Wednesday, 5 June 2019

The infinite integral of $frac{sin x}{x}$ using complex analysis

The problem i came across is the evaluation of $$\int_0^\infty\frac{\sin x}{x}\,dx$$ I chose the function $f(z) = \dfrac{e^{iz}}{z}$ and took a contour of $[\varepsilon , R ] + [R , R+iy] + [-R+iy , R+iy] + [-R,-R+iy]+[-R, -\varepsilon]$ . The problem is how do I continue now to find integrals on each of these segments ?

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real analysis - How to find $lim_{hrightarrow 0}frac{sin(ha)}{h}$

How to find $\lim_{h\rightarrow 0}\frac{\sin(ha)}{h}$ without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...