# k and v = v 1 j + u 3 i + v 2

Size: px
Start display at page:

Download "k and v = v 1 j + u 3 i + v 2"

Transcription

1 ORTHOGONAL FUNCTIONS AND FOURIER SERIES Orthogonl functions A function cn e considered to e generliztion of vector. Thus the vector concets like the inner roduct nd orthogonlity of vectors cn e extended to functions. Inner roduct Consider the vectors u = u 1 i + u 2 j + u 3 k nd v = v 1 i + v 2 j + v 3 k in R 3, then the inner roduct or dot roduct of u nd v is rel numer, sclr, defined s (u,v) = u 1 v 1 + u 2 v 2 + u 2 v 2 = 3 u k v k k=1

2 The inner roduct ossesses the following roerties (u,v) = u 1 v 1 + u 2 v 2 + u 2 v 2 = 3 u k v k k=1 (i) (u,v) = (v, u) (ii) (ku, v) = k(u, v) (iii) (u, u) = 0 if u = 0 nd (u, u) > 0 if u 0 (iv) (u + v, w) = (u, w) + (v, w)

3 Suose tht f 1 nd f 2 re iecewise continuous functions defined on n intervl [, ]. The definite integrl on the intervl of the roduct f 1 (x) f 2 (x) ossesses roerties (i) - (iv) ove. Definition: Inner roduct of functions The inner roduct of two functions f 1 nd f 2 on n intervl [, ] is the numer ( f 1, f 2 ) = f 1 (x) f 2 (x) dx

4 Definition: Orthogonl functions Two functions f 1 nd f 2 re sid to e orthogonl on n intervl [, ] if ( f 1, f 2 ) = f 1 (x) f 2 (x) dx = 0 Exmle: f 1 (x) = x 2 nd f 2 (x) = x 3 re orthogonl on the intervl [ 1, 1] since ( f 1, f 2 ) = 1 1 x 2. x 3 dx = x6 1 = 0

5 Orthogonl sets We re rimrily interested in n infinite sets of orthogonl functions. Definition: Orthogonl set A set of rel-vlued functions {φ 0 (x), φ 1 (x), φ 2 (x),...} is sid to e orthogonl on n intervl [, ] if (φ m, φ n ) = φ m (x) φ n (x) dx = 0, m n

6 Orthonorml sets The norm u of vector u cn e exressed using the inner roduct: (u, u) = u 2 u = (u, u) Similrly the squre norm of function φ n is φ n 2 = (φ n, φ n ), nd so the norm is φ n = (φ n, φ n ). In other words, the squre norm nd the norm of function φ n in n orthogonl set {φ n (x)} re, resectively, φ n 2 = φ 2 n(x) dx nd φ n = φ 2 n(x) dx

7 Exmle 1: Orthogonl set of functions Show tht the set {1, cos x, cos 2x,...} is orthogonl on the intervl [, ]: φ 0 = 1, φ n = cos nx (φ 0, φ n ) = = 1 sin nx n φ 0 (x) φ n (x) dx = nd for m n, using the trigonometric identity (φ m, φ n ) = φ m (x) φ n (x) dx = cos nx dx = 1 [sin n sin( n)] = 0 n cos mxcos nx dx = 1 [cos(m + n)x + cos(m n)x] dx 2 = 1 sin(m + n)x sin(m n)x + = 0 2 m + n m n

8 Exmle 2: Norms Find the norms of the functions given in the Exmle 1 ove. φ 0 2 = dx = 2 φ 0 = 2 φ n 2 = cos 2 nx dx = 1 [1 + cos 2nx] dx = 2 φ n = Any orthogonl set of nonzero functions {φ n (x)}, n = 0, 1, 2... cn e normlized, i.e. mde into n orthonorml set. Exmle: An orthonorml set on the intervl [, ]: 1, cos x cos 2x,,... 2

9 Vector nlogy Suose v 1, v 2, nd v 3 re three mutully orthogonl nonzero vectors in R 3. Such n orthogonl set cn e used s sis for R 3, tht is, ny three-dimensionl vector cn e written s liner comintion u = c 1 v 1 + c 2 v 2 + c 3 v 3 where c i, i = 1, 2, 3 re sclrs clled the comonents of the vector. Ech comonent cn e exressed in terms of u nd the corresonding vector v i : (u,v 1 ) = c 1 (v 1,v 1 ) + c 2 (v 2,v 1 ) + c 3 (v 3,v 1 ) = c 1 v c c 3.0 (u,v 2 ) = c 2 v 2 2 (u,v 3 ) = c 3 v 3 2

