ECE Spring Prof. David R. Jackson ECE Dept. Notes 20
|
|
- Jerome Bond
- 5 years ago
- Views:
Transcription
1 ECE 6341 Sprig 016 Prof. David R. Jackso ECE Dept. Notes 0 1
2 Spherical Wave Fuctios Cosider solvig ψ + k ψ = 0 i spherical coordiates z φ θ r y x
3 Spherical Wave Fuctios (cot.) I spherical coordiates we have ψ = ψ ψ ψ ψ r siθ = + + r r r r siθ θ θ r si θ φ Hece we have r ψ ψ ψ siθ k 0 (1) + ψ + + = r r r r siθ θ θ r si θ φ Usig separatio of variables, let ( ) ( ) ψ = Rr ( ) H θ Φ φ () 3
4 Spherical Wave Fuctios (cot.) After substitutig Eq. () ito Eq. (1), divide by ψ : 1 R 1 H r siθ + rr r r rsiθh θ θ 1 1 Φ + + k = r si θ Φ φ 0 At this poit, we caot yet say that all of the depedece o ay give variable is oly withi oe ter. 4
5 Spherical Wave Fuctios (cot.) Next, ultiply by r si θ : si θ R siθ H r + siθ R r r H θ θ 1 Φ + + kr θ = Φ φ si 0 (3) Sice the uderlied ter is the oly oe which depeds o, It ust be equal to a costat, φ Hece, set 1 Φ φ Φ = (4) 5
6 Spherical Wave Fuctios (cot.) Hece, cos φ Φ ( φ ) = (5) si φ I geeral, w (ot a iteger). Now divide Eq. (3) by si θ ad use Eq. (4), to obtai 1 R 1 H r + siθ R r r Hsiθ θ θ si + kr = θ 0 (6) 6
7 Spherical Wave Fuctios (cot.) The uderlied ters are the oly oes that ivolve θ ow. This tie, the separatio costat is custoarily chose as ( 1) + 1 d siθ dh = ( + 1 ) (7) Hsiθ dθ dθ si θ I geeral, (ot a iteger) To siplify this, let x = cosθ dx = siθ dθ d d = siθ dθ dx ad deote yx ( ) = H( θ ) 7
8 Spherical Wave Fuctios (cot.) Usig x = cosθ d d siθ dθ = dx ( ) 1/ siθ = 1 x 1 d siθ dh = + 1 Hsiθ dθ dθ si θ ( ) y 1 ( 1 x ) 1/ ( ) 1/ d ( ) 1/ ( )( ) 1/ 1 x 1 x 1 1 x y dx + ( + 1) = 0 ( 1 x ) 8
9 Spherical Wave Fuctios (cot.) Cacelig ters, y 1 ( 1 x ) 1/ ( ) 1/ d ( ) 1/ ( )( ) 1/ 1 x 1 x 1 1 x y dx + ( + 1) = 0 ( 1 x ) Multiplyig by y, we have d dx ( ) 1 x y + ( + 1) y= 0 (8) ( 1 x ) 9
10 Spherical Wave Fuctios (cot.) Eq. (8) is the associated Legedre equatio. The solutios are represeted as yx ( ) P = Q ( x) ( x) Associated Legedre fuctio of the first kid. Associated Legedre fuctio of the secod kid. = order, = degree If = 0, Eq. (8) is called the Legedre equatio, i which case yx ( ) 0 P ( x) P( x) = = 0 Q ( ) Q ( x) x Legedre fuctio of the first kid. Legedre fuctio of the secod kid. 10
11 Spherical Wave Fuctios (cot.) Hece: H ( θ ) P = Q (cos θ ) (cos θ ) To be as geeral as possible: w H ( θ ) P = Q w w (cos θ ) (cos θ ) 11
12 Spherical Wave Fuctios (cot.) Substitutig Eq. (7) ito Eq. (6) ow yields 1 d R dr r + + kr = dr dr ( ) 1 0 Next, let x = kr dx d dr = = k dr d k dx ad deote yx () = Rr () 1
13 Spherical Wave Fuctios (cot.) We the have 1 d R dr r + + kr = dr dr ( ) 1 0 d x = kr = k dr d dx k k ( + ) + x = 1 d x dy y dx k dx 1 0 d ( xy ) + x ( + 1) y = 0 dx 13
14 Spherical Wave Fuctios (cot.) or d ( xy ) + x ( + 1) y = 0 dx + + ( + 1) = 0 x y xy x y spherical Bessel equatio Solutio: b (x) Note the lower case b. 14
