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PARSEVAL'S IDENTITY FOR FOURIER INTEGRALS Figure 4.21. A damped sine wave. Its Fourier transform g…!† is g…!† ˆp 1 2 Z 1 1 Z 1 ˆ p 1 2 ˆ 1 p 2 2 1 f …t†e i!t dt e t=T e i!t sin ! 0 tdt 1 !‡! 0 i=T 1 ! ! 0 i=T If f …t† is a radiated electric ®eld, the radiated power is proportional to j f …t†j 2 and the total energy radiated is proportional to R 1 0 jf …t† j 2 dt. This is equal to R 1 0 jg…!† j 2 d! by Parseval's identity. Then jg…!†j 2 must be the energy radiated per unit frequency interval. Parseval's identity can be used to evaluate some de®nite integrals. As an example, let us revisit Example 4.8, where the given function is ( f …x† ˆ 1 jxj < a 0 jxj > a and its Fourier transform is By Parseval's identity, we have Z 1 1 r 2 sin !a g…!† ˆ ! : ff …x† g 2 dx ˆ 187 Z 1 1 fg…!† g 2 d!: :
FOURIER SERIES AND INTEGRALS This is equivalent to from which we ®nd Z a a …1† 2 dx ˆ Z 1 0 2 Z 1 1 2 sin 2 !a ! 2 d!; sin 2 !a ! 2 d! ˆ a 2 : The convolution theorem for Fourier transforms The convolution of the functions f …x† and H…x†, denoted by f H, is de®ned by f H ˆ Z 1 1 f …u†H…x u†du: …4:55† If g…!† and G…!† are Fourier transforms of f …x† and H…x† respectively, we can show that Z 1 1 Z 1 g…!†G…!†e i!x d! ˆ f …u†H…x u†du: …4:56† 2 1 1 This is known as the convolution theorem for Fourier transforms. It means that the Fourier transform of the product g…!†G…!†, the left hand side of Eq. (55), is the convolution of the original function. The proof is not dicult. We have, by de®nition of the Fourier transform, Then g…!† ˆp 1 2 Z 1 1 f …x†e i!x dx; G…!† ˆp 1 2 Z 1 1 g…!†G…!† ˆ 1 Z 1 Z 1 f …x†H…x 0 †e i!…x‡x 0† dxdx 0 : 2 1 1 H…x 0 †e i!x 0 dx 0 : …4:57† Let x ‡ x 0 ˆ u in the double integral of Eq. (4.57) and we wish to transform from (x, x 0 )to(x; u). We thus have dxdx 0 ˆ @…x; x 0 † @…x; u† dudx; where the Jacobian of the transformation is @x @x @…x; x 0 † @…x; u† ˆ @x @u @x 0 @x 0 ˆ 1 0 0 1 ˆ 1: @x @u 188
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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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Page 157 and 158: MATRIX ALGEBRA 3.18 If A ~ ~B ˆ 0,
Page 159 and 160: 4 Fourier series and integrals Four
Page 161 and 162: FOURIER SERIES AND INTEGRALS Fourie
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Page 215 and 216: LINEAR VECTOR SPACES Figure 5.1. A
Page 217 and 218: LINEAR VECTOR SPACES We now proceed
Page 219 and 220: LINEAR VECTOR SPACES Linear combina
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Page 249 and 250: FUNCTIONS OF A COMPLEX VARIABLE and
Page 251 and 252: FUNCTIONS OF A COMPLEX VARIABLE We
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FUNCTIONS OF A COMPLEX VARIABLE Fig
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FUNCTIONS OF A COMPLEX VARIABLE ver
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FUNCTIONS OF A COMPLEX VARIABLE z 0
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FUNCTIONS OF A COMPLEX VARIABLE If
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FUNCTIONS OF A COMPLEX VARIABLE How
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FUNCTIONS OF A COMPLEX VARIABLE Sim
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FUNCTIONS OF A COMPLEX VARIABLE Di
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FUNCTIONS OF A COMPLEX VARIABLE fun
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FUNCTIONS OF A COMPLEX VARIABLE Sin
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FUNCTIONS OF A COMPLEX VARIABLE pro
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FUNCTIONS OF A COMPLEX VARIABLE Alt
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FUNCTIONS OF A COMPLEX VARIABLE Exa
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FUNCTIONS OF A COMPLEX VARIABLE Exa
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FUNCTIONS OF A COMPLEX VARIABLE For
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FUNCTIONS OF A COMPLEX VARIABLE Sol
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FUNCTIONS OF A COMPLEX VARIABLE Rat
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FUNCTIONS OF A COMPLEX VARIABLE Not
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FUNCTIONS OF A COMPLEX VARIABLE The
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FUNCTIONS OF A COMPLEX VARIABLE (b)
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FUNCTIONS OF A COMPLEX VARIABLE to
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FUNCTIONS OF A COMPLEX VARIABLE Sol
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FUNCTIONS OF A COMPLEX VARIABLE the
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FUNCTIONS OF A COMPLEX VARIABLE Thu
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FUNCTIONS OF A COMPLEX VARIABLE If
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FUNCTIONS OF A COMPLEX VARIABLE the
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FUNCTIONS OF A COMPLEX VARIABLE onl
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FUNCTIONS OF A COMPLEX VARIABLE To
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FUNCTIONS OF A COMPLEX VARIABLE Tak
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FUNCTIONS OF A COMPLEX VARIABLE 6.2
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7 Special functions of mathematical
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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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