Exterior Point - If a point is neither interior nor boundary, then it is called
exterior point of S.
(vii)Open and closed Set - If a point set contains only interior points then it is
called an open set. For example a set defined by z < r is an open set. If,
however, z £ r, then the set is called a closed set.
(viii)Bounded Set - A set S is said to be bounded if there exists a finite positive
number k such that z £ k for every point z belonging to S. If a set is not
bounded, then it is called unbounded set. For example, z -z < 5is a
0
bounded set whereas z -z > 5 is unbounded.
0
(ix) Connected Set - A set S is said to be connected if any two of its points are
joined by a polygonal line (a path which consist of finitely many straight line
segments), all of whose points belong to S. For example, the set z < 2 is a
connected set.
(x) Domain - An open connected set is called a domain. For example, the annulus
3< z < 4 is a domain. Thus any neighborhood is a domain.
Boundary of a domain is the collection of all boundary points of S. If boundary
is included to an open domain, then it is called a closed domain.
(xi) Region - A domain together with all, some or none of its boundary points is
called a region.
1.3 Some Basic Results
1. Polar Form of Complex Number
x = r cos q, y = r sin q, z = x + iy = r (cos q + i sin q)
2. Euler 's Theorem
(a) eiq = cos q + i sin q (b) e-iq= (cos q - i sin q) (c) z= reiq=r (cos q+ i sin q)
3. Exponential Function
(a) ez = ex+iy = ex. iiy = ex(cos y+ i sin y)
(b) eiq = cos2q+sin2q =1
(c) ez = ex+iy = ex . eiy = ex
(d) ez=1= enpaif and only if z =2np i, n is any integer.
(e) e2np i = cos 2np + i sin 2np = 1,
Thus, we can write ez = ez+2n pi, n is any integer.
4. Trigonometric Functions
eix – e-ix eix +e-ix
(a) sin x = (b) cos x =
2i 2
Functions of Complex Variable– Analytic Functions 5
5. Hyperbolic Functions
ex – e-x ex +e-x
(a) sinh x = (b) cosh x =
2 2
(c) cosh x + sinh x = ex (d) cosh x - sinh x = e–x
(e) cosh2 x - sinh2 x = 1
6. Trigonometric and hyperbolic function in terms of z=x+iy
1 1 1
(i) (a) sin z = [eiz – e–iz]= [ei(x+iy) – e-i(x+iy)] = [e-yeix – eye-ix]
2 2 2
1
= [e-y(cosx+isinx)-ey(cosx-isinx)]
2
= cos x cosh y + i sin x sinh y
1
(b) cos z = [ei(x+iy) +e-i(x+iy)] = cos x cosh y– i sin x sinh y
2
(c) sin (- z) = - sin z and cos (- z) = cos z
(d) sin z = sinz
1 1
(ii) (a) sin iz = [ei(iz) – e-i(iz)]= [e-z – ez]=isinh z
2i 2i
(b) cos iz = cosh z
(c) sinh z = - i sin iz = - i (sin{i(x+iy)}
= i [sin(y - ix)] = i [sin y cosh x - i cos y sinh x]
= sinh x cos y + i cosh x sin y
(d) cosh z = cosh x cos y + i sinh x sin y
7. Logarithmic Functions
(a) If z = reiq and w = u+iv in z = ew , then we get reiq = eu+iv
or rei(q+2np) = eu+iv = ew [sinceeiq = ei(q+2np)]
Thus ew = eu = r or u = lnr
Hence logarithm of real variable r = z and v = q+2np, n is any integer,
or w = ln z = ln r + i (q+2np), n is any integer .
(b) Ln z = ln z +iq, -p< q £ p (principal argument)
(c) Ln z = ln (x2+y2)1/2 + i tan -1 y/x
(d) Ln (1) = 0, ln (1) = 2np i.
(c) Ln (-1) = pi, ln (-1) = (2n+1) pi
6 Engineering Mathematics-III
1.4 Functions of a Complex Variable
If a symbol z takes any one of the values of a set of complex numbers, then z is called a
complex variable. If for each value of the complex variable z (=x+iy) in a given region
R, there exist one or more values of w (=u+iv) then w is said to be a function of z and
is written as w=u(x, y)+iv (x, y)= f(z) where u, v are real functions of x and y.
If to each value of z, there exists one and only one value of w, then w is said to be a
single valued function of z otherwise a multi-valued function; e.g., w=(1/z) is a
single valued function whereas w = z is a multi-valued function of z.
Limit - A number l is said to be the limit of a function w = f (z) as z tends to z and is
0
denoted by
lim w = lim f(z)= l
zfiz0 zfiz0
if for every e > 0, there exist a number d > 0 depending upon e such that
f(z)- l < e whenever 0 < z -z < d, z „ z .
0 0
Here z may approach z along any path, straight or curved since the two points
0
joining z and z in a complex plane can be joined by infinite number of curves.
0
When we say z fi z we mean xfi x and y fi y.
0 0 o
Illustrative Examples
z3 -8
Ex. 1 : Find the lim .
zfi2 z -2
Sol : Since z „ 2, we have z3 -23 = (z -2)(z2 +2z +4) = z2 +2z +4
z -2 z -2
Therefore, lim z2+2z+4 = 12
zfi2
Further, z2+2z+4 = (x + iy)2 +2 (x + iy) +4
= x2 + 2x - y2 + i(2y + 2xy) + 4
= (x2 + 2x - y2 + 4) +i(2y + 2xy)
Since z fi 2 implies x + iy fi 2 or x fi 2, y fi 0
Hence , lim z2 +2z +4= lim (x2 +2x- y2 +4)+i(2y +2xy) =12
xfi2 xfi2
yfi0 yfi0
Continuity of f (z). A single-valued function w= f (z) is said to be continuous at
z=z , if lim f(z) = f(z ) . f(z) is said to be continuous in any region R of the z-plane,
0 0
zfiz
0
if it is continuous at every point of the region R. When w = f (z) = u (x, y)+iv (x, y) is
continuous at z=z , then u (x, y) and v (x, y) are also continuous at z = z , i.e. at x =
0 0
x and y = y ; and conversely if u (x, y) and v (x, y) are continuous at (x , y ), then
0 0 0 0
f(z) will be continuous at z = z .
0