In my notes, I have an example for solving for the Airy function in the equation: $$\frac{d^2y}{dx^2}-xy=0$$ So he uses the contour integral to represent the solution (with $C$ as a yet unspecified contour): $$y(x)=\int_C e^{zx} \ \tilde{y}(z) dz$$ Then $$y''(x)=\int_C z^2 e^{zx}\ \tilde{y}(z) dz$$
The equation can be written as: $$\frac{d^2y}{dx^2}-xy=\int_C dz \ (z^2-x)e^{zx}\ \tilde{y}(z)=0$$
Now I don't understand how he got the following: $$=\Big[e^{zx}\tilde{y}(z)\Big]_{\partial C} - \int_C dz \ (z^2+\frac{d}{dz})e^{zx}\ \tilde{y}(z)=0$$
where $\partial C$ is the edge of the contour.