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Integrating around a triangular contour for Fresnel integral

$$\int^\infty_0\frac{\cos(x)}{\sqrt{x}}\,dx =\int^\infty_0\frac{\sin(x)}{\sqrt{x}}\,dx =\sqrt{\frac{\pi}{2}}$$ $$\textit{solution}$$ Consider the following function $$f(z)=z^{-1/2}\,e^{iz}$$ Where we choose the principle root for \( z^{-1/2}=e^{-1/2\log(z)}\). By integrating around the following contour $$\int_{C_r}f(z)\,dz+\int_{r}^R f(x)\,dx+\int_{\gamma}f(z)\,dz+\int^{iR}_{ir}f(x)\,dx = 0$$ Taking the integral around the small quarter circle with $r\to 0$ $$\left| … Continue reading

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