Sturm-Liouville Expansions of the Delta
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Gauge Institute Journal H. Vic Dannon Sturm-Liouville Expansions of the Delta Function H. Vic Dannon [email protected] May, 2014 Abstract We expand the Delta Function in Series, and Integrals of Sturm-Liouville Eigen-functions. Keywords: Sturm-Liouville Expansions, Infinitesimal, Infinite- Hyper-Real, Hyper-Real, infinite Hyper-real, Infinitesimal Calculus, Delta Function, Fourier Series, Laguerre Polynomials, Legendre Functions, Bessel Functions, Delta Function, 2000 Mathematics Subject Classification 26E35; 26E30; 26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15; 46S20; 97I40; 97I30. 1 Gauge Institute Journal H. Vic Dannon Contents 0. Eigen-functions Expansion of the Delta Function 1. Hyper-real line. 2. Hyper-real Function 3. Integral of a Hyper-real Function 4. Delta Function 5. Convergent Series 6. Hyper-real Sturm-Liouville Problem 7. Delta Expansion in Non-Normalized Eigen-Functions 8. Fourier-Sine Expansion of Delta Associated with yx"( )+=l yx ( ) 0 & ab==0 9. Fourier-Cosine Expansion of Delta Associated with 1 yx"( )+=l yx ( ) 0 & ab==2 p 10. Fourier-Sine Expansion of Delta Associated with yx"( )+=l yx ( ) 0 & a = 0 11. Fourier-Bessel Expansion of Delta Associated with n2 -1 ux"( )+- (l 4 ) ux ( ) = 0 & ab==0 x 2 12. Delta Expansion in Orthonormal Eigen-functions 13. Fourier-Hermit Expansion of Delta Associated with ux"( )+- (l xux2 ) ( ) = 0 for any real x 14. Fourier-Legendre Expansion of Delta Associated with 2 Gauge Institute Journal H. Vic Dannon 11 11 uu"(ql )++ [ ] ( q ) = 0 -<<pq p 4 cos2 q , 22 15. Fourier-Legendre Expansion of Delta Associated with 112 11 ymy"(ql )++- [ ( ) ] (q ) = 0 -<<pq p 4 cos2 q , 22 16. Fourier-Bessel Expansion of Delta Associated with n2 -1 ux"( )+- (l 4 ) ux ( ) = 0 & 0 <<xb x 2 17. Fourier-Bessel Expansion of Delta Associated with yx"( )+- (l xyx ) ( ) = 0 , 0 <<¥x & a = 0 18. Fourier-Bessel Expansion of Delta Associated with 1 yx"( )+- (m xyx ) ( ) = 0, 0 <<¥x & ap= 2 19. Delta Expansion in an Integral over a Continuum of Eigen- Functions 20. Fourier-Cosine Integral of Delta Associated with yx"( )+=l yx ( ) 0 , 0 <<¥x 21. Weber Formula for Fourier-Bessel Integral of Delta n2 -1 Associated with yx"( )+- (l 4 ) yx ( ) = 0 , ax¥<< , a > 0 x 2 22. Hankel Formula for Fourier-Bessel Integral of Delta n2 -1 Associated with yx"( )+- (l 4 ) yx ( ) = 0 , 0 <<¥x x 2 23. Fourier-Bessel Integral of Delta Associated with yx"( )++ (l xyx ) ( ) = 0 , 0 £<¥x 24. Fourier-Bessel Integral of Delta Associated with 3 Gauge Institute Journal H. Vic Dannon yx"( )++ (l e2x ) yx ( ) = 0 , -¥ <x < ¥ 25. Fourier-Bessel Integral of Delta Associated with yx"( )+- (l e2x ) yx ( ) = 0 , -¥ <x < ¥ References 4 Gauge Institute Journal H. Vic Dannon 0. Eigen-Functions Expansion of the Delta Function Unlike Taylor’s expansion of a function that requires its derivatives of any order, Fourier Integral or Fourier Series representation require no derivatives. Then, the function’s projections on the orthogonal sequence of eigen-functions sum up to the function. It is little known that the Fourier Series representation, and the Fourier Integral representation result from an expansion of the Delta Function: For instance, the Fourier Integral representation, x=¥ f ()xfx=-ò ()(xd x )d x x =-¥ xw=¥ =¥ ixwx()- = f ()xw1 eddx òò2p xw=-¥ =-¥ wx=¥ =¥ -iixwx w = 1 f ()xxededw 2p òò wx=-¥ =-¥ F()w Here, the Fourier Transform F()w is the projection of f ()x on eixw . 5 Gauge Institute Journal H. Vic Dannon The eigen-function expansions of the Delta Function are at the root of the eigen-function expansion of any function. The Delta Function can be defined only as a hyper-real function in infinitesimal Calculus. We proceed with the definition of the Hyper-real line. 6 Gauge Institute Journal H. Vic Dannon 1. Hyper-real Line Each real number a can be represented by a Cauchy sequence of rational numbers, r(,rr123, ,...) so that rn a . The constant sequence (aaa, , ,...) is a constant hyper-real. In [Dan2] we established that, 1. Any totally ordered set of positive, monotonically decreasing to zero sequences (iii123 , , ,...) constitutes a family of infinitesimal hyper-reals. 2. The infinitesimals are smaller than any real number, yet strictly greater than zero. 3. Their reciprocals (11, ,1 ,...) are the infinite hyper-reals. ii123i 4. The infinite hyper-reals are greater than any real number, yet strictly smaller than infinity. 5. The infinite hyper-reals with negative signs are smaller than any real number, yet strictly greater than -¥. 