Download e-book for iPad: A Course of Higher Mathematics. Volume IV by V. I. Smirnov and A. J. Lohwater (Auth.)

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

ISBN-10: 1483167232

ISBN-13: 9781483167237

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Additional resources for A Course of Higher Mathematics. Volume IV

Sample text

With this definition of the adjoint equation the condition for (79) to be soluble must be written as b Ju)(i)/(i)d« = 0, a (84) where co(s) is any solution of equation (83). The treatment of [7], on the basis of which the fundamental theorems were proved, was first given by Predholm in 1903. The theorems proved above are entirely analogous to the theorems on the solution of systems of linear algebraic equations [ΙΠ, 8, 9 and 10]. 11. Fredholm minors. The above argument enables us to obtain the complete set of linearly independent eigenfunctions of the equation with kernel K(st t) corresponding to a given eigenvalue.

The treatment of [7], on the basis of which the fundamental theorems were proved, was first given by Predholm in 1903. The theorems proved above are entirely analogous to the theorems on the solution of systems of linear algebraic equations [ΙΠ, 8, 9 and 10]. 11. Fredholm minors. The above argument enables us to obtain the complete set of linearly independent eigenfunctions of the equation with kernel K(st t) corresponding to a given eigenvalue. We shall merely give the results, without dwelling on the prooff.

Hence a number N exists such t h a t \iP (ΧΊ - fg(x')\ < e with p and q > N, (β) if x' is a point of the finite set r / . We shall assume t h a t the point x' appearing in formula (a) is a point of finite set τ', and we write down the inequality \fP (x) - /,(*) I < I fP (*) - fP (*')| + \fP (*') - /, (*')l + + \fq(x')-fq(X)\, (y) which follows directly from (a). For any position of x on [a, b] we can indicate an x' belonging to τ ' such t h a t \fn(x) — fn{x') | < ε for any n. This x' will be one end of the subinterval to which x belongs.

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


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