91Ó°ÊÓ

The Dirichlet theorem for Fourier series formula gives an expansion of a periodic function f (x) in terms of an horizonless sum of sines and cosines. It's used to putrefy any periodic function or periodic signal into the sum of a group of straightforward oscillating functions, videlicet sines and cosines.

The given functions are

$\begin{array}{l}\underset{0}{\overset{\mathrm{∞}}{∫}}\frac{1−\cos \mathrm{Ï€}\mathrm{Î\pm }}{\mathrm{Î\pm }}\sin \mathrm{Î\pm }d\mathrm{Î\pm }=\frac{\mathrm{Ï€}}{2}\\ \underset{0}{\overset{\mathrm{∞}}{∫}}\frac{1−\cos \mathrm{Ï€}\mathrm{Î\pm }}{\mathrm{Î\pm }}\sin \mathrm{Î\pm }d\mathrm{Î\pm }=\frac{\mathrm{Ï€}}{4}\end{array}$

There need to prove the given functions using problem 17.

The problem 17 says that

$\begin{array}{c}f\left(x\right)=\frac{2}{\mathrm{Ï€}}{∫}_0^{\mathrm{∞}}\frac{1−\cos \left(\mathrm{Î\pm }\mathrm{Ï€}\right)}{\mathrm{Î\pm }}\sin \left(\mathrm{Î\pm }x\right)d\mathrm{Î\pm }\\ =\left\{\begin{array}{l}\begin{array}{ll}−1& ,x∈[−\mathrm{Ï€},0]\end{array}\\ \begin{array}{ll}1& ,x∈[0,\mathrm{Ï€}]\end{array}\\ \begin{array}{ll}0,& ,\mathrm{else}\end{array}\end{array}\right.\end{array}$

Firstly, For$x=1$:

$\begin{array}{c}1=\frac{2}{\mathrm{Ï€}}{∫}_0^{\mathrm{∞}}\frac{1−\cos \left(\mathrm{Î\pm }\mathrm{Ï€}\right)}{\mathrm{Î\pm }}\sin \mathrm{Î\pm }d\mathrm{Î\pm }\\ \frac{\mathrm{Ï€}}{2}={∫}_0^{\mathrm{∞}}\frac{1−\cos \left(\mathrm{Î\pm }\mathrm{Ï€}\right)}{\mathrm{Î\pm }}\sin \mathrm{Î\pm }d\mathrm{Î\pm }\end{array}$

For $x=\mathrm{Ï€}$Dirichlet theorem says that the series converges to $\frac{1}{2}$, so

$\begin{array}{l}\frac{1}{2}=\frac{2}{\mathrm{Ï€}}{∫}_0^{\mathrm{∞}}\frac{1−\cos \left(\mathrm{Î\pm }\mathrm{Ï€}\right)}{\mathrm{Î\pm }}\sin \left(\mathrm{Î\pm }\mathrm{Ï€}\right)d\mathrm{Î\pm }\\ \frac{\mathrm{Ï€}}{4}={∫}_0^{\mathrm{∞}}\frac{1−\cos \left(\mathrm{Î\pm }\mathrm{Ï€}\right)}{\mathrm{Î\pm }}\sin \left(\mathrm{Î\pm }\mathrm{Ï€}\right)d\mathrm{Î\pm }\end{array}$

Thus, using $x=\mathrm{Ï€}$and $x=1$in the problem 17 the given functions are proved.