Equation indicial

For some differential equations, the two roots and S2 of the indicial equation differ by an integer. Under this circumstance, there are two possible outcomes (a) steps 1 to 6 lead to two independent solutions, or (b) for the larger root 5i, steps 1 to 6 give a solution mi, but for the root S2 the recursion relation gives infinite values for the coefficients a beyond some specific value of k and therefore these steps fail to provide a second solution. For some other differential equations, the two roots of... 

Equating the coefficient of to zero, we obtain the indicial equation... 

This equation cannot be solved by expansion in series, as the coefficients of S(p) and its first derivative result in a singularity at p = 0. Because this point is regular, the substitution Sip) = ps (p) is suggested. If the coefficient of p 2 is set equal to zero, the resulting indicial equation is... 

This equation, which determines the powers of the Frobenius series is called the indicial equation and it has solutions m = 0orm = l. The coefficient of the next power must also vanish ... 

For m = 0 the indicial equation yields one function /( )/ 1/ 2 that is approximately equal to a constant for small values of , and one finds that there is another function /( )/ /2 that is approximately proportional to ln for small values of . Similarly one obtains two functions g(jg)/g1 2 with corresponding behavior for small values of g. Of these functions we accept only those that are approximately equal to a constant (different from zero) for small values of and g, respectively, since otherwise x would not be finite everywhere. 

Unless b = 1 the roots of the indicial equation are distinct and provided the ratio an+i(c)/an(c) does not become infinite then this analysis leads to two independent solutions. (A careful calculation is also required if the ratio becomes indeterminate). We will refer to these solutions as the unexceptional cases. 

Since we are interested only in acceptable wave functions, we shall ignore negative values of s. For this reason we could assume F( ) to contain only positive powers of (. Occasionally, however, the indicial equation leads to non-integral values of s, in which case the treatment is greatly simplified by the substitution 17-20. 

The first of these, 17-22a, is the indicial equation. From it we see that s is equal to - -m or —to, inasmuch as a0 is not equal to zero. In order to obtain a solution of the form of Equation 17-20 which is finite at the origin, we must have s positive, so that we choose s = + to. This value of s inserted in Equation 17-226 leads to the conclusion that ai must be zero. Since the general recursion relation 17-22c connects coefficients whose subscripts differ by two, and since ai is zero, all odd coefficients are zero. The even coefficients may be obtained in terms of Oo by the use of 17-22c. 

The equation has singular points at i = 1, both of which are regular points (sec Sec. 17), so that it is necessary to discuss the indicial equation at each of these points. In order to study the behavior near z = +1, it is... 

The singular points, which are regular points, have now been shifted to the points 0 and 1 of x, so that the indicial equation must be obtained at each of these points. Making the substitution T(x) = x G(x), we find by the procedure of Section 17 that s equals % K — M, while the substitution... 

The finite nuclear models of Section 3.1 have Zq = 0 so that U ) is finite at r = 0. The indicial equation reduces to... 

There are 27 terms in the sum, but each term vanishes unless i = j. Whenever a Kronecker delta is present and summation occurs over both indicies, equate both indicies and remove the summation over one of the indicies. Hence,... 

The possible values of a are obtained from the indicial equation, which is based on tbe presumption that ao is the first nonzero coefficient in the series (8.121). 

Setting A = 0 in the recursion relation and noting that a-2 = 0 (ao is the first nonvanishing coefficient), we obtain the indicial equation... 

As stated earlier, the first stage is to find the values for c, through an indicial equation. This is obtained by inspecting the coefficients of the lowest powers in the respective series expansions. Consider Eq. 3.29 with P x), Qix) given by Eqs. 3.30 and 3.31 first, perform the differentiations... 

The indicial equation was found by inspecting the coefficients of the lowest... 

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