By V. I. Smirnov and A. J. Lohwater (Auth.)
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Additional resources for A Course of Higher Mathematics. Volume IV
N(s, £) for any n. The proof of (56) therefore reduces to the proof of (55x), where d*(sf t) is the right-hand side of (56). We notice first of all that, if we transpose two of the xt or two of the yi in the symbol on the left-hand side of (49), the value of the determinant on the right-hand side only changes sign, because the operation amounts to interchanging two rows or columns of this determinant. On expanding the determinant of (56) by elements of 7] FREDHOLM'ß DETERMINANT 29 the first row and taking into account the remark just made, we can write: is, tv .
Our present purpose is to distinguish the relevant class of unbounded kernels. The general theory of integral equations, including both the case of unbounded kernels and the case of an infinite domain of integration, will be given in Vol. V on the basis of a more general concept of the integral (the Lebesgue integral). For definiteness, we shall carry out the discussion for the case of a plane. The whole of it is readily extended to the case of any 7i-dimensional space or of integration over a surface.
Such skew-symmetric kernels only have pure imaginary eigenvalues. 4. A further example may be mentioned, of a degenerate kernel having no eigenvalues. Let Here, n = 1 and the only element a# will be Homogeneous systems (91x) and (922) give us xx = yx = 0, and the homogeneous equation only has a zero solution with any λ. The equation for the eigenvalues reduces here to the absurdity 1 = 0. 14. Generalization of the results obtained. We have assumed in our description of the theory of integral equations t h a t the required function (p(s) and the function f(s) are functions of a single independent variable which can vary in an interval [a, 6].
A Course of Higher Mathematics. Volume IV by V. I. Smirnov and A. J. Lohwater (Auth.)