055effdd17534b Applied Engineering Mathematics contents

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Contents

(ix)

Contents Preface ........................................................................................................................... (vii)

Part I

CHAPTER 1 Matrices ............................................................................................ 3 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Introduction ................................................................................................... 3 Definition of Matrix ....................................................................................... 3 Echelon Form of a Matrix ............................................................................ 20 Rank of a Matrix ........................................................................................... 20 Normal Form or Canonical Form ................................................................. 30 Simultaneous Linear Equations .................................................................... 43 Eigen Values and Eigen Vectors .................................................................. 70 Cayley-Hamilton Theorem .......................................................................... 94

CHAPTER 2 Complex Numbers....................................................................... 112 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13

Introduction ................................................................................................ 112 Complex Number ....................................................................................... 113 Equality of Complex Numbers ................................................................... 113 Algebra of Complex Numbers .................................................................... 114 Properties of Addition of Complex Numbers ............................................. 114 Properties of Multiplication of Complex Numbers .................................... 115 Modulus of a Complex Number ................................................................. 117 Conjugate of a Complex Number ............................................................... 117 Argument or Amplitude of a Complex Number ......................................... 118 Polar Form of a Complex Number ............................................................. 120 Properties of Modulii .................................................................................. 121 Properties of Conjugates............................................................................. 123 Properties of Argument .............................................................................. 125 (ix)

Contents

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2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21

Any Root of an Imaginary Number is an Imaginary Number .................... 126 Square Root of a Complex Number ........................................................... 126 Cube Roots of Unity ................................................................................... 127 De Moivre’s Theorem ................................................................................ 129 nth Roots of Unity ...................................................................................... 130 Properties of nth Roots of Unity ................................................................. 131 Argand Plane and Geometrical Representation of Complex Numbers ...... 131 Geometrical Representation of some Complex Numbers .......................... 132

2.22 2.23

Hyperbolic Functions ................................................................................. 153 Gregory’s Series ......................................................................................... 170

2.24

Summation of Series................................................................................... 181

CHAPTER 3 Theory of Equations.................................................................... 196 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14

Introduction ............................................................................................... 196 Theory of Equation ..................................................................................... 196 Establish Cardon’s Method of Solving Cubic Equation ............................. 198 Obtain the Reducing Cubic of the Biquadratic Equation ax4 + 4bx3 + 6cx2 + 4dx + e = 0 .............................................................. 199 Find out the Relation between the Roots of the Biquardratic and Euler’s Cubic .................................................................. 202 Establish a Relation between the Roots of the Biquadratic and the Reducing Cubic .......................................................... 203 State Descarte’s Rule of Signs.................................................................... 205 State and Prove Factor Theorem ................................................................ 205 The Imaginary Roots of an Equation with Real Coefficients always occur in Conjugate Pair ..................................... 206 Apply Descarte’s Rule of Sign to Discuss the Nature of the Roots of the Equation ........................................................... 207 Find the Minimum Number of Imaginary Roots which the Equation 2x7 – x4 + 4x3 – 5 = 0 must Possess............................................. 208 Empirical Law ............................................................................................ 236 Curve Fitting ............................................................................................... 236 Graphical Method ....................................................................................... 236

Contents

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3.15 3.16 3.17 3.18

Determination of other Empirical Laws Reducible to Linear Form ........... 236 Principle of Least Squares .......................................................................... 239 Method of Least Squares ............................................................................ 240 Change of Scale .......................................................................................... 244

Part II

CHAPTER 4 Successive Differentiation........................................................... 253 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11

Definitions and Notations ........................................................................... 253 Formulae for nth Derivative of some Standard Functions ........................... 269 Rules for finding nth Derivatives ................................................................ 279 Method of Determination of the nth Derivative of an Algebraic Rational Function....................................................................... 281 Some Important Results ............................................................................. 283 Power Series ............................................................................................... 302 Taylor’s (Infinite) Series [Elementary Treatment] ..................................... 302 Maclaurin’s Series [Elementary Discussion].............................................. 304 Expansions based on Algebraic and Trigonometrical Series ..................... 306 Differentiation and Integration of Power Series ......................................... 306 Working Rule for Selection of Method for Expansion ............................... 307

