Hermite equation

In the same way as proceeded with the Laguerre s polynomial, the Hermite equation may be solved by the speeifie eomplex integral representation ... 

Therefore, in the light of the Laurent theorem of residues there can be inferred that the complex integral solution of the Hermite equation may produce the Hermite generating function (with the same recipe as was previously done for generating Laguerre function) ... 

The change of variable z = transforms equation (V-8) into a Hermite equation, the lowest eigenvalue of which is... 

The Hermite equation Is named for Charles Hermite, 1822-1901, a great French mathematician who made many contributions to mathematics, Including the proof that e (2.71828...) Is a transcendental irrational number. 

This differential equation is the same as a famous equation known as the Hermite equation (see Appendix F). Hermite solved this equation by assuming that the solution was of the form... 

We cannot simply require all coefficients past a certain point in the series to vanish if this violates the recursion relation. The function would then fail to satisfy the Hermite equation. Let us assume that c +2 is a vanishing coefficient and that c does not vanish. The numerator in the right-hand side of Eq. (15.4-6) must then vanish for n = v ... 

The Time-Independent Schrodinger Equation for the Harmonic Oscillator (the Hermite Equation)... 

It is evident that application of Green s theorem cannot eliminate second-order derivatives of the shape functions in the set of working equations of the least-sc[uares scheme. Therefore, direct application of these equations should, in general, be in conjunction with C continuous Hermite elements (Petera and Nassehi, 1993 Petera and Pittman, 1994). However, various techniques are available that make the use of elements in these schemes possible. For example, Bell and Surana (1994) developed a method in which the flow model equations are cast into a set of auxiliary first-order differentia] equations. They used this approach to construct a least-sciuares scheme for non-Newtonian flow equations based on equal-order C° continuous, p-version hierarchical elements. 

The function u x) is real because w(x) is always positive and u x) is positive because we take the positive square root. If w x) approaches infinity at any point within the range of hermiticity of A (as x approaches infinity, for example), then tpfx) must approach zero such that the ratio (pfx) approaches zero. Equation (3.18) is now multiplied by u x) and ffx) is replaced by u x)4>i(x)... 

It is customary to express the eigenfunctions for the stationary states of the harmonic oscillator in terms of the Hermite polynomials. The infinite set of Hermite polynomials // ( ) is defined in Appendix D, which also derives many of the properties of those polynomials. In particular, equation (D.3) relates the Hermite polynomial of order n to the th-order derivative which appears in equation (4.39)... 

For reference, the Hermite polynomials for = 0 to = 10 are listed in Table 4.1. When needed, higher-order Hermite polynomials are most easily obtained from the recurrence relation (D.5). If only a single Hermite polynomial is wanted and the neighboring polynomials are not available, then equation (D.4) may be used. 

Another expression for the Hermite polynomials may be obtained by expanding g(, s) using equation (A.l)... 

We next derive some recurrence relations for the Hermite polynomials. If we differentiate equation (D.l) with respect to s, we obtain... 

To find the differential equation that is satisfied by the Hermite polynomials, we first differentiate the second recurrence relation (D.6) and then substitute (D.6) with n replaeed by n — 1 to eliminate the first derivative of i ( )... 

To obtain the orthogonality and normalization relations for the Hermite polynomials, we multiply together the generating functions g(, 5) and g( , t), both obtained from equation (D.l), and the factor e and then integrate over ... 

The Hermite polynomials Hn ) form an orthogonal set over the range —oo oo with a weighting factor e . If we equate coefficients of stY on each side of equation (D.12), we obtain... 

A comparison of equation (G.17) with (D.IO) shows that the solutions u( ) are the Hermite polynomials, whose properties are discussed in Appendix D. Thus, the functions [Pg.323]

We shall now use the theorem previously mentioned. Since is hermitian during this rank computation, so too are D and X. The above constraint on, Equation (20), along with its hermiticity property, leads to the following number of conditions on its elements, and therefore ultimately on the P elements ... 

The Hermite polynomials introduced above represent an example of special functions which arise as solutions to various second-order differential equations. After a summary of some of the properties of these polynomials, a brief description of a few others will be presented. The choice is based on their importance in certain problems in physics and chemistry. 

Show that Vi 2 is an asymptotic solution to Eq. (83) that leads to Hermite s equation. 

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