1、Orthogonal Function Expansion 正交函數展開,Introduction of the Eigenfunction ExpansionAbstract SpaceFunction SapceLinear Operator and Orthogonal Function,Introduction -The Eigenfunction Expansion,Consider the equation :,With the b.cs :,The g.s. is where u1,u2 are linearly index. Fucs. And C1,C2 are arb co
2、nsts.,For b.cs :,The condition of nontrivial sol. of c1,c2 to be existed if :,The Euler Column,The g.s.,For the b.c.s : ,And for nontrivial solution,i.e.,So that we obtain the Eigen values,And the corresponding eigenfucs (nontrivial sols) are,According the analysis, we will have unless the end force
3、 P such that:, Function Space,Rn,Q,Z,N,Inner Product Space,Hilbert Space,Normed Space,Metric Space,由RRn(有序性喪失),Abstract Space,完備性:每一Cauchy系列均收斂,Topological Space,Banach Space,Topological Space,Definition,A topological space is a non-empty set E together with a family of subsets of E satisfying the f
4、ollowing axioms:,(3) The intersection of any finite number of sets in X belongs to X i.e.,(1),(2) The union of any number of sets in X belongs to X i.e.,Metric Space,Definition,A metric space is a 2-tuple (X,d) where X is a set and d is a metric on X, that is, a function d: X X R, such that,d(x, y)
5、0 (non-negativity) d(x, y) = 0 if and only if x = y (identity) d(x, y) = d(y, x) (symmetry) d(x, z) d(x, y) + d(y, z) (triangle inequality).,Cauchy Sequence,Definition: Complete space,A sequence (Xn):in a metric space X=(X,d) is said to be Cauchy if for every there is an N=N(e) such that,x is called
6、 the limit of (Xn) and we write,or, simply,for m,nN ,Any Cauchy Sequence in X is convergence,; d 0,Definition: Completeness,Ball and Sphere,Definition:,Given a point and a real number r0, we define three of sets:,(a) (Open ball),(b) (Closed ball),(c) (Sphere),In all three case, x0 is called the cent
7、er and r the radius.,Furthermore, the definition immediately implies that,A mapping from a normed space X into a normed space Y is called an operator. A mapping from X into the scalar filed R or C is called a functional.,The set of all biunded linear operator from a given normed space X into a given
8、 normed space Y can be made into a normed space, which is denoted by B(x,y). Similarly, the set of all bounded linear functionals on X becomes a normed space, which is called the dual space X of X.,Definition (Open set and closed set):,A subset M of a metric space X is said to be open if is contains
9、 a ball about each of its points. A subset K of X is said to be closed if its complement (in X ) is open, that is, Kc=X-K is open.,Normed Space,Definition of Normed Space,Definition of Banach Space,Here a norm on a vector space X is a real-value function on X whose value at an is denoted by,Here x a
10、nd y are arbitrary vector in X and is any scalar,A Banach space is a complete normed space.,A metric d induced by a norm on a normed space X satisfies,Lemma (Translation invariance),(a) d(x+a,y+a)=d(x,y),(b),For all x, y and every scalar,Proof.,Let X be the vector space of all ordered pairs of real
11、numbers. Show norms on X are defined by,The sphere,In a normed space X is called the unit sphere.,Unit Sphere in LP,If a normed space X contains a sequence (en) with the property that every there is a uniquie sequence of scalars (an) such that,Then (en) is called basis for X. series,Which has the su
12、m x is then called the expansion of x,A inner product space is a vector space X with an inner product define on X.,Here, the inner product is the mapping of into the scale filed, such that,Inner Product Space,Hilbert Space,Examples of finite-dimensional Hilbert spaces include,1. The real numbers wit
13、h the vector dot product of and,2. The complex numbers with the vector dot product of and the complex conjugate of .,Hence inner product spaces are normed spaces, and Hilber spaces are Banach spaces.,A Hilbert space is a complete inner product space.,(Norm),(Metric),Euclidean space Rn,The space Rn i
14、s a Hilbert space with inner product define by,Where,Space L2a,b,Hilbert sequence space l2,With the inner product,The norm,Space lp,The space lp with is not inner product space, hence not a Hilbert space,Orthonormal Sets and Sequences,Orthogonality of elements plays a basis role in inner product and