CHAPTER 5 Partial Differentiation................................................................. 322 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Functions of Two Real Variables ............................................................... 322 Partial Derivatives ...................................................................................... 322 Partial Derivatives of Second and Higher Orders....................................... 325 Total Differential ....................................................................................... 341 Total Differential Co-efficient .................................................................... 343 An Important Result .................................................................................. 344 Differentiation of Implicit Functions .......................................................... 344 Homogeneous Function .............................................................................. 345

5.9

Euler’s Theorem on Homogeneous Functions of Two Independent Variables ........................................................................ 346

Contents

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5.10

Euler’s Theorem on Homogeneous Function of Three Independent Variables ...................................................................... 348

5.11

Extension of Eluer’s Theorem in the Cases of Any Number of Independent Variables ...................................................... 351

5.12

To obtain the following Results if u be a Homogenous Function of Two Independent Variables x and y and of Degree n ................................ 351

5.13

Laplace’s Equation ..................................................................................... 353

5.14

An Interesting Observation......................................................................... 353

5.15

Jacobian ...................................................................................................... 354

5.16

Exact Differential ....................................................................................... 374

5.17

An Important Theorem ............................................................................... 375

5.18

Maxima and Minima of Functions of Two Variables ................................ 380 5.18.1

Lagrange’s Method of Undetermined Multipliers ...................... 384

CHAPTER 6 Reduction Formulae ................................................................... 394

CHAPTER 7 Curve Tracing.............................................................................. 433

CHAPTER 8 Areas of Plane Curves (Quadrature) ........................................ 454

CHAPTER 9 Lengths of Plane Curves (Rectification) ....................................... 496

CHAPTER 10 Volumes and Surfaces of Solids of Revolution ......................... 534

CHAPTER 11 Centroid (Centre of Gravity) and Moment of Inertia ............. 580

Contents

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CHAPTER 12 Beta and Gamma Functions ....................................................... 617

CHAPTER 13 Multiple Integrals ........................................................................ 634 13.1

13.2

Change of Order of Integration (Change of Variables) .............................. 643 13.1.1 Changing from Cartesion to Polar co-ordinates .......................... 643 13.1.2 Change of Order of Integration ................................................... 646 Triple Integration ........................................................................................ 650

CHAPTER 14 Vector Calculus ........................................................................... 661 14.1

14.2

Vector Differentiation................................................................................. 661 14.1.1 Vector Point Function and Vector Field ..................................... 661 14.1.2 Differentiation of a Vector .......................................................... 662 14.1.3 Application to Space Curves ....................................................... 663 Gradient of Scalar Function........................................................................ 664 14.2.1 The Vector differential Operator ‘DEL’ or ‘NABLA’, denoted as ‘  ’ is defined by.................................................................... 664 14.2.2 14.2.3 14.2.4 14.2.5

Gradient ....................................................................................... 664 Physical significance of ‘grad ’ ................................................ 664 Directional Derivative ................................................................. 665 Some basic properties of the Gradient ........................................ 665

14.3

The Divergence of a Vector Function ........................................................ 673 14.3.1 Definition .................................................................................... 673 14.3.2 Physical Significance of the Divergence ..................................... 673 14.3.3 Some properties of Divergence ................................................... 673 14.3.4 Solenoidal Vectors ...................................................................... 674

14.4

Curl of a Vector Function ........................................................................... 679 14.4.1 If A is a Differential Vector Function, then curl A is defined as, curl A =  × A ..................................................................... 679 14.4.2 Physical Significance of Curl ...................................................... 680 14.4.3 Irrotational Vector ....................................................................... 680

Contents

(xiv) 14.4.4 Properties .................................................................................... 680 14.4.5 Conservative Vector Field........................................................... 681

14.5

Laplacian Operator :2 ............................................................................... 689 14.5.1 Definition .................................................................................... 689 14.5.2 Vector Identities ........................................................................ 689 14.5.3 Operation of  on Product of Two Functions ............................ 692

14.6

Vector Integration ....................................................................................... 697 14.6.1 Ordinary Integration of Vectors .................................................. 697 14.6.2 Line Integrals .............................................................................. 698 14.6.3 Physical Applications .................................................................. 699 14.6.4 Theorem ...................................................................................... 699

