ISSN:
1420-8903
Keywords:
Primary 35B40
;
Secondary 30D05
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary The purpose of this paper is to solve the following Pythagorean functional equation:(e p(x,y) ) 2 ) = q(x,y) 2 + r(x, y) 2, where each ofp(x,y), q(x, y) andr(x, y) is a real-valued unknown harmonic function of the real variablesx, y on the wholexy-planeR 2. The result is as follows. Theorem. Suppose that each of p(x, y), q(x, y) and r(x, y) is a real-valued unknown harmonic function on R 2.The only systems of harmonic solutions of (1) are (i) $$\left\{ {\begin{array}{*{20}c} {p(x,y) = \log \left| {E(z)} \right|} \\ {q(x,y) = \operatorname{Re} (E(z))} \\ {r(x,y) = \operatorname{Im} (E(z))} \\ \end{array} } \right.$$ and (ii) $$\left\{ {\begin{array}{*{20}c} {p(x,y) = \log \left| {E(z)} \right|} \\ {q(x,y) = \operatorname{Re} (E(z)){\mathbf{ }},} \\ {r(x,y) = - \operatorname{Im} (E(z))} \\ \end{array} } \right.$$ In other words, there exists an entire function E(z) such that p(x, y) = log|E(z)|, q(x, y) = Re(E(z))and either r(x, y) = Im(E(z))or r(x, y) = −Im(E(z))and p(x, y), q(x, y) and r(x, y) satisfy (1).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02112300
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