Let u, v R be two real valued functions defined on an open set C...

 

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Let u, v → R be two real valued functions defined on an open set C R² satisfying the following: (i) All the partial derivatives of u and u up to second order exist and are continuous at all points of . (ii) The functions u and u satisfy the Cauchy-Riemann equations: Əv dy du Əx = 0 du ду = = Show u and u (both) satisfy Laplace's equation: Ju J²u + Əx²2 Əy² Əv дх and Fu əx² J'v dy² = 0. Let u, v → R be two real valued functions defined on an open set C R² satisfying the following: (i) All the partial derivatives of u and u up to second order exist and are continuous at all points of . (ii) The functions u and u satisfy the Cauchy-Riemann equations: Əv dy du Əx = 0 du ду = = Show u and u (both) satisfy Laplace's equation: Ju J²u + Əx²2 Əy² Əv дх and Fu əx² J'v dy² = 0.