Word Count : 419

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To prove that \( u = x^2 - y^2 \) and \( v = \frac{y}{x^2 + y^2} \) are harmonic functions, we need to verify that both \( u \) and \( v \) satisfy Laplace's equation: \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \quad \text{and} \quad \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0 \] ### 1. Proof for \( u = x^2 - _______ __________ _________ ____ ____ _____ ______ ______.
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