Mathematical Methods for Physicists: A concise introduction - Site Map

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SERIES REPRESENTATIONS OF ANALYTIC FUNCTIONS Series representations of analytic functions We now turn to a very important notion: series representations of analytic functions. As a prelude we must discuss the notion of convergence of complex series. Most of the de®nitions and theorems relating to in®nite series of real terms can be applied with little or no change to series whose terms are complex. Complex sequences A complex sequence is an ordered list which assigns to each positive integer n a complex number z n : z 1 ; z 2 ; ...; z n ; ...: The numbers z n are called the terms of the sequence. For example, both i; i 2 ; ...; i n ; ... or 1 ‡ i; …1 ‡ i†=2; …1 ‡ i†=4; …1 ‡ i†=8; ... are complex sequences. The nth term of the second sequence is (1 ‡ i†=2 n1 . A sequence z 1 ; z 2 ; ...; z n ; ... is said to be convergent with the limit l (or simply to converge to the number l) if, given ">0, we can ®nd a positive integer N such that jz n lj
FUNCTIONS OF A COMPLEX VARIABLE Solution: Given any number ">0, we must ®nd N such that 1 ‡ z n 1
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Mathematical Methods for Physicists
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PUBLISHED BY CAMBRIDGE UNIVERSITY P
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Contents Preface xv 1 Vector and te
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CONTENTS 3 Matrix algebra 100 De®n
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CONTENTS The adjoint operators 219
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CONTENTS Sturm±Liouville systems 3
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CONTENTS The Runge±Kutta method 47
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Preface This book evolved from a se
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VECTOR AND TENSOR ANALYSIS Figure 1
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VECTOR AND TENSOR ANALYSIS Vector a
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VECTOR AND TENSOR ANALYSIS We can g
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VECTOR AND TENSOR ANALYSIS Using th
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VECTOR AND TENSOR ANALYSIS The trip
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VECTOR AND TENSOR ANALYSIS will be
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VECTOR AND TENSOR ANALYSIS orthonor
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VECTOR AND TENSOR ANALYSIS A(u) is
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VECTOR AND TENSOR ANALYSIS a circle
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VECTOR AND TENSOR ANALYSIS The vect
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VECTOR AND TENSOR ANALYSIS operate
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VECTOR AND TENSOR ANALYSIS We ®rst
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VECTOR AND TENSOR ANALYSIS system w
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VECTOR AND TENSOR ANALYSIS we have
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VECTOR AND TENSOR ANALYSIS with sim
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VECTOR AND TENSOR ANALYSIS where A
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VECTOR AND TENSOR ANALYSIS Z P2 P 1
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VECTOR AND TENSOR ANALYSIS name is
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VECTOR AND TENSOR ANALYSIS while th
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VECTOR AND TENSOR ANALYSIS Next we
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VECTOR AND TENSOR ANALYSIS This imp
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VECTOR AND TENSOR ANALYSIS Proof:
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VECTOR AND TENSOR ANALYSIS Here we
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VECTOR AND TENSOR ANALYSIS (2) Addi
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VECTOR AND TENSOR ANALYSIS A genera
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VECTOR AND TENSOR ANALYSIS two poin
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VECTOR AND TENSOR ANALYSIS which sh
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2 Ordinary di€erential equations
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ORDINARY DIFFERENTIAL EQUATIONS The
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ORDINARY DIFFERENTIAL EQUATIONS Sol
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ORDINARY DIFFERENTIAL EQUATIONS It
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ORDINARY DIFFERENTIAL EQUATIONS whe
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ORDINARY DIFFERENTIAL EQUATIONS Gen
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ORDINARY DIFFERENTIAL EQUATIONS and
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ORDINARY DIFFERENTIAL EQUATIONS (1)
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ORDINARY DIFFERENTIAL EQUATIONS Now
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ORDINARY DIFFERENTIAL EQUATIONS Obv
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ORDINARY DIFFERENTIAL EQUATIONS Sol
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ORDINARY DIFFERENTIAL EQUATIONS Ord
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ORDINARY DIFFERENTIAL EQUATIONS Cas
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ORDINARY DIFFERENTIAL EQUATIONS Int
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ORDINARY DIFFERENTIAL EQUATIONS of
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ORDINARY DIFFERENTIAL EQUATIONS and
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ORDINARY DIFFERENTIAL EQUATIONS Fig
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3 Matrix algebra As vector methods
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MATRIX ALGEBRA Four basic algebra o
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MATRIX ALGEBRA then ~A ~B ˆ ˆ 2
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MATRIX ALGEBRA Figure 3.1. Coordina
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MATRIX ALGEBRA in®nite sum of matr
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MATRIX ALGEBRA The product of two s
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MATRIX ALGEBRA ~B ~ A ˆ ~I. Multip
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MATRIX ALGEBRA but so that ~A 1 …
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MATRIX ALGEBRA But if ~ A is an in
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MATRIX ALGEBRA Taking the dot produ
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MATRIX ALGEBRA change the handednes
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MATRIX ALGEBRA preserves the norm o
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MATRIX ALGEBRA Then ~A 0 R 0 ˆ…
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MATRIX ALGEBRA Once the eigenvalues
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MATRIX ALGEBRA Example 3.15 Show th
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0 1 x 11 x 1i x 1n x 21 x 2i x
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MATRIX ALGEBRA Since 0 1 1 B @ 1 0
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MATRIX ALGEBRA We now proceed to pr
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MATRIX ALGEBRA distance between any
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MATRIX ALGEBRA This matrix equation
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~A ~B ˆ a11 B ~ ! a 12 ~B ˆ a 21
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MATRIX ALGEBRA 3.18 If A ~ ~B ˆ 0,
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4 Fourier series and integrals Four
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FOURIER SERIES AND INTEGRALS Fourie
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FOURIER SERIES AND INTEGRALS Soluti
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FOURIER SERIES AND INTEGRALS Figure
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FOURIER SERIES AND INTEGRALS but b
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FOURIER SERIES AND INTEGRALS Parsev
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FOURIER SERIES AND INTEGRALS the A
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FOURIER SERIES AND INTEGRALS Orthog
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FOURIER SERIES AND INTEGRALS Substi
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FOURIER SERIES AND INTEGRALS and th
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FOURIER SERIES AND INTEGRALS We sho
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FOURIER SERIES AND INTEGRALS and th
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FOURIER SERIES AND INTEGRALS contin
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FOURIER SERIES AND INTEGRALS amount
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FOURIER SERIES AND INTEGRALS Using
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FOURIER SERIES AND INTEGRALS It is
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FOURIER SERIES AND INTEGRALS Parsev
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FOURIER SERIES AND INTEGRALS This i
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FOURIER SERIES AND INTEGRALS Soluti
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FOURIER SERIES AND INTEGRALS The de
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FOURIER SERIES AND INTEGRALS By the
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FOURIER SERIES AND INTEGRALS 4.10 F
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FOURIER SERIES AND INTEGRALS 4.22 V
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LINEAR VECTOR SPACES Figure 5.1. A
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LINEAR VECTOR SPACES We now proceed
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LINEAR VECTOR SPACES Linear combina
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LINEAR VECTOR SPACES is a basis for
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LINEAR VECTOR SPACES then Eq. (5.10
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LINEAR VECTOR SPACES Step 3. Clearl
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LINEAR VECTOR SPACES then hujvi ˆh
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SPECIAL FUNCTIONS OF MATHEMATICAL P
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THE CALCULUS OF VARIATIONS Figure 8
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THE CALCULUS OF VARIATIONS This is
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THE CALCULUS OF VARIATIONS p Lettin
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THE CALCULUS OF VARIATIONS provided
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THE CALCULUS OF VARIATIONS and Eq.