10 Hence nd c 1 = (u,v 1) v 1 2 c 2 = (u,v 2) v 2 2 c 3 = (u,v 3) v 3 2 u = (u,v 1) v 1 2 v 1 + (u,v 2) v 2 2 v 2 + (u,v 3) v 3 2 v 3 = 3 n=1 (u,v n ) v n 2 v n

11 Orthogonl series exnsion Suose {φ n (x)} is n infinite orthogonl set of functions on n intervl [, ]. If y = f (x) is function defined on the intervl [, ], we cn determine set of coefficients c n, n = 0, 1, 2,... for which f (x) = c 0 φ 0 (x) + c 1 φ 1 (x) + c 2 φ 2 (x) c n φ n (x) +... (1) using the inner roduct. Multilying the exression ove y φ m (x) nd integrting over the intervl [, ] gives f (x) φ m (x) dx = = c 0 φ 0 (x) φ m (x) dx + c 1 = c 0 (φ 0, φ m ) + c 1 (φ 1, φ m ) c n (φ n, φ m ) +... φ 1 (x) φ m (x) dx c n φ n (x) φ m (x) dx +...

12 By orthogonlity, ech term on r.h.s. is zero excet when m = n, in which cse we hve f (x) φ n (x) dx = c n φ2 n(x) dx The required coefficients re then In other words f (x) = n=0 c n = c n φ n (x) = n=0 f (x) φ n (x) dx, n = 0, 1, 2... φ2 n(x) dx f (x) φ n (x) dx φ n (x) 2 φ n (x) = n=0 ( f, φ n ) φ n (x) 2 φ n(x)

13 Definition: Orthogonl set / weight function A set of rel-vlued functions {φ 0 (x), φ 1 (x), φ 2 (x),...} is sid to e orthogonl with resect to weight function w(x) on n intervl [, ] if w(x) φ m (x) φ n (x) dx = 0, m n The usul ssumtion is tht w(x) > 0 on the intervl of orthogonlity [, ]. For exmle, the set {1, cos x, cos 2x,...} is orthogonl w.r.t. the weight function w(x) = 1 on the intervl [, ].

14 If {φ n (x)} is orthogonl w.r.t. weight function w(x) on the intervl [, ], them multilying the exnsion (1), f (x) = c 0 φ 0 (x) + c 1 φ 1 (x)..., yw(x) nd integrting y rts yields c n = f (x) w(x) φ n (x) dx φ n (x) 2 where φ n (x) 2 = w(x) φ 2 n(x) dx

15 The series f (x) = n=0 c n φ n (x) (2) with the coefficients given either y c n = f (x) φ n (x) dx φ n (x) 2 or c n = f (x) w(x) φ n (x) dx φ n (x) 2 (3) is sid to e n orthogonl series exnsion of f or generlized Fourier series. Comlete sets We shll ssume tht n orthogonl set {φ n (x)} is comlete. Under this ssumtion f cn not e orthogonl to ech φ n of the orthogonl set.

16 Fourier series Trigonometric series The set of functions 1, cos x, cos 2 x, cos 3 x,..., sin x, sin 2 x, sin 3 x,... is orthogonl on the intervl [, ]. We cn exnd function f defined on [, ] into the trigonometric series f (x) = n cos n x + n sin n x n=1 (4)

17 Determining the ocefficients 0, 1, 2,..., 1, 2,... : We multily y 1 (the first function in our orthogonl set) nd integrte oth sides of the exnsion (4) from to f (x) dx = 0 dx + n cos n 2 xdx+ n sin n xdx n=1 Since cos(nx/) nd sin(nx/), n 1, re orthogonl to 1 on the intervl, the r.h.s. reduces s follows Solving for 0 yields f (x) dx = = 1 dx = 0 2 x = 0 f (x) dx (5)

18 Now, we multily (4) y cos(mx/) nd integrte f (x) cos m xdx = n=1 By orthogonlity, we hve cos m xdx n cos m x cos n xdx+ n cos m xdx= 0, m > 0 cos m x sin n xdx= 0 cos m x cos n xdx= δ mn where the Kronecker delt δ mn = 0 if m n, nd δ mn = 1 if m = n. cos m x sin n xdx

19 Thus the eqution (6) ove reduces to f (x) cos n xdx= n nd so n = 1 f (x) cos n xdx (6)

20 Finlly, multilying (4) y sin(mx/), integrting nd using the orthogonlity reltions we find tht n = 1 sin m xdx= 0, m > 0 sin m x cos n xdx= 0 sin m x sin n xdx= δ mn f (x) sin n xdx (7) The trigonometric series (4) with coefficients 0, n, nd n defined y (5), (6) nd (7), resectively re sid to e the Fourier series of the function f. The coefficients otined from (5), (6) nd (7) re referred s Fourier coefficients of f.