15 Spherical Wave Fuctios (cot.) ad let Hece ( ) Deote = 1/ b x x g x ( ) yx ( ) = b x ( ) 1/ 1 3/ b ( x) = x g x g b ( x) = x g x g x g + x g 4 1/ 3/ 3/ 5/ 3 x g x g + x g + x g x g 4 ( ) 3/ 1/ 1/ 1/ 1/ ( ) 1/ + x + 1 x g( x) = 0 15
16 Spherical Wave Fuctios (cot.) Multiply by 1/ x 3 x g xg + g + ( xg g) + x ( + 1 ) g( x) = 0 4 Cobie these ters or 1 xg + xg g + xg ( + 1) g = 0 4 Use ( + 1) = + 4 Cobie these ters Defie γ
17 Spherical Wave Fuctios (cot.) ( ) We the have γ x g + xg + x g = This is Bessel s equatio of order γ. 0 Hece g Jγ ( x) = Yγ ( x) so that b 1 J+ 1/( x) π = x Y + 1/( x) added for coveiece 17
18 Spherical Wave Fuctios (cot.) Defie π j( x) J+ 1/( x) x π y( x) Y+ 1/( x) x The j ( ) () ( ) kr R r = b kr = y ( ) kr 18
19 Suary ( ) ( ) ψ + k ψ = 0 ( φ ) ( φ ) j kr P (cos ) cos θ ψ = y kr Q (cos ) si θ π b( x) = B+ 1/( x) x I geeral, w 19
20 Properties of Spherical Bessel Fuctios = π b( x) = B+ 1/( x) x Bessel fuctios of half-iteger order are give by closed-for expressios. J ( x) k + k ( 1) x! ( 1) k = 0 k! ( + k)! z =Γ z+ = This becoes a closed-for expressio! = + 1/ 0
21 Properties of Spherical Bessel Fuctios (cot.) Exaples: J1/( x) = si x π x ( ) J 1/( x) = cos x π x ( ) ( x) si J3/ ( x) = cos x π x x ( x) ( ) cos J 3/ ( x) = + si x π x x ( ) J( x)cos( π ) J ( x) Y ( x) si( π ) 1/ 1/ ( ) Y ( x) = J x 3/ 3/ ( ) Y ( x) = J x 1
22 Properties of Spherical Bessel Fuctios (cot.) Start with: k ( ) ( ) Proof for = 1/ J1/( x) = si x π x ( ) k ( ) ( ) + k 1/+ k 1 x 1 x J ( x) = J1/( x) = k= 0 k! + k! k= 0 k! 1/ + k! Hece k ( 1) ( k) x x J1/( x) = k = 0 k! 1/ +! k + 1
23 Properties of Spherical Bessel Fuctios (cot.) Exaie the factorial expressio: Note: x = x( x ) ( ) =! 1! 1/! π / 1 + 1/! = + 1/ 1/ 3/... 3/! ( k ) ( k )( k )( k ) ( ) ( k 1/)( k 1/)( k 3/ )...( 3/ )( π /) = + 1 = + = = = ( k 1)( k 1)( k 3 )...( 3 )( π / ) ( k 1 )!( π /) ( k)( k )( k ) 1 + k ( k 1 )!( π /) ( k)( k )( k ) 1 + k k! ( k 1 )!( π /) k k k k 3
24 Properties of Spherical Bessel Fuctios (cot.) Hece ( 1) x x J1/( x) = k k = 0 ( ) ( ) 1 k + 1! π / k! k k! k k + 1 Hece, we have k ( 1) ( k + ) π x x J1/( x) = k = 0 1! k + 1 k + 1 4
25 Properties of Spherical Bessel Fuctios (cot.) k ( 1) ( k + ) π x x J1/( x) = k = 0 1! k + 1 k + 1 or k = 0 k ( 1) ( k + ) π x J 1/( x ) = x 1! k + 1 We the recogize that π x J x x ( ) si 1/ = ( ) 5
26 Properties of Legedre Fuctios Relatio to Legedre fuctios (whe w = = iteger): ( ) / d P ( x) = 1 x P( x) dx ( ) / d Q ( x) = 1 x Q( x) dx These also hold for. For w (ot a iteger) the associated Legedre fuctio is defied i ters of the hypergeoetric fuctio. 6
27 Properties of Legedre Fuctios (cot.) Rodriguez s forula (for = ): 1 d P x x! dx ( ) ( = 1) Legedre polyoial (a polyoial of order ) ( ) ( ) 0 P0 x = x 1 = 1 1 d P1 ( x) = ( x 1) = x dx 1 d 1 P x = x 1 = 3x 1 8 dx ( ) ( ) ( ) 7
28 Properties of Legedre Fuctios (cot.) P x = > Note: ( ) 0, This follows fro these two relatios: ( ( ) 1 ) / d P x = x P( x) dx 1 d ( ) ( 1) P x = x = polyoial of order! dx 8
