CVRC4

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Evaluate \int _{0}^{\infty }{\frac  {\sin x}{x}}dx\,

Use the function {\frac  {e^{{ix}}}{x}}\, in the integrand and take the imaginary part later.

\int _{0}^{\infty }{\frac  {e^{{ix}}}{x}}dx={\frac  {1}{2}}{\mathrm  {p.v.}}\int _{{-\infty }}^{\infty }{\frac  {e^{{ix}}}{x}}dx\,

Use this formula:

{\mathrm  {p.v.}}\int _{{-\infty }}^{\infty }f(x)dx=\pi i\sum _{{k=1}}^{m}{\mathrm  {Res}}\{f(z);b_{k}\}+2\pi i\sum _{{k=1}}^{n}{\mathrm  {Res}}\{f(z);a_{k}\}\, where a_{k}\, are not real and b_{k}\, are.

{\frac  {1}{2}}{\mathrm  {p.v.}}\int _{{-\infty }}^{\infty }{\frac  {e^{{ix}}}{x}}dx={\frac  {1}{2}}\pi i\,

\int _{0}^{\infty }{\frac  {\sin x}{x}}dx={\mathrm  {Im}}{\frac  {1}{2}}{\mathrm  {p.v.}}\int _{{-\infty }}^{\infty }{\frac  {e^{{ix}}}{x}}dx={\mathrm  {Im}}{\frac  {\pi i}{2}}={\frac  {\pi }{2}}\,


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