21 Definition: Fourier series The Fourier series of function f defined on the intervl (, ) is given y f (x) = n cos n x + n sin n x where n=1 0 = 1 n = 1 n = 1 (8) f (x) dx (9) f (x) cos n xdx (10) f (x) sin n xdx (11)

22 Exmle 1: Exnsion in Fourier series f (x) = With = we hve from (9) nd (10) tht 0 = 1 f (x) dx = dx + 0 n = 1 f (x) cos nx dx = 1 = 1 sin nx ( x) + 1 n 0 n 0 = 1 cos nx n n 0 cos n + 1 = n 2 = 1 ( 1)n n 2 0, < x < 0 x, 0 x < ( x) dx = 1 x x dx + ( x) cos nx dx 0 sin nx dx 0 = 2

23 Similrly, we find from (11) n = 1 n 0 ( x) sin nx dx = 1 n The function f (x) is thus exnded s f (x) = 1 ( 1) n 4 + n=1 n 2 cos nx + 1 n sin nx (12) We lso note tht 1 ( 1) n = 0, n even 2, n odd.

24 Convergence of Fourier series Theorem: Conditions for convergence Let f nd f e iecewise continuous on the intervl (, ); tht is, let f nd f e continuous excet t finite numer of oints in the intervl nd hve only finite discontinuities t these oints. Then the Fourier series of f on the intervl converges to f (x) t oint of continuity. At oint of discontinuity, the Fourier series converges to the verge f (x+) + f (x ), 2 where f (x+) nd f (x ) denote the limit of f t x from the right nd from the left, resectively.

25 Exmle 2: Convergence of oint of discontinuity The exnsion (12) of the function (Exmle 1) 0, < x < 0 f (x) = x, 0 x < will converge to f (x) for every x from the intervl (, ) excet t x = 0 where it will converge to f (0+) + f (0 ) 2 = = 2.

26 Periodic extension Oserve tht ech of the functions in the sis set 1, cos x, cos 2 x, cos 3 x,..., sin x, sin 2 x, sin 3 x,... hs different fundmentl eriod 2/n, n 1, ut since ositive integer multile of eriod is lso eriod, we see tht ll the functions hve in common the eriod 2. Thus the r.h.s. of f (x) = n cos n x + n sin n x n=1 is 2-eriodic; indeed 2 is the fundmentl eriod of the sum. We conclude tht Fourier series not only reresents the function on the intervl (, ) ut lso gives the eriodic extension of f outside this intervl.

27 We cn now ly the Theorem on conditions for convergence to the eriodic extension or simly ssume the function is eriodic, f (x + T) = f (x), with eriod T = 2 from the outset. When f is iecewise continuous nd the right- nd left-hnd derivtives exist t x = nd x =, resectively, then the Fourier series converges to the verge [ f ( )+ f (+)]/2 t these oints nd lso to this vlue extended eriodiclly to ±3, ±5, ±7, nd so on. Exmle: The Fourier series of the function f (x) in the Exmle 1 converges to the eriodic extension of the function on the entire x-xis. At 0, ±2, ±4,..., nd t ±, ±3, ±5,..., the series converges to the vlues f (0+) + f (0 ) 2 = 2 nd f (+) + f ( ) 2 = 0

28 Sequence of rtil sums It is interesting to see how the sequence of rtil sums {S N (x)} of Fourier series roximtes function. For exmle S 1 (x) = 4, S 2(x) = cos x + sin x, S 3(x) = cos x + sin x + 1 sin 2x 2

### DEFINITION The inner product of two functions f 1 and f 2 on an interval [a, b] is the number. ( f 1, f 2 ) b DEFINITION 11.1.