29 Properties of Legedre Fuctios (cot.) w=, = ( ( ) 1 ) / d Q x = x Q( x) dx Q ( ) x = ifiite series, ot a polyoial (ay blow up) ( 1) Q ± = (see ext slide) The Q fuctios all ted to ifiity as x ± 1 θ 0, π Recall : x = cosθ 9
30 Properties of Legedre Fuctios (cot.) Lowest-order Q fuctios: Q 0 ( x) 1 1+ x = l 1 x Q Q 1 ( x) ( x) x 1+ x = l 1 1 x 3 x l x = x 4 1 x 30
31 Properties of Legedre Fuctios (cot.) Negative idex idetities: ( ) ( ) P x P x ( + 1) = (This idetity also holds for.) ( ) π ( ) ( ) ( ) Q + x = P x + Q x ( 1) 1 31
32 Properties of Legedre Fuctios (cot.) P ( x) Q ( x) = (see Harrigto, Appedix E) ifiite series = ifiite series P ( x) ( ) ( ) ( ) ( ) ( ) ( ) ( ) N ( ) N 1 +! 1 x si π 1! +! 1 x = = 0!! π = + 1! N = largest iteger less tha or equal to. P P (1) = 1 ( 1) = Q ( x) ( ) cos( π ) ( ) ( π ) π P x P x = si Q ( ± 1) = Both are valid solutios, which are liearly idepedet for (see ext slide) 3
33 Properties of Legedre Fuctios (cot.) Proof that a valid solutio is P ( x) d dx The ( ) 1 x P ( x) + ( + 1) P ( x) 0 ( 1 x ) = Let t = x d ( 1 t )( 1 ) P ( t) + ( + 1) P ( t) 0 = dt ( 1 t ) d dx d = dt or (t x) d dx ( ) 1 x P ( x) + ( + 1) P ( x) 0 ( 1 x ) = Hece, a valid solutio is P ( x) 33
34 Properties of Legedre Fuctios (cot.) P ( x) ad P ( x) are two liearly idepedet solutios. Valid idepedet solutios: P Q (cos θ ) (cos θ ) or P P (cos θ ) ( cos θ ) We have a choice which set we wish to use. 34
35 Properties of Legedre Fuctios (cot.) = P ( x) = ( 1) P ( x) (They are liearly depedet.) I this case we ust use P Q (cos θ ) (cos θ ) 35
36 Properties of Legedre Fuctios (cot.) Suary of z-axis properties (x = cos (θ )) = P 1 = 1 P 1 = 1 ( ) ( ) Q ( ± 1) = Q ( ± 1) = P 1 = 1 P 1 = ( ) ( ) ( ) 36
37 Properties of Legedre Fuctios (cot.) z P ( ) x allowed P ( x) allowed y x x = cosθ Q ( ) ( ) x ad Q x are ot allowed o ± zaxis. 37
38 Properties of Legedre Fuctios (cot.) Outside or iside sphere z z x y Iside hollow coe y Oly P (x) is allowed x Both P (x) ad P (x) are allowed 38
39 Properties of Legedre Fuctios (cot.) z Outside coe x y Oly P (x) is allowed 39
40 Properties of Legedre Fuctios (cot.) z x y Oly P (x) is allowed Iside iverted coe Note: The physics is the sae as for the upright coe, but the atheatical for of the solutio is differet! 40
41 Properties of Legedre Fuctios (cot.) z y x Both P (x) ad P (x) are allowed Outside iverted coe 41
42 Properties of Legedre Fuctios (cot.) z Outside bicoe x y P (x) ad P (x) are allowed Q (x) ad Q (x) are allowed 4
ECE Spring Prof. David R. Jackson ECE Dept. Notes 8
ECE 6341 Sprig 16 Prof. David R. Jackso ECE Dept. Notes 8 1 Cylidrical Wave Fuctios Helmholtz equatio: ψ + k ψ = ψ ρφ,, z = A or F ( ) z z ψ 1 ψ 1 ψ ψ ρ ρ ρ ρ φ z + + + + k ψ = Separatio of variables:
More informationNotes 19 Bessel Functions
ECE 638 Fall 17 David R. Jackso Notes 19 Bessel Fuctios Notes are from D. R. Wilto, Dept. of ECE 1 Cylidrical Wave Fuctios Helmholtz equatio: ψ + k ψ = I cylidrical coordiates: ψ 1 ψ 1 ψ ψ ρ ρ ρ ρ φ z
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationBESSEL EQUATION and BESSEL FUNCTIONS
