Abstract
In the following we shall explain the basic ideas of the theory of integral operators generating solutions of linear partial differential equations with analytic coefficients.
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That is, solutions which are real for real values of x and y.
(I. 3. 4) = formula (4) of Chapter I, § 3.
In two dimensions these equations reduce to the Cauchy-Riemann equations (with the usual roles of x and y interchanged).
A harmonic vector corresponding to the associate f(u, ζ) is a vector (H 1, H 2, H 3) whose H 1-component has f(u, ζ) as its B 2 -associate. The B 2 -associates of H 2 and H 3 are equal to f(u, ζ) multiplied by simple polynomials in ζ and ζ -1. H 2 and H 3 are determined by H 1 up to the real and the imaginary parts of an analytic function of the complex variable y + iz. See for details p. 81ff.
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© 1969 Springer-Verlag Berlin Heidelberg
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Bergman, S. (1969). Introduction. In: Integral Operators in the Theory of Linear Partial Differential Equations. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol N. F., 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-38025-3_1
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DOI: https://doi.org/10.1007/978-3-662-38025-3_1
Publisher Name: Springer, Berlin, Heidelberg
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