caa a22 4500 465825230 CHVBK 20180323112147.0 cr unu---uuuuu 170327e19901201xx s 000 0 eng 10.1007/BF02112300 doi (NATIONALLICENCE)springer-10.1007/BF02112300 Haruki H. Department of Pure Mathematics, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada aut A new Pythagorean functional equation [Elektronische Daten] [H. Haruki] 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). Birkhäuser Verlag, 1990 Primary 35B40 nationallicence Secondary 30D05 nationallicence aequationes mathematicae Birkhäuser-Verlag 40/1(1990-12-01), 271-280 0001-9054 40:1<271 1990 40 10 https://doi.org/10.1007/BF02112300 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02112300 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Haruki H. Department of Pure Mathematics, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada aut NATIONALLICENCE 773 0- aequationes mathematicae Birkhäuser-Verlag 40/1(1990-12-01), 271-280 0001-9054 40:1<271 1990 40 10 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer