My question essentially has to do with the derivative of a Contour Integral's parameterized curve. $$\frac{\partial}{\partial x} \oint_{\partial \Omega(x)} f(n, x) \; \mathrm{d}n$$ to be exact. Where $\partial \Omega(x)$ is a Jordan curve which is differentiable for any $x \in \mathbb{C}$, and $f(n, x)$ is any function integrable around $\partial \Omega(x)$ in respect to $n$.
My work has essentially gotten down to:
$$\frac{\partial}{\partial x} \oint_{\partial \Omega(x)} f(n, x) \; \mathrm{d}n = \int_{\partial \Omega(x)} f_x(n, x) \; \mathrm{d}n$$ $$+\int_{0}^{2\pi} \gamma_x(\theta, x) \gamma_\theta(\theta, x) f_n(\gamma(\theta, x), x) \; \mathrm{d}\theta$$ $$+\int_{0}^{2\pi} \gamma_{\theta x}(\theta, x) f(\gamma(\theta, x), x) \; \mathrm{d}\theta.$$
Where $\gamma$ is the parameterized curve of $\partial \Omega$, and $f_x(n, x) = \frac{\partial f(n, x)}{\partial x}$. I am unsure of how to simplify this further or if this is even a decent approach. Does anybody have a good resource for this?
My goal is to write this derivative as multiple contour integrals, without using $\gamma$ within them.
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