Let $O$ be a non-empty subset of $\mathbb{C}$ (the complex plane), and let $g : O \to \mathbb{C}$ be a holomorphic function that satisfies the differential equation $g^{\prime\prime}(z) - zg(z) = 0$ for every $z \in O$. Prove that for all differentiable curves $U$ in $O$ with the same endpoints, the value $$\int_U g(x)^2 dx$$ is the same.
Complex derivatives of line integrals
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complex-analysis
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0Where did you come across this? It seems like it might be homework... So, where are you stuck? – 2011-01-24
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0not even sure where to begin :-S – 2011-01-24
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5The differential equation in question is the [Airy equation](https://en.wikipedia.org/wiki/Airy_function). Its two linear independent solutions (functions $\mathrm{Ai}(z)$ and $\mathrm{Bi}(z)$) are holomorphic on the whole complex plane. Hence $g(z)^2$ is also a holomorphic function and the integral has the same value by the residue theorem. – 2013-06-23