398 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 11.1 ORTHOGONAL FUNCTIONS REVIEW MATERIAL The notions of generlized vectors nd vector spces cn e found in ny liner lger text. INTRODUCTION The concepts

### (9) P (x)u + Q(x)u + R(x)u =0

STURM-LIOUVILLE THEORY 7 2. Second order liner ordinry differentil equtions 2.1. Recll some sic results. A second order liner ordinry differentil eqution (ODE) hs the form (9) P (x)u + Q(x)u + R(x)u =0

### Abstract inner product spaces

WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the

### MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further

### ENGI 9420 Lecture Notes 7 - Fourier Series Page 7.01

ENGI 940 ecture Notes 7 - Fourier Series Pge 7.0 7. Fourier Series nd Fourier Trnsforms Fourier series hve multiple purposes, including the provision of series solutions to some liner prtil differentil

### Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner

### Green function and Eigenfunctions

Green function nd Eigenfunctions Let L e regulr Sturm-Liouville opertor on n intervl (, ) together with regulr oundry conditions. We denote y, φ ( n, x ) the eigenvlues nd corresponding normlized eigenfunctions

### McGill University Math 354: Honors Analysis 3 Fall 2012 Solutions to selected Exercises. g(x) 2 dx 1 2 a

McGill University Mth 354: Honors Anlysis 3 Fll 2012 Assignment 1 Solutions to selected Exercises Exercise 1. (i) Verify the identity for ny two sets of comlex numers { 1,..., n } nd { 1,..., n } ( n )

### Best Approximation in the 2-norm

Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion

### Chapter 28. Fourier Series An Eigenvalue Problem.

Chpter 28 Fourier Series Every time I close my eyes The noise inside me mplifies I cn t escpe I relive every moment of the dy Every misstep I hve mde Finds wy it cn invde My every thought And this is why

### (4.1) D r v(t) ω(t, v(t))

1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution

### LECTURE 10: JACOBI SYMBOL

LECTURE 0: JACOBI SYMBOL The Jcobi symbol We wish to generlise the Legendre symbol to ccomodte comosite moduli Definition Let be n odd ositive integer, nd suose tht s, where the i re rime numbers not necessrily

### Wave Equation on a Two Dimensional Rectangle

Wve Eqution on Two Dimensionl Rectngle In these notes we re concerned with ppliction of the method of seprtion of vriles pplied to the wve eqution in two dimensionl rectngle. Thus we consider u tt = c

### LINEAR ALGEBRA APPLIED

5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order

### Chapter 6 Techniques of Integration

MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln

### Math 473: Practice Problems for the Material after Test 2, Fall 2011

Mth 473: Prctice Problems for the Mteril fter Test, Fll SOLUTION. Consider the following modified het eqution u t = u xx + u x. () Use the relevnt 3 point pproximtion formuls for u xx nd u x to derive

### AMATH 731: Applied Functional Analysis Fall Additional notes on Fréchet derivatives

AMATH 731: Applied Functionl Anlysis Fll 214 Additionl notes on Fréchet derivtives (To ccompny Section 3.1 of the AMATH 731 Course Notes) Let X,Y be normed liner spces. The Fréchet derivtive of n opertor

### Duke Math Meet

Duke Mth Meet 01-14 Power Round Qudrtic Residues nd Prime Numers For integers nd, we write to indicte tht evenly divides, nd to indicte tht does not divide For exmle, 4 nd 4 Let e rime numer An integer

Chter 8 Qudrtic Residues 8. Qudrtic residues Let n>be given ositive integer, nd gcd, n. We sy tht Z n is qudrtic residue mod n if the congruence x mod n is solvble. Otherwise, is clled qudrtic nonresidue

### NOTES ON HILBERT SPACE

NOTES ON HILBERT SPACE 1 DEFINITION: by Prof C-I Tn Deprtment of Physics Brown University A Hilbert spce is n inner product spce which, s metric spce, is complete We will not present n exhustive mthemticl

### Inner-product spaces

Inner-product spces Definition: Let V be rel or complex liner spce over F (here R or C). An inner product is n opertion between two elements of V which results in sclr. It is denoted by u, v nd stisfies:

### Discrete Least-squares Approximations

Discrete Lest-squres Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve

### Math Theory of Partial Differential Equations Lecture 2-9: Sturm-Liouville eigenvalue problems (continued).

Mth 412-501 Theory of Prtil Differentil Equtions Lecture 2-9: Sturm-Liouville eigenvlue problems (continued). Regulr Sturm-Liouville eigenvlue problem: d ( p dφ ) + qφ + λσφ = 0 ( < x < b), dx dx β 1 φ()

### Chapter 6. Riemann Integral

Introduction to Riemnn integrl Chpter 6. Riemnn Integrl Won-Kwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl

### g i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f

1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where

### 7.3 Problem 7.3. ~B(~x) = ~ k ~ E(~x)=! but we also have a reected wave. ~E(~x) = ~ E 2 e i~ k 2 ~x i!t. ~B R (~x) = ~ k R ~ E R (~x)=!