15、 Hilbert spaces.The vectors form a basis for R3, so that every has a unique representation.,Continuous functions,Let X be the inner product space of all real-valued continuous functions on 0,2 with inner product defined by,An orthogonal sequence in X is (un), where,Another orthogonal sequence in X i
16、s (vn), where,Hence an orthonormal sequence sequence is (en),From (vn) we obtain the orthonormal sequence ( ) where,Homework,1. Does d(x,y)=(x-y)2 define a metric on the set of all real numbers?,2. Show that defines a metric on the set of all real numbers.,3. Let Show that the open interval (a,b) is
17、 an incomplete subspace of R, whereas the closed interval a,b is complete.,4. Prove that the eigenfunction and eigenvalue are orthogonalization and real for the the Sturm-Lioville System.,6. For the very special case and , the self-adjoint eigenvalue equation becomes,5. Show the following when linea
18、r second-order difference equation is expressed in self-adjoint form:,(a) The Wronskian is equal to constant divided by the initial coefficient p,(b) A second solution is given by,Use this obtain a “second” solution of the following,(a ) Legendres equation,(b ) Laguerres equation,(c ) Hermites equat
19、ion,Function Space,(A) L2 a,b space:,Space of real fucs. f(x) which is define on a,b and square integrable i.e .,In the language of vector space, we say that,“any n linearly indep vectors form a basis in E ”space”. Similarly, in function spaceIt is possible to choose a set of basis function such tha
20、t any function, satisfying Appropriate condition can be expressed as a linear combination to a basis in L2a,bCertainly, any such set of fucs. Must have infinitely many numbers; that is, such a L2a,b comprises infinitely many dimensions.,(B) Schwarz Inequality:,Given f(x), g(x) in L2a,b, Define,Proof
21、:,(C) Linear Dependence, Independence:,Criterion: A set of fucs.,In L2 a,b is linear dep.(indep.) if its,If its Gramian (G) vanishes (does not vanish), where,The proof is the same as in linear vector space.,(D) The orthogonal System,A set of real fucs.,.is called an orthogonal set of fucs.,In L2a,b
22、if these fucs. are define in L2a,b if all the integral,exist and are zero for all pairs of distinct,Properties of Complete System,Theorem: Let f(x), F(x) be defined on L2 a,b for which,Then we have,Proof:,Since f+F, and f-F are square integer able, from the completeness relation,Theorem:,Every squar
23、e integer able fnc. f(x) is uniquely determined (except for its value at a finite number of points) by its Fourier series.,Proof:,Suppose there are two fucs. f(x),g(x) having the identical Fourier series representation,i.e.,Then using we find,g(x)=f(x) at the pts of continuity of the integrand,g(x)
24、and f(x) coincide everywhere, except possibly at a finite number of pts. of discontinuity,Proof: Since,And let,We can prove f(x)=g(x) at every point.,Theorem:,An continuous fuc. f(x) which is orthogonal to all the fucs. of the complete system must be identically zero.,Proof: Since,Assume x2x1,Take,T
25、heorem:,The fourier series of every square integer able fuc. f(x) can be integrated term by term. In other words, if,Then,Where x1,x2 are any points on the inteval a,b,The Sturm-Liouville Problem,Self-adjoint Operator,For a linear operator L the analog of a quadratic form for a matrix is the integra
26、l,Because of the analogy with the transposed matrix, it is convenient to define the linear operator,Comparing the integrands,The operator L is said to be self-adjoint.,As the adjoint operator L. The necessary and sufficient condition that,The Sturm-Liouville Boundary Value Problem,A differential equ
27、ation defined on the interval having the form of,and the boundary conditions,is called as Sturm-Liouville boundary value problem or Sturm-Liouville system, where , ; the weighting function r(x)0 are given functions; a1, a2 , b1, b2 are given constants; and the eigenvalue is an unspecified parameter.