14.7

Surface Integrals ......................................................................................... 714 14.7.1 Definition .................................................................................... 714 14.7.2 Definition of Surface Integral as the Limit of a Sum .................. 716 14.7.3 Evaluation of a Surface Integral .................................................. 716 14.7.4 Physical Interpretation of Surface Integrals ................................ 717

14.8

Volume Integrals ........................................................................................ 726 14.8.1 Expression of Volume Integral as the Limit of a Sum ................ 726

14.9

Green’s Theorem in the Plane .................................................................... 732 14.9.1 Green’s Theorem ...................................................................... 732 14.9.2 Vector Notation of Green’s Theorem ......................................... 733 14.9.3 Physical Interpretation of Green’s Theorem ............................... 734 14.9.4 Application of Green’s Theorem to the Evaluation of Area of a Simple Closed Curve ................................................... 734

14.10

Gauss Divergence Theorem........................................................................ 746

14.11

Stoke’s Theorem ......................................................................................... 759

Part III

CHAPTER 15 Fourier Series .............................................................................. 779 15.0

Introduction ............................................................................................... 779

Contents

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15.1

Periodic Functions ..................................................................................... 779 15.1.1 Even and Odd Functions ............................................................. 780

15.2

Fourier Series ............................................................................................. 784 15.2.1 Introduction ................................................................................. 784 15.2.2 Trigonometric Series .................................................................. 784 15.2.3 Theorem ..................................................................................... 784 15.2.4 Fourier Series .............................................................................. 787 15.2.5 Convergence of Fourier Series .................................................... 787 15.2.6 Dirichlet's Theorem ..................................................................... 787 15.2.7 Summary .................................................................................... 788

15.3

Change of Interval [Functions with Arbitrary Periods] ............................. 813 15.3.1 Euler's formulae for functions having a period '2l' where l is any Positive Number ................................................................... 813

15.4

Fourier Series of Even and Odd Functions ................................................ 826 15.4.1 Even functions ............................................................................ 826 15.4.2 Odd functions ............................................................................. 827

15.5

Half Range Series ...................................................................................... 836 15.5.1 Half Range Sine Series................................................................ 836 15.5.2 Half Range Cosine Series............................................................ 837

Part IV

CHAPTER 16 Differential Equations of First Order and their Applications 853 16.1

Introduction ................................................................................................ 853

16.2

Formation of an Ordinary D.E .................................................................... 855

16.3

The Differential Equations of the First Order and of the First Degree ....... 860 16.3.1 Separation of Variables ............................................................... 860 16.3.2 Homogeneous Equations............................................................. 863 16.3.3 Non-Homogeneous Differential Equations ................................ 868

16.4

Linear Differential Equations ..................................................................... 874 16.4.1 Non-linear Differential Equation of First Order ......................... 880

Contents

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16.5

Exact Differential Equations ..................................................................... 886 16.5.1 Integrating Factors ..................................................................... 888 16.5.2 First Method to find an Integrating Factor .................................. 889 16.5.3 Second Method to Find the Integrating Factor............................ 890 16.5.4 Third Method to Find the Integrating Factor .............................. 892 16.5.5 Fourth Method to Find the Integrating Factor ............................. 894 16.5.6 Fifth Method for Finding Integrating Factor ............................... 896

16.6

Equations of First Order but of Higher Degree .......................................... 899 16.6.1 Equations which can be solved for p........................................... 899 16.6.2 Equations Solvable for y ............................................................. 906 16.6.3 Equation Solvable for x ............................................................... 906 16.6.4 Clairaut’s Equation...................................................................... 916 16.6.5 Lagrange’s Equations .................................................................. 919 16.6.6 Equations reducible to Clairaut’s Form ...................................... 920

16.7

Applications to Geometry, Law of Natural Growth and Newton’s Law of Cooling .......................................................................... 922

16.8

Applications to Electrical Circuits .............................................................. 929

16.9

Applications to Heat Conduction (or Heat Flow) Problems ....................... 930

16.10

Applications to Problems of Newton’s Law of Cooling ............................ 932

