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I am trying to find the fourier transform of $$\frac{\sin(ax)}{x}$$ for $a >0$. This is clearly an even function so we only need to do the real part, but I could not evaluate $$\int_{-\infty}^{\infty} \cos(kx) \sin(ax) \frac{dx}{x}. $$ Can someone please help me? I tried wolfram alpha but it only gave an answer which is not very helpful. |
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We have that $$\cos(kx)\sin(ax) = \frac{\sin((a-k)x)}{2}+\frac{\sin((a+k)x)}{2}$$ So put in a different form we have that $$\int_{\Bbb{R}}\frac{\cos(kx)\sin(ax)}{x}dx = \int_{\Bbb{R}}\frac{\sin((a-k)x)+\sin((a+k)x)}{2x} ~~dx$$ We also know that because of the identity: $$\int_{\Bbb{R}^+}\frac{f(t)}{t} = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$$ And that $\mathcal{L}[\sin(ax)](s) = \frac a {s^2 +a^2},$ where $\mathcal{L} $ represents the Laplace transform that $$\int_{\Bbb{R}}\frac {\sin(x)}{x} = \int_{\Bbb{R}}\frac a {x^2 + a^2} = \arctan(\frac x a)\Big|_{-\infty}^\infty = \pi \operatorname{sgn}(a)$$ Making the resulting integral: $$\frac{\pi(\operatorname{sgn}(a-k)+\operatorname{sgn}(a+k))}2$$ |
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To avoid confusion, this is the version of the Fourier transform that I will use in my answer: $$f(x)\longmapsto \hat{f}(\omega)=\int_{-\infty}^\infty e^{-i\omega x}f(x)\,dx.$$ One way is to determine the Fourier transform of $\sin(ax)$, and then using the fact that division by $-ix$ corresponds to integration in the frequency domain. The transform of the sine function is $$\sin(ax)=\frac{1}{2i}(e^{iax}-e^{-iax})\longmapsto\hat{f}(\omega)=\frac{1}{2i}(2\pi\delta(\omega-a)-2\pi\delta(\omega+a))=\frac{\pi}{i}(\delta(\omega-a)-\delta(\omega+a)),$$ where $\delta(\omega)$ denotes the Dirac delta. We now get $$g(x)=\frac{\sin(ax)}{x}\longmapsto \hat{g}(\omega)=-\pi\int_{-\infty}^\omega (\delta(u-a)-\delta(u+a))\,du=\pi(H(\omega+a)-H(\omega-a)),$$ where $H(\omega)$ is the Heaviside step function. |
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