7. Problem 7. We hve two semi-innite slbs of dielectric mteril with nd equl indices of refrction n >, with n ir g (n ) of thickness d between them. Let the surfces be in the x; y lne, with the g being

### Families of Solutions to Bernoulli ODEs

In the fmily of solutions to the differentil eqution y ry dx + = it is shown tht vrition of the initil condition y( 0 = cuses horizontl shift in the solution curve y = f ( x, rther thn the verticl shift

### Best Approximation. Chapter The General Case

Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given

### Theoretical foundations of Gaussian quadrature

Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of

### c n φ n (x), 0 < x < L, (1) n=1

SECTION : Fourier Series. MATH4. In section 4, we will study method clled Seprtion of Vribles for finding exct solutions to certin clss of prtil differentil equtions (PDEs. To do this, it will be necessry

### Orthogonal Polynomials and Least-Squares Approximations to Functions

Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny

### (PDE) u t k(u xx + u yy ) = 0 (x, y) in Ω, t > 0, (BC) u(x, y, t) = 0 (x, y) on Γ, t > 0, (IC) u(x, y, 0) = f(x, y) (x, y) in Ω.

Seprtion of Vriles for Higher Dimensionl Het Eqution 1. Het Eqution nd Eigenfunctions of the Lplcin: An 2-D Exmple Ojective: Let Ω e plnr region with oundry curve Γ. Consider het conduction in Ω with fixed

### Sturm-Liouville Theory

LECTURE 1 Sturm-Liouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory

### The Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5

The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle

### 63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

### Advanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015

Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n

### Polynomials and Division Theory

Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

### 8 Laplace s Method and Local Limit Theorems

8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved

### Supplement 4 Permutations, Legendre symbol and quadratic reciprocity

Sulement 4 Permuttions, Legendre symbol nd qudrtic recirocity 1. Permuttions. If S is nite set contining n elements then ermuttion of S is one to one ming of S onto S. Usully S is the set f1; ; :::; ng

### Analytical Methods Exam: Preparatory Exercises

Anlyticl Methods Exm: Preprtory Exercises Question. Wht does it men tht (X, F, µ) is mesure spce? Show tht µ is monotone, tht is: if E F re mesurble sets then µ(e) µ(f). Question. Discuss if ech of the

### Fundamental Theorem of Calculus

Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under

### Mapping the delta function and other Radon measures

Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support

### Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.

Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)

Qudrtic recirocity Frncisc Bozgn Los Angeles Mth Circle Octoer 8, 01 1 Qudrtic Recirocity nd Legendre Symol In the eginning of this lecture, we recll some sic knowledge out modulr rithmetic: Definition

### Orthogonal Polynomials

Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils

### QUADRATIC RESIDUES MATH 372. FALL INSTRUCTOR: PROFESSOR AITKEN

QUADRATIC RESIDUES MATH 37 FALL 005 INSTRUCTOR: PROFESSOR AITKEN When is n integer sure modulo? When does udrtic eution hve roots modulo? These re the uestions tht will concern us in this hndout 1 The

### USA Mathematical Talent Search Round 1 Solutions Year 25 Academic Year

1/1/5. Alex is trying to oen lock whose code is sequence tht is three letters long, with ech of the letters being one of A, B or C, ossibly reeted. The lock hs three buttons, lbeled A, B nd C. When the

### Review of Gaussian Quadrature method

Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

PRIMES AND QUADRATIC RECIPROCITY ANGELICA WONG Abstrct We discuss number theory with the ultimte gol of understnding udrtic recirocity We begin by discussing Fermt s Little Theorem, the Chinese Reminder

### PHYSICS 116C Homework 4 Solutions

PHYSICS 116C Homework 4 Solutions 1. ( Simple hrmonic oscilltor. Clerly the eqution is of the Sturm-Liouville (SL form with λ = n 2, A(x = 1, B(x =, w(x = 1. Legendre s eqution. Clerly the eqution is of