BESSEL EQUATION ad BESSEL FUNCTIONS Bessel s Equatio Summary of Bessel Fuctios d y dy y d + d + =. If is a iteger, the two idepedet solutios of Bessel s Equatio are J J, Bessel fuctio of the first kid,
More informationSOLUTION SET VI FOR FALL [(n + 2)(n + 1)a n+2 a n 1 ]x n = 0,
4. Series Solutios of Differetial Equatios:Special Fuctios 4.. Illustrative examples.. 5. Obtai the geeral solutio of each of the followig differetial equatios i terms of Maclauri series: d y (a dx = xy,
More informationA second look at separation of variables
A secod look at separatio of variables Assumed that u(x,t)x(x)t(t) Pluggig this ito the wave, diffusio, or Laplace equatio gave two ODEs The solutios to those ODEs subject to boudary ad iitial coditios
More informationNotes 8 Singularities
ECE 6382 Fall 27 David R. Jackso Notes 8 Sigularities Notes are from D. R. Wilto, Dept. of ECE Sigularity A poit s is a sigularity of the fuctio f () if the fuctio is ot aalytic at s. (The fuctio does
More informationPARTIAL DIFFERENTIAL EQUATIONS SEPARATION OF VARIABLES
Diola Bagayoko (0 PARTAL DFFERENTAL EQUATONS SEPARATON OF ARABLES. troductio As discussed i previous lectures, partial differetial equatios arise whe the depedet variale, i.e., the fuctio, varies with
More informationDe Moivre s Theorem - ALL
De Moivre s Theorem - ALL. Let x ad y be real umbers, ad be oe of the complex solutios of the equatio =. Evaluate: (a) + + ; (b) ( x + y)( x + y). [6]. (a) Sice is a complex umber which satisfies = 0,.
More informationRay Optics Theory and Mode Theory. Dr. Mohammad Faisal Dept. of EEE, BUET
Ray Optics Theory ad Mode Theory Dr. Mohammad Faisal Dept. of, BUT Optical Fiber WG For light to be trasmitted through fiber core, i.e., for total iteral reflectio i medium, > Ray Theory Trasmissio Ray
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationDifferent kinds of Mathematical Induction
Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}
More informationSummer MA Lesson 13 Section 1.6, Section 1.7 (part 1)
Suer MA 1500 Lesso 1 Sectio 1.6, Sectio 1.7 (part 1) I Solvig Polyoial Equatios Liear equatio ad quadratic equatios of 1 variable are specific types of polyoial equatios. Soe polyoial equatios of a higher
More informationx !1! + 1!2!
4 Euler-Maclauri Suatio Forula 4. Beroulli Nuber & Beroulli Polyoial 4.. Defiitio of Beroulli Nuber Beroulli ubers B (,,3,) are defied as coefficiets of the followig equatio. x e x - B x! 4.. Expreesio
More informationName Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions
Nae Period ALGEBRA II Chapter B ad A Notes Solvig Iequalities ad Absolute Value / Nubers ad Fuctios SECTION.6 Itroductio to Solvig Equatios Objectives: Write ad solve a liear equatio i oe variable. Solve
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More information1. Hydrogen Atom: 3p State
7633A QUANTUM MECHANICS I - solutio set - autum. Hydroge Atom: 3p State Let us assume that a hydroge atom is i a 3p state. Show that the radial part of its wave fuctio is r u 3(r) = 4 8 6 e r 3 r(6 r).