16.11

Applications to Problems on Chemical Reactions...................................... 933

CHAPTER 17 Higher Order Linear Differential Equations and their Applications ........................................................................ 935 17.1

Linear Differential Equation ....................................................................... 935 17.1.1 Complementary Function (C.F) .................................................. 936

17.2

Particular Integral ....................................................................................... 942 17.2.1 Methods of finding P.I. ............................................................... 945

17.3

Cauchy’s Homogeneous Linear Equation .................................................. 962

17.4

Legendre’s Linear Equation ....................................................................... 967

17.5

Linear Differential Equations of Second Order-Method of Variation of Parameters .............................................................................. 970

Contents

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17.6

Applications ................................................................................................ 975 17.6.1 Bending of Beams ....................................................................... 975 17.6.2 Boundary Conditions .................................................................. 977 17.6.3 Electrical Circuits ........................................................................ 987 17.6.4 Simple Harmonic Motion ............................................................ 996

CHAPTER 18 Partial Differential Equations .................................................. 1004 18.1

Partial Differential Equations ................................................................... 1004 18.1.1 Introduction ............................................................................... 1004 18.1.2 Formation of a Partial Differential Equation by the Elimination of Arbitrary Constants .......................................... 1005

18.2.

Formation of the Partial Differential Equations by the Elimination of Arbitrary Functions .......................................................... 1008

18.3

First Order linear Partial Differential Equations ..................................... 1013 18.3.1 Lagrange's Equation .................................................................. 1013 18.3.2 Multipliers Method to solve the Lagrange's Partial Differential Equations ............................................................... 1018

18.4

Non-linear Partial Differential Equations of Order One........................... 1025 18.4.1 Definitions ................................................................................. 1025 18.4.2 Standard Form I : (Equations involving p and q only) .............. 1026 18.4.3 Standard Form II ....................................................................... 1030 18.4.4 Standard Form III ..................................................................... 1034 18.4.5 Standard Form –IV (CLAIRAUT’S TYPE) ............................ 1037

CHAPTER 19 Solution of Partial Differential Equations .............................. 1070 19.1

Partial Differential Equations – Applications ........................................... 1070 19.1.1 Some of the important partial differential equations which occur in the study of Engineering and Physical problems are ............................................................... 1070

19.2

One-dimensional Wave Equation ............................................................. 1074

Contents

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19.3

One Dimensional Heat Equation ............................................................. 1086

19.4

Heat Equation with Non-homogenous Boundary Conditions .................. 1099

19.5

Two Dimensional Heat Flow (Laplace Equation in Two Dimensions).... 1107

CHAPTER 20 Solution in Series ....................................................................... 1116 20.1

Ordinary and Singular Points of a Differential Equation ......................... 1116

20.2

Working Rule when x = 0 is a Singular Point .......................................... 1116

20.3

Solution of Differential Equation in a Series when x = 0 is an Ordinary Point. i.e., when P0 does not Vanish for x = 0 ........................... 1125

Part V

CHAPTER 21 Functions of a Complex Variable ............................................ 1131 21.1

Introduction .............................................................................................. 1131 21.1.1 Complex Numbers .................................................................... 1131 21.1.2 The Four Fundamental Operations of Complex Numbers ........ 1132 21.1.3 Graphical Representation of Complex Numbers ...................... 1132 21.1.4 Polar Form or Mod-Amp Form of Complex Numbers ............. 1133 21.1.5 Some More Properties of Moduli and Arguments ................... 1135 21.1.6 De Moivre’s Theorem ............................................................... 1136 21.1.7 Vector Representation of Complex Numbers .......................... 1136 21.1.8 Loci of ‘z’ ................................................................................. 1137 21.1.9 Point Sets [Regions in z – plane] .............................................. 1144 21.1.10 The Extended z – plane ............................................................ 1148

21.2

Functions, Limits and Continuity ............................................................ 1148 21.2.1 Variables and Functions ............................................................ 1148 21.2.2 Single and Multiple – valued Functions .................................. 1149 21.2.3 Inverse Functions ...................................................................... 1149 21.2.4 Transformations ....................................................................... 1149 21.2.5 Limits ........................................................................................ 1150

