Abstract
Formula1 (0. 1. 4) permits us to associate with each solution U of the differential equation (0. 1. 1b) an analytic function g(z) of a complex variable. Conversely, the question arises of determining from a given g(z) the corresponding solution of the equation (0. 1. 1b). This can be done using the so-called integral operator of the first kind which will be discussed in §§ 1–3. We shall express solutions U in terms of an arbitrary function of a complex variable f(z). Next, f(z) will be expressed in terms of a function g(z) which essentially coincides with U(z, 0), see (2. 1) and (2–5).
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References
(0. 1. 4) = formula (4) of the introduction. (2. 1) = formula (1) of § 2 of the present chapter.
U” (0,..., 0) = an n dimensional neighborhood of the point (0,..., 0).
If t -1 E z * is continuous for t = 0 then s1 can pass through t = 0.
Equation (2) is more complicated than (6), but one solution of (2) generates a family of solutions of (7).
In some previous papers the symbol p has been used instead of C 2 .
U(z, z*) denotes the function of two independent variables z and z* which we obtain when we continue U(x + iy, x-iy) to complex values of the variables x and y.
The approach discussed here (and in Section 6) for the equation Δ ψ + ψ = 0 has already been developed in [B.4], [B.6].
In [V.3]–[V.11] applications of the integral operator to the solution of boundary value problems are discussed. Since these applications do not lie within the scope of the present survey, they are not reviewed here, but for a few exceptions. A summary of investigations in [V.3]–[V.9], [V.11] is given in [V.10].
We assume that the coefficients A, C of L are entire functions. If this is not the case, the results of this section have to be modified in an obvious manner.
In the case of multiply connected domains the same result holds. However, Re [c2(g)] could be multivalued, while g is single valued.
See [L.15] p. 123ff.
As long as x and y are real, z* = z̄, but if x and y are allowed to assume complex values z and z* become two independent complex variables.
However, it should be emphasized that the functions B x (z 1 , z 2 )appearing in (11) will, in general, depend on n.
Eichler’s considerations refer to a more general class of equations, namely, N can be singular. See for details Chapter V.
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© 1969 Springer-Verlag Berlin Heidelberg
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Bergman, S. (1969). Differential equations in two variables with entire coefficients. 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_2
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DOI: https://doi.org/10.1007/978-3-662-38025-3_2
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