More informationChapter 9: Numerical Differentiation
178 Chapter 9: Numerical Differetiatio Numerical Differetiatio Formulatio of equatios for physical problems ofte ivolve derivatives (rate-of-chage quatities, such as velocity ad acceleratio). Numerical
More informationEigenvalues and Eigenvectors
5 Eigevalues ad Eigevectors 5.3 DIAGONALIZATION DIAGONALIZATION Example 1: Let. Fid a formula for A k, give that P 1 1 = 1 2 ad, where Solutio: The stadard formula for the iverse of a 2 2 matrix yields
More informationMa 530 Infinite Series I
Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li
More informationChapter Vectors
Chapter 4. Vectors fter readig this chapter you should be able to:. defie a vector. add ad subtract vectors. fid liear combiatios of vectors ad their relatioship to a set of equatios 4. explai what it
More informationUNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS 116C. Problem Set 4. Benjamin Stahl. November 6, 2014
UNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS 6C Problem Set 4 Bejami Stahl November 6, 4 BOAS, P. 63, PROBLEM.-5 The Laguerre differetial equatio, x y + ( xy + py =, will be solved
More informationPHYC - 505: Statistical Mechanics Homework Assignment 4 Solutions
PHYC - 55: Statistical Mechaics Homewor Assigmet 4 Solutios Due February 5, 14 1. Cosider a ifiite classical chai of idetical masses coupled by earest eighbor sprigs with idetical sprig costats. a Write
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationIntegrals of Functions of Several Variables
Itegrals of Fuctios of Several Variables We ofte resort to itegratios i order to deterie the exact value I of soe quatity which we are uable to evaluate by perforig a fiite uber of additio or ultiplicatio
More informationProblem 4: Evaluate ( k ) by negating (actually un-negating) its upper index. Binomial coefficient
Problem 4: Evaluate by egatig actually u-egatig its upper idex We ow that Biomial coefficiet r { where r is a real umber, is a iteger The above defiitio ca be recast i terms of factorials i the commo case
More informationThe Binomial Multi- Section Transformer
4/4/26 The Bioial Multisectio Matchig Trasforer /2 The Bioial Multi- Sectio Trasforer Recall that a ulti-sectio atchig etwork ca be described usig the theory of sall reflectios as: where: ( ω ) = + e +
More informationBernoulli Polynomials Talks given at LSBU, October and November 2015 Tony Forbes
Beroulli Polyoials Tals give at LSBU, October ad Noveber 5 Toy Forbes Beroulli Polyoials The Beroulli polyoials B (x) are defied by B (x), Thus B (x) B (x) ad B (x) x, B (x) x x + 6, B (x) dx,. () B 3
More informationJacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a
Jacobi sybols efiitio Let be a odd positive iteger If 1, the Jacobi sybol : Z C is the costat fuctio 1 1 If > 1, it has a decopositio ( as ) a product of (ot ecessarily distict) pries p 1 p r The Jacobi
More informationOrthogonal Functions
Royal Holloway Uiversity of odo Departet of Physics Orthogoal Fuctios Motivatio Aalogy with vectors You are probably failiar with the cocept of orthogoality fro vectors; two vectors are orthogoal whe they
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationThe Binomial Multi-Section Transformer
4/15/2010 The Bioial Multisectio Matchig Trasforer preset.doc 1/24 The Bioial Multi-Sectio Trasforer Recall that a ulti-sectio atchig etwork ca be described usig the theory of sall reflectios as: where:
More informationPHYSICS 116A Homework 2 Solutions
PHYSICS 6A Homework 2 Solutios I. [optioal] Boas, Ch., 6, Qu. 30 (proof of the ratio test). Just follow the hits. If ρ, the ratio of succcessive terms for is less tha, the hits show that the terms of the
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationTopic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or
Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad
More information5.6 Binomial Multi-section Matching Transformer
4/14/2010 5_6 Bioial Multisectio Matchig Trasforers 1/1 5.6 Bioial Multi-sectio Matchig Trasforer Readig Assiget: pp. 246-250 Oe way to axiize badwidth is to costruct a ultisectio Γ f that is axially flat.
