\[
V(x, y)=e^{x} \sin y-x^{2}+y^{2}+2 x y
\]
Calculer \( \Delta V(x, y) \). Conclure
Déterminer la fonction \( U(x, y) \) telle que \( f(z)=U(x, y)+i V(x, y) \) soit analytique et vérifie la condition \( f(0)=1 \).

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Answer

\( \Delta V(x, y) = \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} = (e^x \sin y - 2) + (-e^x \sin y + 2) = 0 \)

Steps

Step 1 ：\( \frac{\partial V}{\partial x} = e^x \sin y - 2x + 2y \)

Step 2 ：\( \frac{\partial V}{\partial y} = e^x \cos y + 2y + 2x \)

Step 3 ：\( \Delta V(x, y) = \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} = (e^x \sin y - 2) + (-e^x \sin y + 2) = 0 \)