A Contribution to the Theory of Linear Homogeneous Geometric Difference Equation
A Contribution to the Theory of Linear Homogeneous Geometric Difference Equation
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(l-^, (1) converges for |a;|>/? at least. By an analytic continuation performed by aid of the difference equation itself we may obtain information about the analytic character of this function for |a:|<^ also. Substituting the ex- pression (1) for u{x) in the difference equation (5) of § 2 transformed by aid of (3) of § 2 we obtain the following difference equation for v{x) h^i(l-a:r)(l--^|... (l--^^' where P/i(ar) is a linear combination of the functions Gii — \(i=n — h, n — h-{-l, ... N) with coefficients that are polynomials in x, which do not depend on r. (It may be remembered that we have supposed G„(a:)=l, compare pp. — so- il and 12. ) The functions Gi(x) are by hypothesis holomorphic in the neighbourhood of infinity. Now we suppose, moreover, that they admit of an analytic continuation to every point of the x-plane, perhaps with the exception of the origin, and that their only singularities are at the following points ^V^2^^3'--'^V-'; ' (3) which may occur in finite or infinite'number.
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You can find similar books in the "Read Also" column, or choose other free books by Folke Ryde to read online
To quickly assess the difficulty of the text, read a short excerpt:
(l-^, (1) converges for |a;|>/? at least. By an analytic continuation performed by aid of the difference equation itself we may obtain information about the analytic character of this function for |a:|<^ also. Substituting the ex- pression (1) for u{x) in the difference equation (5) of § 2 transformed by aid of (3) of § 2 we obtain the following difference equation for v{x) h^i(l-a:r)(l--^|... (l--^^' where P/i(ar) is a linear combination of the functions Gii — \(i=n — h, n — h-{-l, ... N) with coefficients that are polynomials in x, which do not depend on r. (It may be remembered that we have supposed G„(a:)=l, compare pp. — so- il and 12. ) The functions Gi(x) are by hypothesis holomorphic in the neighbourhood of infinity. Now we suppose, moreover, that they admit of an analytic continuation to every point of the x-plane, perhaps with the exception of the origin, and that their only singularities are at the following points ^V^2^^3'--'^V-'; ' (3) which may occur in finite or infinite'number.
What to read after A Contribution to the Theory of Linear Homogeneous Geometric Difference Equation?
You can find similar books in the "Read Also" column, or choose other free books by Folke Ryde to read online
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