Verwandte Artikel zu Theory of Differential Equations Volume 4
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Inhaltsangabe
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1902 Excerpt: ...possessed by the equation. The further conditions, sufficient to.determine whether a regular integral should or should not be associated with any root of the indicial equation, can be obtained in a form, which is mainly descriptive for the equation of general order and can be rendered completely explicit for any particular given equation. Let the equation be WM-i« ft 1Я1 «if--,-I-... + qmw-0, of characteristic index n. Let E (в) be the indicial function, and let a-be one of its zeros, so that Then, if a regular integral is to be associated with r, it must be of the form u-a (c, + c,« + c,1-I-... + c,' +...). This expression, when substituted in the equation, muet satisfy it identically, so that, after substitution, the coefficient of «+' must vanish for every value of p: and therefore /. (P) cr +/I (p)cr» +... +/, (p)V+f-0, where the number of terms in this difference-relation depends upon the actual forms of 9,, q,,.... qm. Of the coefficients/,, /lf-,/r, the first is which is of degree m--n in p; of the remainder, one at least, viz. fr-m, is of degree m in p, where g has the same significance м in §76. The successive use of this difference-relation, together with the equations for the earlier coefficients, the first of which is leads to the value« of all the quantities сй + e,, for the successive values of p; mod thus л formal expression for м is obtained that satisfies the equation. If, however, the expression is an infinite series, it has no functional significance when it diverges: that this frequently, even generally, is the case, may be inferred as follows. For if c„+i + o, with indefinite increase of /i, tends to a limit that is not infinite, so also would c„t,+...
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Reseña del editor
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1902 Excerpt: ...possessed by the equation. The further conditions, sufficient to.determine whether a regular integral should or should not be associated with any root of the indicial equation, can be obtained in a form, which is mainly descriptive for the equation of general order and can be rendered completely explicit for any particular given equation. Let the equation be WM-i« ft 1Я1 «if--,-I-... + qmw-0, of characteristic index n. Let E (в) be the indicial function, and let a-be one of its zeros, so that Then, if a regular integral is to be associated with r, it must be of the form u-a (c, + c,« + c,1-I-... + c,' +...). This expression, when substituted in the equation, muet satisfy it identically, so that, after substitution, the coefficient of «+' must vanish for every value of p: and therefore /. (P) cr +/I (p)cr» +... +/, (p)V+f-0, where the number of terms in this difference-relation depends upon the actual forms of 9,, q,,.... qm. Of the coefficients/,, /lf-,/r, the first is which is of degree m--n in p; of the remainder, one at least, viz. fr-m, is of degree m in p, where g has the same significance м in §76. The successive use of this difference-relation, together with the equations for the earlier coefficients, the first of which is leads to the value« of all the quantities сй + e,, for the successive values of p; mod thus л formal expression for м is obtained that satisfies the equation. If, however, the expression is an infinite series, it has no functional significance when it diverges: that this frequently, even generally, is the case, may be inferred as follows. For if c„+i + o, with indefinite increase of /i, tends to a limit that is not infinite, so also would c„t,+...
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