6. The sum of a real number with an infinitesimal is a non-constant hyper-real. 7. The Hyper-reals are the totality of constant hyper-reals, a family of infinitesimals, a family of infinitesimals with 7 Gauge Institute Journal H. Vic Dannon negative sign, a family of infinite hyper-reals, a family of infinite hyper-reals with negative sign, and non-constant hyper-reals. 8. The hyper-reals are totally ordered, and aligned along a line: the Hyper-real Line. 9. That line includes the real numbers separated by the non- constant hyper-reals. Each real number is the center of an interval of hyper-reals, that includes no other real number. 10. In particular, zero is separated from any positive real by the infinitesimals, and from any negative real by the infinitesimals with negative signs, -dx . 11. Zero is not an infinitesimal, because zero is not strictly greater than zero. 12. We do not add infinity to the hyper-real line. 13. The infinitesimals, the infinitesimals with negative signs, the infinite hyper-reals, and the infinite hyper-reals with negative signs are semi-groups with respect to addition. Neither set includes zero. 14. The hyper-real line is embedded in ¥ , and is not homeomorphic to the real line. There is no bi-continuous one-one mapping from the hyper-real onto the real line. 8 Gauge Institute Journal H. Vic Dannon 15. In particular, there are no points on the real line that can be assigned uniquely to the infinitesimal hyper-reals, or to the infinite hyper-reals, or to the non-constant hyper- reals. 16. No neighbourhood of a hyper-real is homeomorphic to an n ball. Therefore, the hyper-real line is not a manifold. 17. The hyper-real line is totally ordered like a line, but it is not spanned by one element, and it is not one-dimensional. 9 Gauge Institute Journal H. Vic Dannon 2. Hyper-real Function 2.1 Definition of a hyper-real function f()x is a hyper-real function, iff it is from the hyper-reals into the hyper-reals. This means that any number in the domain, or in the range of a hyper-real f ()x is either one of the following real real + infinitesimal real – infinitesimal infinitesimal infinitesimal with negative sign infinite hyper-real infinite hyper-real with negative sign Clearly, 2.2 Every function from the reals into the reals is a hyper-real function. 10 Gauge Institute Journal H. Vic Dannon 3. Integral of a Hyper-real Function In [Dan3], we defined the integral of a Hyper-real Function. Let f()x be a hyper-real function on the interval [,ab]. The interval may not be bounded. f ()x may take infinite hyper-real values, and need not be bounded. At each ax££b, there is a rectangle with base [,xx-+dx dx ], height f()x , and area 22 f ()xdx. We form the Integration Sum of all the areas for the x ’s that start at x= a, and end at xb= , å f ()xdx. xabÎ[,] If for any infinitesimal dx , the Integration Sum has the same hyper-real value, then f ()x is integrable over the interval [,ab]. Then, we call the Integration Sum the integral of f()x from xa= , to x= b, and denote it by 11 Gauge Institute Journal H. Vic Dannon xb= ò f ()xdx. xa= If the hyper-real is infinite, then it is the integral over [,ab], If the hyper-real is finite, xb= ò fxdx( )= real part of the hyper-real . xa= 3.1 The countability of the Integration Sum In [Dan1], we established the equality of all positive infinities: We proved that the number of the Natural Numbers, Card , equals the number of Real Numbers, Card = 2Card , and we have 22Card Card Card=====( Card ) .... 2 2 ... º¥. In particular, we demonstrated that the real numbers may be well-ordered. Consequently, there are countably many real numbers in the interval [,ab], and the Integration Sum has countably many terms. While we do not sequence the real numbers in the interval, the summation takes place over countably many f ()xdx. The Lower Integral is the Integration Sum where f ()x is replaced 12 Gauge Institute Journal H. Vic Dannon by its lowest value on each interval [,xx-+dx dx ] 22 æö ç ÷ 3.2 å ç inff (tdx )÷ ç xtx-££+dx dx ÷ xabÎ[,]èø22 The Upper Integral is the Integration Sum where f ()x is replaced by its largest value on each interval [,xx-+dx dx ] 22 æö ç ÷ ç supf (tdx )÷ 3.3 å ç ÷ ç xtx-££+dx dx ÷ xabÎ[,]èø22 If the integral is a finite hyper-real, we have 3.4 A hyper-real function has a finite integral if and only if its upper integral and its lower integral are finite, and differ by an infinitesimal. 13 Gauge Institute Journal H. Vic Dannon 4. Delta Function In [Dan5], we have defined the Delta Function, and established its properties 1. The Delta Function is a hyper-real function defined from the ïìü1 ï hyper-real line into the set of two hyper-reals íýïï0, . The îþïïdx hyper-real 0 is the sequence 0, 0, 0, ..