### Chapter 5. , r = r 1 r 2 (1) µ = m 1 m 2. r, r 2 = R µ m 2. R(m 1 + m 2 ) + m 2 r = r 1. m 2. r = r 1. R + µ m 1

Tor Kjellsson Stockholm University Chpter 5 5. Strting with the following informtion: R = m r + m r m + m, r = r r we wnt to derive: µ = m m m + m r = R + µ m r, r = R µ m r 3 = µ m R + r, = µ m R r. 4

### Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24

Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the

### The Dirac distribution

A DIRAC DISTRIBUTION A The Dirc distribution A Definition of the Dirc distribution The Dirc distribution δx cn be introduced by three equivlent wys Dirc [] defined it by reltions δx dx, δx if x The distribution

### a 2 +x 2 x a 2 -x 2 Figure 1: Triangles for Trigonometric Substitution

I.B Trigonometric Substitution Uon comletion of the net two sections, we will be ble to integrte nlyticlly ll rtionl functions of (t lest in theory). We do so by converting binomils nd trinomils of the

### ODE: Existence and Uniqueness of a Solution

Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =

### Chapter 8: Methods of Integration

Chpter 8: Methods of Integrtion Bsic Integrls 8. Note: We hve the following list of Bsic Integrls p p+ + c, for p sec tn + c p + ln + c sec tn sec + c e e + c tn ln sec + c ln + c sec ln sec + tn + c ln

### Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.

Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one

### Sections 5.2: The Definite Integral

Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)

### The Riemann Integral

Deprtment of Mthemtics King Sud University 2017-2018 Tble of contents 1 Anti-derivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Anti-derivtive Function Definition Let f : I R be function

### Taylor Polynomial Inequalities

Tylor Polynomil Inequlities Ben Glin September 17, 24 Abstrct There re instnces where we my wish to pproximte the vlue of complicted function round given point by constructing simpler function such s polynomil

### Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

### Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

### 10 Vector Integral Calculus

Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve

### 10 Elliptic equations

1 Elliptic equtions Sections 7.1, 7.2, 7.3, 7.7.1, An Introduction to Prtil Differentil Equtions, Pinchover nd Ruinstein We consider the two-dimensionl Lplce eqution on the domin D, More generl eqution

### Fourier series. Preliminary material on inner products. Suppose V is vector space over C and (, )

Fourier series. Preliminry mteril on inner products. Suppose V is vector spce over C nd (, ) is Hermitin inner product on V. This mens, by definition, tht (, ) : V V C nd tht the following four conditions

### LECTURE 3. Orthogonal Functions. n X. It should be noted, however, that the vectors f i need not be orthogonal nor need they have unit length for

ECTURE 3 Orthogonl Functions 1. Orthogonl Bses The pproprite setting for our iscussion of orthogonl functions is tht of liner lgebr. So let me recll some relevnt fcts bout nite imensionl vector spces.

### 1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.

1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt

### UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence

### Summary: Method of Separation of Variables

Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section

### Name of the Student:

SUBJECT NAME : Discrete Mthemtics SUBJECT CODE : MA 2265 MATERIAL NAME : Formul Mteril MATERIAL CODE : JM08ADM009 Nme of the Student: Brnch: Unit I (Logic nd Proofs) 1) Truth Tble: Conjunction Disjunction

### 7. Indefinite Integrals

7. Indefinite Integrls These lecture notes present my interprettion of Ruth Lwrence s lecture notes (in Herew) 7. Prolem sttement By the fundmentl theorem of clculus, to clculte n integrl we need to find

### Integration Techniques

Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u

### Chapter 6. Infinite series

Chpter 6 Infinite series We briefly review this chpter in order to study series of functions in chpter 7. We cover from the beginning to Theorem 6.7 in the text excluding Theorem 6.6 nd Rbbe s test (Theorem

### Integrals - Motivation

Integrls - Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is non-liner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but

### MATH 174A: PROBLEM SET 5. Suggested Solution

MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous rel-vlued function on I), nd let L 1 (I) denote the completion

### Review of Riemann Integral

1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.

### STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH

STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH XIAO-BIAO LIN. Qudrtic functionl nd the Euler-Jcobi Eqution The purpose of this note is to study the Sturm-Liouville problem. We use the vritionl problem s

### Chapter 3 Single Random Variables and Probability Distributions (Part 2)

Chpter 3 Single Rndom Vriles nd Proilit Distriutions (Prt ) Contents Wht is Rndom Vrile? Proilit Distriution Functions Cumultive Distriution Function Proilit Densit Function Common Rndom Vriles nd their

### The Regulated and Riemann Integrals

Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

### Lecture 1. Functional series. Pointwise and uniform convergence.

1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

### The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

### 1 The Lagrange interpolation formula

Notes on Qudrture 1 The Lgrnge interpoltion formul We briefly recll the Lgrnge interpoltion formul. The strting point is collection of N + 1 rel points (x 0, y 0 ), (x 1, y 1 ),..., (x N, y N ), with x

### MA Handout 2: Notation and Background Concepts from Analysis

MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,

### 21.6 Green Functions for First Order Equations

21.6 Green Functions for First Order Equtions Consider the first order inhomogeneous eqution subject to homogeneous initil condition, B[y] y() = 0. The Green function G( ξ) is defined s the solution to

### STURM-LIOUVILLE PROBLEMS

STURM-LIOUVILLE PROBLEMS Mrch 8, 24 We hve seen tht in the process of solving certin liner evolution equtions such s the het or wve equtions we re led in very nturl wy to n eigenvlue problem for second

### Consequently, the temperature must be the same at each point in the cross section at x. Let:

HW 2 Comments: L1-3. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the

### Some integral inequalities on time scales

Al Mth Mech -Engl Ed 2008 29(1:23 29 DOI 101007/s10483-008-0104- c Editoril Committee of Al Mth Mech nd Sringer-Verlg 2008 Alied Mthemtics nd Mechnics (English Edition Some integrl ineulities on time scles

### Chapter 3 The Schrödinger Equation and a Particle in a Box

Chpter 3 The Schrödinger Eqution nd Prticle in Bo Bckground: We re finlly ble to introduce the Schrödinger eqution nd the first quntum mechnicl model prticle in bo. This eqution is the bsis of quntum mechnics

### Hilbert Spaces. Chapter Inner product spaces

Chpter 4 Hilbert Spces 4.1 Inner product spces In the following we will discuss both complex nd rel vector spces. With L denoting either R or C we recll tht vector spce over L is set E equipped with ddition,

### Mrgolius 2 In the rticulr cse of Ploue's constnt, we tke = 2= 5+i= 5, nd = ;1, then ; C = tn;1 1 2 = ln(2= 5+i= 5) ln(;1) More generlly, we would hve

Ploue's Constnt is Trnscendentl Brr H. Mrgolius Clevelnd Stte University Clevelnd, Ohio 44115.mrgolius@csuohio.edu Astrct Ploue's constnt tn;1 ( 1 2) is trnscendentl. We rove this nd more generl result

### The Fundamental Theorem of Calculus

The Fundmentl Theorem of Clculus MATH 151 Clculus for Mngement J. Robert Buchnn Deprtment of Mthemtics Fll 2018 Objectives Define nd evlute definite integrls using the concept of re. Evlute definite integrls

### Math 554 Integration

Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we

### df dt f () b f () a dt

Vector lculus 16.7 tokes Theorem Nme: toke's Theorem is higher dimensionl nlogue to Green's Theorem nd the Fundmentl Theorem of clculus. Why, you sk? Well, let us revisit these theorems. Fundmentl Theorem

### STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS Throughout, we let [, b] be bounded intervl in R. C 2 ([, b]) denotes the spce of functions with derivtives of second order continuous up to the endpoints. Cc 2

### Bases for Vector Spaces

Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything

### A BRIEF INTRODUCTION TO UNIFORM CONVERGENCE. In the study of Fourier series, several questions arise naturally, such as: c n e int

A BRIEF INTRODUCTION TO UNIFORM CONVERGENCE HANS RINGSTRÖM. Questions nd exmples In the study of Fourier series, severl questions rise nturlly, such s: () (2) re there conditions on c n, n Z, which ensure

### Numerical Methods I Orthogonal Polynomials

Numericl Methods I Orthogonl Polynomils Aleksndr Donev Cournt Institute, NYU 1 donev@cournt.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fll 2014 Nov 6th, 2014 A. Donev (Cournt Institute) Lecture IX

Mth 452 - Advnced Clculus II Line Integrls nd Green s Theorem The min gol of this chpter is to prove Stoke s theorem, which is the multivrible version of the fundmentl theorem of clculus. We will be focused