91Ó°ÊÓ

An expansion of a periodic function, f (x), in terms of a horizonless sine and cosine sum is provided by the formula for the Fourier series. Any periodic function or periodic signal is putrefied using this technique into the sum of a collection of simple oscillating functions, such as sines and cosines.

The given function is

$f\left(x\right)=\left\{\begin{array}{l}\begin{array}{ll}1,& 0<x<1\end{array}\\ \begin{array}{ll}0,& x>1\end{array}\end{array}\right.$

There need to represent the given as Fourier sine integral and Fourier cosine integral.

The Fourier cosine transform formula is

$f_c\left(x\right)=\sqrt{\frac{2}{\mathrm{Ï€}}}{∫}_0^{\mathrm{∞}}{g}_c\left(x\right)\cos \mathrm{Î\pm }xd\mathrm{Î\pm }$

The cosine integral is:

$\begin{array}{c}g\left(\mathrm{Î\pm }\right)=\sqrt{\frac{2}{\mathrm{Ï€}}}{∫}_0^{\mathrm{∞}}f\left(x\right)\cos \left(\mathrm{Î\pm }x\right)dx\\ =\sqrt{\frac{2}{\mathrm{Ï€}}}{∫}_0^1\cos \left(\mathrm{Î\pm }x\right)dx\\ ={\sqrt{\frac{2}{\mathrm{Ï€}}}\frac{\sin \left(\mathrm{Î\pm }x\right)}{\mathrm{Î\pm }}|}_0^1\\ =\sqrt{\frac{2}{\mathrm{Ï€}}}\frac{\sin \mathrm{Î\pm }}{\mathrm{Î\pm }}\end{array}$

Substitute the value of $g\left(\mathrm{Î\pm }\right)$in Fourier cosine transform formula$f_c\left(x\right)=\frac{2}{\mathrm{Ï€}}{∫}_0^{\mathrm{∞}}\frac{\sin \mathrm{Î\pm }}{\mathrm{Î\pm }}\cos \left(\mathrm{Î\pm }x\right)d\mathrm{Î\pm }$

The Fourier sine transform formula is

$f_s\left(x\right)=\sqrt{\frac{2}{\mathrm{Ï€}}}{∫}_0^{\mathrm{∞}}{g}_s\left(\mathrm{Î\pm }\right)\sin \mathrm{Î\pm }xd\mathrm{Î\pm }$

The sine integral is:

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

Substitute the value of $g\left(\mathrm{Î\pm }\right)$in Fourier sine transform formula

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

Thus, the Fourier cosine transform is $f_c\left(x\right)=\frac{2}{\mathrm{Ï€}}{∫}_0^{\mathrm{∞}}\frac{\sin \mathrm{Î\pm }}{\mathrm{Î\pm }}\cos \left(\mathrm{Î\pm }x\right)d\mathrm{Î\pm }$and the Fourier sine transform is$f_s\left(x\right)=\frac{2}{\mathrm{Ï€}}{∫}_0^{\mathrm{∞}}\frac{1−\cos \mathrm{Î\pm }}{\mathrm{Î\pm }}\sin \left(\mathrm{Î\pm }x\right)d\mathrm{Î\pm }$.