The Addition Theorem

Equating the coefficients of equal powers of z on both sides, we find the combined (joined) distribution of order two 

Because of the convolution form of (4.13), it is obvious that the sum of h+j(0, n a, b) over n equals 1. As before, the distribution addition theorem (4.13) is obtained by setting ai+U2 = a and by summing the orders i +J to obtain the order of the convoluted distribution. 

Having determined the probability distribution for m = 0,h(0, n a, b), ve proceed to the general case m / 0. For this purpose, the w-dependent factor of (4.1) must be expanded in po ver of w . This can be done by analogy to the expansion of Gi (0, z a, b), taking into account the following assignment  

With these substitutions, the kernel of Gi (0, z a, b) can be brought to the form 

Note that by the assignment (4.14), the parameter a becomes complex in the 

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The two expansions discussed so far appear to be quite different. In the multistate Gaussian model, different functions are centered at different values of AU. In the Gram-Charlier expansion, all terms are centered at (AU)0. The difference, however, is smaller that it appears. In fact, one can express a combination of Gaussian functions in the form of (2.56) taking advantage of the addition theorem for Hermite polynomials [44], Similarly, another, previously proposed representation of Pq(AU) as a r function [45] can also be transformed into the more general form of (2.56). 

We proceed to estahUsh the addition theorem for four-dimensional spherical harmonics. Equation (9.19) is an identity with respect to r. Expanding the integrand in powers of r... 

In comparison, if one uses the transformation group of the Schrodinger equation as well as the addition theorem (9.31) for the eigenfunctions, the summation is easy to carry out the whole summation (9.33) is easier to calculate than one single term. 

To calculate angular distributions for an actual collision, F -fR) must be written in terms of angles with respect to the external frame, that is, with respect to the relative heavy-particle velocity direction. This is done by applying the addition theorem of spherical harmonics... 

Clebsch-Gordan coefficients have already occurred several times in our considerations in the Introduction (formula (2)) while generalizing the quasispin concept for complex electronic configurations, while defining a relativistic wave function (formulas (2.15) and (2.16)), in the addition theorem of spherical functions (5.5) and in the definition of tensorial product of two tensors (5.12). Let us discuss briefly their definition and properties. There are a number of algebraic expressions for the Clebsch-Gordan coefficients [9, 11], but here we shall present only one ... 

As an example, we calculate the coefficient D2. As from the addition theorem... 

The functions /0 and /j are originally defined in terms of the polar angle that is 0 = arccos(e h). Thus, before performing integration, one needs to transform both integrands to the same coordinate set. Doing this with the aid of the addition theorem for Legendre polynomials, one finds... 

The addition theorem relates spherical harmonics with different arguments. 

Applying the addition theorem, the Legendre polynomials P/(cos0p) can be expressed in terms of products of the spherical harmonics as below... 

When the perturbation from more ligands is to be taken into account, the addition theorem for spherical harmonics is usually used to develop the expression of the electrostatic model when more ligands are involved. This can be done very simply by means of the concepts of the angular-overlap model. 

Eq. (39 a) is in effect the addition theorem for spherical harmonics. This can be seen by rewriting it as... 

According to the addition theorem of the cosine function, the interferogram may be separated into two parts... 

The expansion of Eq. (62) can be inverted to express the spherical harmonics in terms of spheroconal harmonics, whose completeness in turn leads to the addition theorem in the form... 

In going from the = 2 to the = 3 eigenfunctions, the raising actions of the p operators is implemented. The translations into spherical or sphero-conal harmonics follow by using coordinate transformation equations and the addition theorem [5, 6]. 

Section 2.6 recognizes that for the hydrogen atom, its Hamiltonian also commutes with and H correspondingly, it also admits solutions with Lame spheroconal harmonics polynomial eigenfunctions. It also shares the same radial eigenfunction with the familiar solution with spherical harmonics, and additionally both can be obtained from a common generating function and both satisfy the addition theorem. 

The doubly-odd europium isotopes in the range Eu are well described by the dj proton state coupled to the different neutron-shell model states, discussed above in connection ivith the odd-neution nuclei d in Eu, h Q in Eu and in i4 -i5ogjj gy usjug g-factors of the neighbouring odd-A nuclei and the additivity theorem, tiie magnetic moments of these doubly-odd europium isotopes are weU reproduced. The strongly deformed Eu and Eu are shown to be due to the configurations 3 (p[413 5/2] n[505 11/2]) and 0 (p[413 5 ] n[642 5/2]), respectively. 

With the properties of J (icr) in mind (i.e. continuity, differentiability, orthogonality to core states) and the addition theorem at r-R1 = S y, we are satisfied that NL(K>r-R) is everywhere continuous and differentiable, and furthermore orthogonal to the core states of all muffin-tin wells except that centred at R. 

With the addition theorem (8.7) the cancellation required will take place if... 

We note that the augmented spherical Neumann Ntjl(r/St) and Bessel J (r) functions are in this case defined from a tail of radial dependence (r/S ) rather than (r/S) as in the energy-dependent orbital (8.1). The addition theorem (6.13,8.7) which represents the expansion in the sphere at q of the tail of the muffin-tin orbital centred at q must therefore be changed to include the correct radial dependences, i.e. (r/St). From (6.13,8.7) we find... 

The addition theorem states that if f(t) and g(t) have the Fourier transforms F(w) and G(w), then the function f(t)+g(t) has the Fourier transform F(uj)+G(tu). This follows easily from the definition of the transform. 

In accord with the addition theorem of modified Bessel functions we have ... 

Thus, the ealeulation of the matrix (4) reduees to the ealeulation of the matrices S S, and S°. It is very diffieult to calculate all these matriees for closely paeked media comprising scatterers comparable to the wavelength. In this case, the matrix S° can contribute significantly to the matrix (4), and all the matriees must be calculated with the eoefficients of the addition theorem in the form (7). These coefficients describe all pecuharities of the waves in the vicinity of the scatterers including the near-field effects. For low-density media, when the distances between the particles r. aj, (where the... 

The reflection matrix for a closely packed medium composed of wavelengthsized scatterers can be represented as a sum of matrices (10) with the coefficients of the addition theorem (7). These coefficients describe all the details of the field in the vicinity of any scatterer, including the near-field effects [26]. We consider the manifestations of these effects quahtatively using the field configuration near a spherical scatterer as the simplest example. 

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