# How many poles does the Laplace Transform of a square wave have? a) 0 b) 1 c) 2 d) Infinitely Manhy

How many poles does the Laplace Transform of a square wave have?
a) 0
b) 1
c) 2
d) Infinitely Manhy
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Solution:
The Fourier series for square wave is $f\left(x\right)=\frac{4}{\pi }\sum _{i=1,3,5,\dots }^{\mathrm{\infty }}\frac{1}{n}\mathrm{sin}\left(\frac{n\pi x}{L}\right)$
The Laplace transform is $L\left\{\mathrm{sin}\left(ax\right)\right\}=\frac{a}{{x}^{2}+{a}^{2}}$
The function is $f\left(x\right)=\frac{4}{\pi }\left[\mathrm{sin}\left(\frac{\pi x}{L}\right)+\frac{1}{3}\mathrm{sin}\left(\frac{3\pi x}{L}\right)+\frac{1}{5}\mathrm{sin}\left(\frac{5\pi x}{L}\right)+\dots \right]$
Apply Laplace transform:
Conclusion:
$Lf\left(x\right)=\frac{4}{\pi }\left[L\left\{\mathrm{sin}\left(\frac{\pi x}{L}\right)\right\}+\frac{1}{3}L\left\{\mathrm{sin}\left(\frac{3\pi x}{L}\right)\right\}+\frac{1}{5}L\left\{\mathrm{sin}\left(\frac{5\pi x}{L}\right)\right\}+\dots \right]$
$=\frac{4}{\pi }\left[\frac{\frac{\pi }{L}}{{x}^{2}+{\left(\frac{\pi }{L}\right)}^{2}}+\frac{1}{3}\left(\frac{\frac{3\pi }{L}}{{x}^{2}+{\left(\frac{3\pi }{L}\right)}^{2}}\right)+\frac{\frac{5\pi }{L}}{{x}^{2}+{\left(\frac{5\pi }{L}\right)}^{2}}+\dots \right]$
$=\frac{4}{\pi }\left[\frac{\frac{\pi }{L}}{{x}^{2}+{\left(\frac{\pi }{L}\right)}^{2}}+\frac{\frac{\pi }{L}}{{x}^{2}+{\left(\frac{3\pi }{L}\right)}^{2}}+\frac{\frac{\pi }{L}}{{x}^{2}+{\left(\frac{5\pi }{L}\right)}^{2}}\dots \right]$
$=\frac{4}{L}\left[\frac{1}{{x}^{2}+{\left(\frac{\pi }{L}\right)}^{2}}+\frac{1}{{x}^{2}+{\left(\frac{3\pi }{L}\right)}^{2}}+\frac{1}{{x}^{2}+{\left(\frac{5\pi }{L}\right)}^{2}}+\dots \right]$
Hence, as there are no real singularities, therefore the number of poles are 0.