Contents

(xix) 21.2.6 Theorems on Limits (Without Proof) ........................................ 1151 21.2.7 Continuity ................................................................................. 1153 21.2.8 Theorems on Continuity (without proof) .................................. 1154

21.3

Differentiability – Analytic Functions ..................................................... 1157 21.3.1 Differentiability-Derivative....................................................... 1157 21.3.2 Differentiability  Continuity ................................................ 1158 21.3.3 Elementary Properties of Derivatives ....................................... 1158 21.3.4 Analytic Functions .................................................................... 1160 21.3.5 Cauchy – Riemann Equations [C – R Equations] .................... 1160 21.3.6 Cauchy – Riemann Equations in Polar Coordinates ................ 1163 21.3.7 Complex Form of C – R Equations ........................................... 1164 21.3.8 Harmonic Functions ................................................................. 1164 21.3.9 Harmonic Conjugates ................................................................ 1165 21.3.10 Singular Point (Singularity) ..................................................... 1165 21.3.11 Maximum – Modulus Theorem ................................................ 1165

21.4

Methods of Construction of an Analytic Function f(z) = u + iv ............... 1179 21.4.1 Method of finding v given u ...................................................... 1179 21.4.2 To Find u Given v ..................................................................... 1180 21.4.3 Milne – Thompson Method ....................................................... 1180 21.4.4 Given (u – v) or (u + v), to find f (z) ......................................... 1181 21.4.5 Application of Analytic Functions ............................................ 1181

21.5

Elementary Functions .............................................................................. 1197 21.5.1 Polynomial Functions................................................................ 1197 21.5.2 Rational Algebraic Functions ................................................... 1197 21.5.3 Exponential Function, ( e z ) ...................................................... 1197 21.5.4 Trigonometric Functions ........................................................... 1198 21.5.5 Hyperbolic Functions ................................................................ 1199 21.5.6 Logarthmic Function: (Inverse of Exponential Functions) ....... 1201 21.5.7 Inverse Trigonometric Functions .............................................. 1202 21.5.8 Inverse Hyperbolic Functions ................................................... 1203 21.5.9 The Exponential Functions, z a ............................................... 1204

Contents

(xx)

CHAPTER 22

Complex Integration ................................................................. 1206 22.1

Introduction of Line Integral .................................................................... 1206 22.1.1 Properties of Contour Integrals ................................................ 1207

22.2

Curves and Regions .................................................................................. 1215 22.2.1 Definitions ................................................................................. 1215 22.2.2 Cauchy’s Integral Theorem (Cauchy’s Theorem) ..................... 1216 22.2.3 Cauchy’s Theorem for Doubly Connected Regions ................. 1217 22.2.4 Cauchy’s Integral Formulae (C.I.F.) ......................................... 1218 22.2.5 Cauchy’s Integral Formula for Multiply Connected Region..... 1220

22.3

Complex Variable ..................................................................................... 1250 22.3.1 Absolute Convergence .............................................................. 1250 22.3.2 Uniform Convergence ............................................................... 1250 22.3.3 Power Series ............................................................................. 1250 22.3.4 Taylor’s Theorem ...................................................................... 1251 22.3.5 Laurent’s Theorem .................................................................... 1252

22.4

Zeros of an Analytic Function ........................................................ 1255 22.4.1 Singularity of a Function ........................................................... 1256

22.5

Contour Integration ................................................................................ 1258 22.5.1 Introduction – Residue of f(z) ................................................... 1258 22.5.2 Residue at a Pole of Order m .................................................... 1258 22.5.3 Residue at a Simple Pole (m = 1) ............................................. 1259 22.5.4 Evaluation of Integrals using Residues ................................... 1260 22.5.5 Cauchy’s Residue Theorem .................................................... 1261

22.6

Application of Residue Theorem to Evaluate Some Real Definite Integrals ..................................................... 1267 22.6.1 Type I: Integration around a Unit Circle .................................. 1267

CHAPTER 23

Fourier Transforms .................................................................. 1276 23.1

