Read e-book online A Treatise on the Theory of Bessel Functions PDF

By G. N. Watson

ISBN-10: 1603861041

ISBN-13: 9781603861045

An Unabridged, Digitally Enlarged Printing, to incorporate: The Tabulation Of Bessel capabilities - Bibliography - Index Of Symbols - checklist Of Authors Quoted, And A accomplished Index

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Extra resources for A Treatise on the Theory of Bessel Functions

Example text

J = U J . - ( z i - ai)k, pour 1 # i, j, u; = u, o'= du', d z , du; dz, . du:, dz,, . Alors 012 a ( z , - a,)u;(z) + . . + ( z n - a,)ui(z) = 1 - h(z). I l existe une forme différentielle $ de degré n - 1 sur C", à coeficients dans A, telle que Supp $ c Supp h et o - co' = d($ d z , dz, . . dz,). En effet = (dui + k d z , + ( z j - a j ) d k ) ( d u j- k d z , = (du, duj - ( z , - a,) dui dk - - ( z , - a i ) d k ) d z l . . dz, ( z j - a j )duj dk)dz, . . dz, 36 Chap. 1, ALGÈBRES NORMÉES 54 donc d z , .

Dz,). Pour tout z E Cn, posons de sorte que clji = -clij, et Supp Pi c (Supp h) u (Supp h'). On a Posons u:'(z) = u:(z) - Pi(z), u Alors u" = u = ( u , , . . , u,J, U" = . (u; , . . u;;). + 2 Vij i <; où vij est l'application de Cn dans An dont la i-ème composante est (zj - aj)aij, dont la j-ème composante est no 3 37 CALCUL FONCTIONNEL HOLOMORPHE et dont les autres composantes sont nulles. D'après le lemme 4, il existe une forme différentielle S/, de degré n - 1 sur Cn, à coefficients dans A, telle que Supp S/, c Supp h et w - du;' dz, .

Exemples. -Reprenons les exemples du 5 2, no 2. Dans les exemples 1 (algèbre des fonctions continues tendant vers O à l'infini sur un espace localement compact R) et 2 (algèbre des fonctions n fois dérivables sur (0, l)), A est régulière (cf. 5 3, no 3, exemples). On verra (chap. II, 5 3, prop. Dans l'exemple 5 (algèbre des fonctions continues dans le disque lzl < 1 et analytiques à l'intérieur), A n'est pas régulière (5 7, exerc. 6). -Soit A une algèbre de Banach unifère commutative régulière.

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A Treatise on the Theory of Bessel Functions by G. N. Watson

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