28、,The Regular Sturm-Liouville Equation,It is a special kind of boundary value problem which consists of a second-order homogeneous linear differential equation and linear homogeneous boundary conditions of the form,where the p, q and r are real and continuous functions such that p has a continuous de
29、rivative, and p(x) 0, r(x) 0 for all x on a real interval a x b; and is a parameter independent of x. L is the linear homogeneous differential operator defined by L(y) = p(x)y+q(x)y.And two supplementary boundary conditions,where A1 , A2 , B1 and B2 are real constants such that A1 and A2 not both ze
30、ro and B1 and B2 are not both zero.,A1y(a)+A2y(a) = 0 B1y(b)+B2y(b) = 0 .,Definition 1.1 : Consider the Sturm-Liouville problem consisting of the differentail equation and supplementary conditions. The value of the parameter in for which there exists nontrivial solution of the problem is called the
31、eigenvalue of the problem. The corresponding nontrivial solution,is called the eigenfunction of the problem. The Sturm-Liouville problem is also called an eigenvalue problem.,The Nonhomogeneous Sturm-Liouville Problems,And as in regular Sturm-Liouville problems we assume that p, p, q, and r are cont
32、inuous on a x b and p(x) 0, r(x) 0 there.We solve the problem by making use of the eigenfunctions of the corresponding homogeneous problem consisting of the differential equation,Consider boundary value problem consisting of the nonhomogeneous differential equation,Ly = - p(x)y+q(x)y = w(x)y+f(x),wh
33、ere is a given constant and f is a given function on a a x b and the boundary conditions,A1y(a)+A2y(a)=0 B1y(b)+B2y(b)=0 .,The Bessels Differential Equation,In the Sturm-Liouville Boundary Value Problem, there is an important special case called Bessels Differential Equation which arises in numerous
34、 problems, especially in polar and cylindrical coordinates. Bessels Differential Equation is defined as: .,where is a non-negative real number. The solutions of this equation are called Bessel Functions of order n. Although the order n can be any real number, the scope of this section is limited to
35、non-negative integers, i.e., , unless specified otherwise. Since Bessels differential equation is a second order ordinary differential equation, two sets of functions, the Bessel function of the first kind Jn(x) and the Bessel function of the second kind (also known as the Weber Function) Yn(x) , ar
36、e needed to form the general solution:,Five Approaches,The Bessel functions are introduced here by means of a generating function. Other approaches are possible. Listing the various possibilities, we have.,Gram-Schmidt Orthogonalization,We now demand that each solution be multiplied by,We star with
37、n=0, letting,Then normalize,The presence of the new un(x) will guarantee linear independence.,Fro n=1, let,This demand of orthogonality leads to,As is normalized to unity, we have,Fixing the value of a10. Normalizing, we have,We demand that be orthogonal to,Where,The equation can be replaced by,And
38、aij becomes,The coefficients aij are given by,If some order normalization is selected,Orthogonal polynomial Generated by Gram-Schmidt Orthogonalization of,2. Series solution of Bessels differential equation,Using y for dy/dx and for d2y/dx2 . Again, assuming a solution of the form,Inserting these co
39、efficients in our assumed series solution, we have,Inserting these coefficients in our assumed series solution, we have,With the result that.,3. Generating function,Expanding this function in a Laurent series, we obtain,It is instructive to compare.,The coefficient of tn, Jn(x), is defined to be Bes
40、sel function of the first kind of integral order n. Expanding the exponential, we have a product of Maclaurin series in xt/2 and x/2t, respectively.,The coefficient tn is then,For a given s we get tn(n=0) from r=n+s;,Bessel function J0(x), J1(x) and J2(x),4. Contour integral: Some writers prefer to
41、start with contour integral definitions of the Hankel function, and develop the Bessel function Jv(x) from the Hankel functions.,The integral representation,(Schlsefli integral),may easily be established as a Cauchy integral for v=n, that is , an integer. Recognizing that the numerator is the genera
42、ting function and integrating around the origin,Cut line,5. Direct solution of physical problems, Fraunhofer diffraction with a circular aperture illusterates this. Incidentally, can be treated by series expansion if desired. Feynman develop Bessel function from a consideration of cavity resonators.
43、,The parameter B us given by,In the theory of diffraction through a circular aperture we encounter the integral,Feynman develop Bessel function from a consideration of cavity resonators. (Homework 1),The intensity of the light in the diffraction pattern is proportional to 2 and,Fro green light Hence
44、, if a=0.5 cm,(2) Using only the generating function,Explicit series form Jn(x), shoe that Jn(x) has odd or even parity according to whether n is odd or even, this,(3) Show by direct differentiation that,Satisfies the two recurrence relations,And bessels differential equation,Homework,(4) Show that,Thus generating modified Bessel function In(x),(5) The chebyshev polynomials (typeII) are generated,Using the techniques for transforming series, develop a series representation of Un(x),PS:請參考補充講義,