Integral Transforms .................................................................................. 1276 23.1.1 Definition of Integral Transform ............................................... 1276 23.1.2 Fourier Integral Theorem ......................................................... 1277 23.1.3 Fourier Sine and Cosine Integrals ............................................ 1279 23.1.4 Complex Form of Fourier Integrals ......................................... 1280

Contents

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23.2

Fourier Transforms ................................................................................... 1287 23.2.1 Definitions ................................................................................. 1287 23.2.2 Fourier Cosine Transforms ...................................................... 1288 23.2.3 Fourier Sine Transforms .......................................................... 1288 23.2.4 Properties of Fourier Transforms ............................................. 1290

23.3

Finite Fourier Transforms ........................................................................ 1304

23.4

Convolution Theorem ............................................................................... 1310 23.4.1 Convolution of Two Functions ................................................ 1310 23.4.2 Parseval's Identity .................................................................... 1311

23.5

Applications of Fourier Transforms in Initial and Boundary Value Problems ....................................................................... 1320

CHAPTER 24 Laplace Transforms .................................................................. 1338 24.1

Introduction .............................................................................................. 1338 24.1.1 Definition .................................................................................. 1338 24.1.2 Conditions for L.T to exist ........................................................ 1339 24.1.3 Linearity Property ..................................................................... 1339 24.1.4 First Shifting, Second Shifting and Change of Scale Properties ......................................................................... 1340 24.1.5 Laplace Transforms of Standard Functions .............................. 1341 24.1.6 Multiplication by “t” ................................................................. 1346 24.1.7 Division By ‘t’ .......................................................................... 1349 24.1.8 Laplace Transform of Derivatives ............................................ 1352 24.1.9 Laplace Transform of Integrals ................................................. 1354 24.1.10 Laplace Transforms of Periodic Functions ............................... 1357 24.1.11 Laplace Transform of the Unit Step Function (Heavisides Unit Function) ....................................................... 1364 24.1.12 Heaviside Shift Theorem........................................................... 1364 24.1.13 Dirac Delta Function (Unit Impulse Function) ......................... 1365

24.2

Inverse Laplace Transforms ..................................................................... 1368 24.2.1 Table of Inverse Transforms ..................................................... 1369 24.2.2 Method of Partial Fractions ....................................................... 1372 24.2.3 First Shifting Theorem .............................................................. 1376

Contents

(xxii) 24.2.4 24.2.5 24.2.6 24.2.7 24.2.8 24.2.9 24.2.10 24.2.11 24.2.12 24.2.13

Second Shifting Theorem .......................................................... 1379 Change of Scale Property .......................................................... 1383 Inverse Laplace Transform of Derivatives ................................ 1386 Inverse Laplace Transform of Integrals .................................... 1387 Multiplication by powers of s ................................................... 1387 Division by powers of ‘s’ .......................................................... 1387 Theorem ................................................................................... 1388 Convolution ............................................................................... 1389 Convolution Theorem (Convolution Property) ......................... 1390 Application of Laplace Transform to Solutions of Ordinary Differential Equations ............................................... 1412

CHAPTER 25 Z-Transforms and Applications .............................................. 1422 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 25.9 25.10 25.11 25.12 25.13 25.14 25.15 25.16 25.17 25.18 25.19 25.20

Introduction .............................................................................................. 1422 Z-transform ............................................................................................... 1422 Properties of Z-Transforms ...................................................................... 1423 Theorem.................................................................................................... 1429 Change of Scale ........................................................................................ 1429 Shifting Property ...................................................................................... 1430 Inverse Z-Transform ................................................................................. 1431 Solution of Difference Equations ............................................................. 1432 Multiplication by K .................................................................................. 1433 Division by K ........................................................................................... 1434 Initial Value .............................................................................................. 1434 Final Value ............................................................................................... 1434 Partial Sum ............................................................................................... 1435 Convolution .............................................................................................. 1436 Convolution Property of Casual Sequence ............................................... 1436 Transform of Important Sequences .......................................................... 1438 Final Value Theorem ................................................................................ 1440 By Binomial Expansion and Partial Fraction ........................................... 1444 Partial Fractions ........................................................................................ 1447 Inversion by Residue Method ................................................................... 1457

25.21

Solution of Difference Equation ............................................................... 1462