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THE CALCULUS OF VARIATIONS Example
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THE CALCULUS OF VARIATIONS must van
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THE CALCULUS OF VARIATIONS mechanic
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THE CALCULUS OF VARIATIONS Solution
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THE CALCULUS OF VARIATIONS The acti
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THE CALCULUS OF VARIATIONS is to ®
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THE CALCULUS OF VARIATIONS y…"; x
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9 The Laplace transformation The La
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THE LAPLACE TRANSFORMATION This est
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THE LAPLACE TRANSFORMATION or L‰x
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THE LAPLACE TRANSFORMATION Recall L
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THE LAPLACE TRANSFORMATION Let u ˆ
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THE LAPLACE TRANSFORMATION But f
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THE LAPLACE TRANSFORMATION but and
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THE LAPLACE TRANSFORMATION 9.6 Find
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PARTIAL DIFFERENTIAL EQUATIONS prob
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PARTIAL DIFFERENTIAL EQUATIONS If
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PARTIAL DIFFERENTIAL EQUATIONS In t
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PARTIAL DIFFERENTIAL EQUATIONS The
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PARTIAL DIFFERENTIAL EQUATIONS We a
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PARTIAL DIFFERENTIAL EQUATIONS Let
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PARTIAL DIFFERENTIAL EQUATIONS brin
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PARTIAL DIFFERENTIAL EQUATIONS We n
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PARTIAL DIFFERENTIAL EQUATIONS whic
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PARTIAL DIFFERENTIAL EQUATIONS to t
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PARTIAL DIFFERENTIAL EQUATIONS Subs
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PARTIAL DIFFERENTIAL EQUATIONS Now
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PARTIAL DIFFERENTIAL EQUATIONS Figu
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SIMPLE LINEAR INTEGRAL EQUATIONS f
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SIMPLE LINEAR INTEGRAL EQUATIONS Th
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SIMPLE LINEAR INTEGRAL EQUATIONS wh
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SIMPLE LINEAR INTEGRAL EQUATIONS in
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SIMPLE LINEAR INTEGRAL EQUATIONS A
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SIMPLE LINEAR INTEGRAL EQUATIONS no
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SIMPLE LINEAR INTEGRAL EQUATIONS Th
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SIMPLE LINEAR INTEGRAL EQUATIONS wh
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12 Elements of group theory Group t
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ELEMENTS OF GROUP THEORY The same s
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ELEMENTS OF GROUP THEORY Figure 12.
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ELEMENTS OF GROUP THEORY Table 12.3
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ELEMENTS OF GROUP THEORY Obviously
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ELEMENTS OF GROUP THEORY This can b
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ELEMENTS OF GROUP THEORY Group repr
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ELEMENTS OF GROUP THEORY (1) A matr
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ELEMENTS OF GROUP THEORY Equation (
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ELEMENTS OF GROUP THEORY Table 12.6
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ELEMENTS OF GROUP THEORY met. In qu
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ELEMENTS OF GROUP THEORY 0 1 1 0 0
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ELEMENTS OF GROUP THEORY Figure 12.
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ELEMENTS OF GROUP THEORY We can exp
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ELEMENTS OF GROUP THEORY Next, we a
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NUMERICAL METHODS is rather tedious
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NUMERICAL METHODS Figure 13.2. cour
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NUMERICAL METHODS and take g…x†
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NUMERICAL METHODS Solution: Here f
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NUMERICAL METHODS where y i ˆ f
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NUMERICAL METHODS The ®rst-order o
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NUMERICAL METHODS approximate value
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NUMERICAL METHODS Runge±Kutta meth
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NUMERICAL METHODS Equations of high
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NUMERICAL METHODS We now illustrate
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NUMERICAL METHODS 13.14. Find to th
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INTRODUCTION TO PROBABILITY THEORY
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Appendix 1 Preliminaries (review of
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APPENDIX 1 PRELIMINARIES Problem A1
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APPENDIX 1 PRELIMINARIES Example A1
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APPENDIX 1 PRELIMINARIES Since the
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APPENDIX 1 PRELIMINARIES Figure A1.
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APPENDIX 1 PRELIMINARIES How do we
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APPENDIX 1 PRELIMINARIES This follo
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APPENDIX 1 PRELIMINARIES Theorems o
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APPENDIX 1 PRELIMINARIES This is th
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APPENDIX 1 PRELIMINARIES (d) Find t
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APPENDIX 1 PRELIMINARIES Divide the
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APPENDIX 1 PRELIMINARIES Example A1
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APPENDIX 1 PRELIMINARIES Di€erent
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APPENDIX 1 PRELIMINARIES derivative
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Appendix 2 Determinants The determi
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APPENDIX 2 DETERMINANTS where a 11
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APPENDIX 2 DETERMINANTS D ˆ a i1 C
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APPENDIX 2 DETERMINANTS Example A2.
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APPENDIX 2 DETERMINANTS Proof: Expa
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Appendix 3 Table of * F…x† ˆp
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Index Abel's integral equation, 426
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INDEX group theory (contd) symmetry
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INDEX series solution of di€erent
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