An integral formula for the derivatives of solutions of certain elliptic systems

Volume 54 / 1991

H. Renelt Annales Polonici Mathematici 54 (1991), 45-57 DOI: 10.4064/ap-54-1-45-57


 Abstract. As was shown by Polozhii and Shabat, the solutions ƒ of elliptic systems $ƒ = νƒ_{\bar{z}} + μ{\bar{ƒ_z}$ satisfy a generalized Cauchy integral formula. Here we will show that the derivatives fz, too, satisfy an integral formula. This formula, announced already in [6], rests upon the notion of generalized (— l)th powers and represents an astonishing analogue to the classical Cauchy integral formula for the derivative of an analytic function. The proof rests on an integral relation for generalized powers, which is of independent interest and which likewise represents a generalization of a classical relation.


  • H. Renelt

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