More information5.6 Binomial Multi-section Matching Transformer
4/14/21 5_6 Bioial Multisectio Matchig Trasforers 1/1 5.6 Bioial Multi-sectio Matchig Trasforer Readig Assiget: pp. 246-25 Oe way to axiize badwidth is to costruct a ultisectio Γ f that is axially flat.
More informationAppendix F: Complex Numbers
Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write
More informationMIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More informationNotes 12 Asymptotic Series
ECE 6382 Fall 207 David R. Jackso otes 2 Asymptotic Series Asymptotic Series A asymptotic series (as ) is of the form a ( ) f as = 0 or f a + a a + + ( ) 2 0 2 ote the asymptotically equal to sig. The
More information42 Dependence and Bases
42 Depedece ad Bases The spa s(a) of a subset A i vector space V is a subspace of V. This spa ay be the whole vector space V (we say the A spas V). I this paragraph we study subsets A of V which spa V
More informationRADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify
Example 1 Simplify 1.2A Radical Operatios a) 4 2 b) 16 1 2 c) 16 d) 2 e) 8 1 f) 8 What is the relatioship betwee a, b, c? What is the relatioship betwee d, e, f? If x = a, the x = = th root theorems: RADICAL
More informationFor use only in [the name of your school] 2014 FP2 Note. FP2 Notes (Edexcel)
For use oly i [the ame of your school] 04 FP Note FP Notes (Edexcel) Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets For use oly i [the ame of your school] 04 FP Note BLANK PAGE Copyright
More informationChapter 1. Complex Numbers. Dr. Pulak Sahoo
Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler
More information6.4 Binomial Coefficients
64 Bioial Coefficiets Pascal s Forula Pascal s forula, aed after the seveteeth-cetury Frech atheaticia ad philosopher Blaise Pascal, is oe of the ost faous ad useful i cobiatorics (which is the foral ter
More informationJEE ADVANCED 2013 PAPER 1 MATHEMATICS
Oly Oe Optio Correct Type JEE ADVANCED 0 PAPER MATHEMATICS This sectio cotais TEN questios. Each has FOUR optios (A), (B), (C) ad (D) out of which ONLY ONE is correct.. The value of (A) 5 (C) 4 cot cot
More informationPresentation of complex number in Cartesian and polar coordinate system
a + bi, aεr, bεr i = z = a + bi a = Re(z), b = Im(z) give z = a + bi & w = c + di, a + bi = c + di a = c & b = d The complex cojugate of z = a + bi is z = a bi The sum of complex cojugates is real: z +
More informationLecture 2 Appendix B: Some sample problems from Boas, Chapter 1. Solution: We want to use the general expression for the form of a geometric series
Lecture Appedix B: ome sample problems from Boas, Chapter Here are some solutios to the sample problems assiged for Chapter, 6 ad 9 : 5 olutio: We wat to use the geeral expressio for the form of a geometric
More informationInverse Matrix. A meaning that matrix B is an inverse of matrix A.
Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix
More informationCreated by T. Madas SERIES. Created by T. Madas
SERIES SUMMATIONS BY STANDARD RESULTS Questio (**) Use stadard results o summatios to fid the value of 48 ( r )( 3r ). 36 FP-B, 66638 Questio (**+) Fid, i fully simplified factorized form, a expressio
More information3sin A 1 2sin B. 3π x is a solution. 1. If A and B are acute positive angles satisfying the equation 3sin A 2sin B 1 and 3sin 2A 2sin 2B 0, then A 2B
1. If A ad B are acute positive agles satisfyig the equatio 3si A si B 1 ad 3si A si B 0, the A B (a) (b) (c) (d) 6. 3 si A + si B = 1 3si A 1 si B 3 si A = cosb Also 3 si A si B = 0 si B = 3 si A Now,
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationIn algebra one spends much time finding common denominators and thus simplifying rational expressions. For example:
74 The Method of Partial Fractios I algebra oe speds much time fidig commo deomiators ad thus simplifyig ratioal epressios For eample: + + + 6 5 + = + = = + + + + + ( )( ) 5 It may the seem odd to be watig
More informationPhysics 219 Summary of linear response theory
1 Physics 219 Suary of liear respose theory I. INTRODUCTION We apply a sall perturbatio of stregth f(t) which is switched o gradually ( adiabatically ) fro t =, i.e. the aplitude of the perturbatio grows
More informationAVERAGE MARKS SCALING
TERTIARY INSTITUTIONS SERVICE CENTRE Level 1, 100 Royal Street East Perth, Wester Australia 6004 Telephoe (08) 9318 8000 Facsiile (08) 95 7050 http://wwwtisceduau/ 1 Itroductio AVERAGE MARKS SCALING I
More informationApplication 10.5D Spherical Harmonic Waves
Applicatio 10.5D Spherical Haroic Waves I probles ivolvig regios that ejoy spherical syetry about the origi i space, it is appropriate to use spherical coordiates. The 3-diesioal Laplacia for a fuctio
More informationX. Perturbation Theory
X. Perturbatio Theory I perturbatio theory, oe deals with a ailtoia that is coposed Ĥ that is typically exactly solvable of two pieces: a referece part ad a perturbatio ( Ĥ ) that is assued to be sall.
More informationMath 12 Final Exam, May 11, 2011 ANSWER KEY. 2sinh(2x) = lim. 1 x. lim e. x ln. = e. (x+1)(1) x(1) (x+1) 2. (2secθ) 5 2sec2 θ dθ.
Math Fial Exam, May, ANSWER KEY. [5 Poits] Evaluate each of the followig its. Please justify your aswers. Be clear if the it equals a value, + or, or Does Not Exist. coshx) a) L H x x+l x) sihx) x x L
More informationSEQUENCE AND SERIES NCERT
9. Overview By a sequece, we mea a arragemet of umbers i a defiite order accordig to some rule. We deote the terms of a sequece by a, a,..., etc., the subscript deotes the positio of the term. I view of
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationECE 901 Lecture 4: Estimation of Lipschitz smooth functions
ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig
More informationSolutions to Problem Set 8
8.78 Solutios to Problem Set 8. We ow that ( ) ( + x) x. Now we plug i x, ω, ω ad add the three equatios. If 3 the we ll get a cotributio of + ω + ω + ω + ω 0, whereas if 3 we ll get a cotributio of +
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More information8.3 Perturbation theory
8.3 Perturbatio theory Slides: Video 8.3.1 Costructig erturbatio theory Text referece: Quatu Mechaics for Scietists ad gieers Sectio 6.3 (u to First order erturbatio theory ) Perturbatio theory Costructig
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More informationEvaluation of Some Non-trivial Integrals from Finite Products and Sums
Turkish Joural of Aalysis umber Theory 6 Vol. o. 6 7-76 Available olie at http://pubs.sciepub.com/tjat//6/5 Sciece Educatio Publishig DOI:.69/tjat--6-5 Evaluatio of Some o-trivial Itegrals from Fiite Products
More informationIntroduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT
Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Itroductio to Extreme Value Theory Laures de Haa Erasmus Uiversity Rotterdam, NL Uiversity of Lisbo, PT Itroductio to Extreme Value Theory
More informationThe Non-homogeneous Diffusion Equation
The No-hoogeeous Diffusio Equatio The o-hoogeeous diffusio equatio, with sources, has the geeral for, 2 r,t a 2 r,t Fr,t t a 2 is real ad The hoogeeous diffusio equatio, 2 r,t a 2 t r,t ca be solved by
More informationSubject: Differential Equations & Mathematical Modeling-III
Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso
More information[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.
[ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural
More information18.01 Calculus Jason Starr Fall 2005
Lecture 18. October 5, 005 Homework. Problem Set 5 Part I: (c). Practice Problems. Course Reader: 3G 1, 3G, 3G 4, 3G 5. 1. Approximatig Riema itegrals. Ofte, there is o simpler expressio for the atiderivative
More informationSequences and Limits
Chapter Sequeces ad Limits Let { a } be a sequece of real or complex umbers A ecessary ad sufficiet coditio for the sequece to coverge is that for ay ɛ > 0 there exists a iteger N > 0 such that a p a q
More informationON SUPERSINGULAR ELLIPTIC CURVES AND HYPERGEOMETRIC FUNCTIONS
ON SUPERSINGULAR ELLIPTIC CURVES AND HYPERGEOMETRIC FUNCTIONS KEENAN MONKS Abstract The Legedre Family of ellitic curves has the remarkable roerty that both its eriods ad its suersigular locus have descritios
More informationMATH 6101 Fall 2008 Newton and Differential Equations
MATH 611 Fall 8 Newto ad Differetial Equatios A Differetial Equatio What is a differetial equatio? A differetial equatio is a equatio relatig the quatities x, y ad y' ad possibly higher derivatives of
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More informationNumerical Methods in Fourier Series Applications
Numerical Methods i Fourier Series Applicatios Recall that the basic relatios i usig the Trigoometric Fourier Series represetatio were give by f ( x) a o ( a x cos b x si ) () where the Fourier coefficiets
More informationIntroduction to Astrophysics Tutorial 2: Polytropic Models
Itroductio to Astrophysics Tutorial : Polytropic Models Iair Arcavi 1 Summary of the Equatios of Stellar Structure We have arrived at a set of dieretial equatios which ca be used to describe the structure
More information+ {JEE Advace 03} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks: 00. If A (α, β) = (a) A( α, β) = A( α, β) (c) Adj (A ( α, β)) = Sol : We
More informationRandom Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More informationLinearly Independent Sets, Bases. Review. Remarks. A set of vectors,,, in a vector space is said to be linearly independent if the vector equation
Liearly Idepedet Sets Bases p p c c p Review { v v vp} A set of vectors i a vector space is said to be liearly idepedet if the vector equatio cv + c v + + c has oly the trivial solutio = = { v v vp} The
More information( ) (( ) ) ANSWERS TO EXERCISES IN APPENDIX B. Section B.1 VECTORS AND SETS. Exercise B.1-1: Convex sets. are convex, , hence. and. (a) Let.
Joh Riley 8 Jue 03 ANSWERS TO EXERCISES IN APPENDIX B Sectio B VECTORS AND SETS Exercise B-: Covex sets (a) Let 0 x, x X, X, hece 0 x, x X ad 0 x, x X Sice X ad X are covex, x X ad x X The x X X, which
More informationMath 4707 Spring 2018 (Darij Grinberg): midterm 2 page 1. Math 4707 Spring 2018 (Darij Grinberg): midterm 2 with solutions [preliminary version]
Math 4707 Sprig 08 Darij Griberg: idter page Math 4707 Sprig 08 Darij Griberg: idter with solutios [preliiary versio] Cotets 0.. Coutig first-eve tuples......................... 3 0.. Coutig legal paths
More informationSolutions to Final Exam Review Problems
. Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the
More informationMa/CS 6a Class 22: Power Series
Ma/CS 6a Class 22: Power Series By Ada Sheffer Power Series Mooial: ax i. Polyoial: a 0 + a 1 x + a 2 x 2 + + a x. Power series: A x = a 0 + a 1 x + a 2 x 2 + Also called foral power series, because we
More informationSEQUENCES AND SERIES
9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationBertrand s postulate Chapter 2
Bertrad s postulate Chapter We have see that the sequece of prie ubers, 3, 5, 7,... is ifiite. To see that the size of its gaps is ot bouded, let N := 3 5 p deote the product of all prie ubers that are
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationTHE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART II Calculators are NOT permitted Time allowed: 2 hours
THE 06-07 KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART II Calculators are NOT permitted Time allowed: hours Let x, y, ad A all be positive itegers with x y a) Prove that there are
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationExpected Number of Level Crossings of Legendre Polynomials
Expected Number of Level Crossigs of Legedre olomials ROUT, LMNAYAK, SMOHANTY, SATTANAIK,NC OJHA,DRKMISHRA Research Scholar, G DEARTMENT OF MATHAMATICS,COLLEGE OF ENGINEERING AND TECHNOLOGY,BHUBANESWAR,ODISHA
More informationThe natural exponential function
The atural expoetial fuctio Attila Máté Brookly College of the City Uiversity of New York December, 205 Cotets The atural expoetial fuctio for real x. Beroulli s iequality.....................................2
More informationEnumerative & Asymptotic Combinatorics
C50 Eumerative & Asymptotic Combiatorics Notes 4 Sprig 2003 Much of the eumerative combiatorics of sets ad fuctios ca be geeralised i a maer which, at first sight, seems a bit umotivated I this chapter,
More informationMath 475, Problem Set #12: Answers
Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe
More informationChapter 2 The Solution of Numerical Algebraic and Transcendental Equations
Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said
More information