Download A Course of Higher Mathematics. Volume IV by V. I. Smirnov and A. J. Lohwater (Auth.) PDF

By V. I. Smirnov and A. J. Lohwater (Auth.)

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Additional resources for A Course of Higher Mathematics. Volume IV

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X = xm. We now form a new sequence by taking the first function from sequence (I), the second function from sequence (II), the third function from sequence (III), and so on: /(l) (X) = fti (X), /(*> (X) = tf> (*), /(3) {X) = /£» {x)t . . , /(Π)(^)=/(η")(^··· (*) We show t h a t this subsequence is now convergent at any point x = xk. In fact, let us take the point x = xk. e. all the functions /<*> (x) = /^) (*), /(*+« (x) = yg^D ( a . ) , . . e. the sequence of functions (**) is convergent at the point x = xk.

W ) . On replacing t h e integral in (33) b y t h e corresponding R i e m a n n s u m , w e obtain t h e approximate equation: rt«)-/M+* Σ K{8,8q) 1, . , n) . 7] 25 FREDHOLMS DETERMINANT We obtain the following determinant in the denominator on solving this system by Cramer's theorem [THV 8]: DnW 11 - λΚη δ -λΚΆδ, '-λκ'η1δ - λΚι2 δ, 1-λΚ22δ, -λΚ1ηδ -λΚ2ηδ .

On replacing t h e integral in (33) b y t h e corresponding R i e m a n n s u m , w e obtain t h e approximate equation: rt«)-/M+* Σ K{8,8q) 1, . , n) . 7] 25 FREDHOLMS DETERMINANT We obtain the following determinant in the denominator on solving this system by Cramer's theorem [THV 8]: DnW 11 - λΚη δ -λΚΆδ, '-λκ'η1δ - λΚι2 δ, 1-λΚ22δ, -λΚ1ηδ -λΚ2ηδ .

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