Frobenius solution

In particular the even chain terminates when / is even, while the odd chain terminates when l is odd. The corresponding Frobenius solution then simplifies to a polynomial. It is called the Legendre polynomial of degree /, or Pi(x). The modified form of equation (16) becomes... 

A key observation for our purposes here is that the numerical computation of invariant measures is equivalent to the solution of an eigenvalue problem for the so-called Frobenius-Perron operator P M - M defined on the set M. of probability measures on F by virtue of... 

There are two procedures available for solving this differential equation. The older procedure is the Frobenius or series solution method. The solution of equation (4.17) by this method is presented in Appendix G. In this chapter we use the more modem ladder operator procedure. Both methods give exactly the same results. 

The eigenvalues and eigenfimctions of the orbital angular momentum operator may also be obtained by solving the differential equation I ip = Xh ip using the Frobenius or series solution method. The application of this method is presented in Appendix G and, of course, gives the same results... 

Equation (6.24) may be solved by the Frobenius or series solution method as presented in Appendix G. However, in this chapter we employ the newer procedure using ladder operators. 

The Frobenius or series solution method for solving equation (G.l) assumes that the solution may be expressed as a power series in x... 

In Chapters 4, 5, and 6 the Schrodinger equation is applied to three systems the harmonie oseillator, the orbital angular momentum, and the hydrogen atom, respectively. The ladder operator technique is used in each case to solve the resulting differential equation. We present here the solutions of these differential equations using the Frobenius method. 

It is often possible to find a solution of homogeneous differential equations in the form of a power series. According to Frobenius, the power series should have the general form... 

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 ... 

This equation can be solved by the Frobenius method, assuming a series solution... 

For regular singular points, a series solution of the differential equation can be obtained by the method of Frobenius. This is based on the following generalization of the power series expansion ... 

Linear homogeneous equations containing nonconstant coefficients were not treated, except for the elementary Euler-Equidimensional equation, which was reduced to a constant coefficient situation by letting x = exp(r). In the next chapter, we deal extensively with the nonconstant coefficient case, starting with the premise that all continuous solutions are in fact representable by an infinite series of terms, for example ejqj (jc) = 1 + x + x /2 + x /3 + . This leads to a formalized procedure, called the Method of Frobenius, to find all the linearly independent solutions of homogeneous equations, even if coefficients are nonconstant. 

Use the method of Frobenius to find the two linearly independent solutions when a = p = 2. Introducing... 

The method of Frobenius can be applied to Eq. 3.143 (Bessel s equation) to yield two linearly independent solutions, which are widely tabulated as... 

The reader can see, had we applied the method of Frobenius, the second solution would take the form... 

Thus, this function is inadmissible in the domain of 2 = 0, so take Bq = 0. This same conclusion would have been reached following a Frobenius analysis, since the series solution would obviously contain ln(x) as taught in the previous sections. 

Apply the method of Frobenius and show that the only solution that is finite at jc = 0 has the first few terms as... 

We have seen in Chapter 3 that finite difference equations also arise in Power Series solutions of ODEs by the Method of Frobenius the recurrence relations obtained there are in fact finite-difference equations. In Chapters 7 and 8, we show how finite-difference equations also arise naturally in the numerical solutions of differential equations. 

Another exceptional case occurs when repeated roots arise. This can be treated in a manner analogous to the case for repeated roots in the Method of Frobenius (CASE II). Thus, the second solution is obtained by taking the limit... 

This does not react as a diazo-compound unless a mineral acid is added to the solution. Bamberger called it an isodiazo -compound, which isomerises to a diazo-compound. By the action of hot concentrated potassium hydroxide solution on Griess s diazobenzene potassium (see p. 771), Schraube and Schmidt obtained an isomeric compound giving the diazo-reaction with acid. Von Pechmann and L. Frobenius found that the silver salt of /)-nitrodiazo-benzene with methyl iodide does not give a nitrosamine but the methyl ether CeH4(N02)-N N-0CH3, so that the nitrosamine formula appeared to represent a tautomeric form of diazobenzene. 

There are tabulated solutions of Bessel s equation [6,9,10,11], and the standard approach is to solve by comparison. Equation 3.76 to Equation 3.78 could be solved by using the Frobenius series (Equation 3.65), but that approach may be too work intensive. 

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