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Consider the following integration equation: $\int_{0}^{\infty}f(x) \cos\left(\alpha x\right)dx=\frac{e^{-\alpha}\sin\left(\alpha\right)}{\alpha}$

I want to compute $f(x)$. For this, I use Fourier integration. But the results missed. Please help me.

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  • $\begingroup$ what is the result? $\endgroup$ – tired Oct 17 '16 at 14:59
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    $\begingroup$ by the way, the lhs is an even function of $\alpha$ whereas the rhs hasn't this symmetry. this smells a bit fishy.. $\endgroup$ – tired Oct 17 '16 at 15:05
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    $\begingroup$ Did you mean to have a $|\alpha$ in the exponent on the right hand side? In any case, under the assumption that the question is sloppily written, you can extend $f$ to $x<0$ by making it even, then the above is the Fourier transform of the modified $f$ (or a scaled version), and the problem reduces down to a Fourier inversion. $\endgroup$ – copper.hat Oct 17 '16 at 15:52
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We will restrict ourselves to $\xi>0$. Let us extend $f$ even across the $y$-axis to a symmetric function $f(x) = f(-x)$. Then we see \begin{align} \sqrt{2\pi}\hat f(\xi)=&\ \int^\infty_{-\infty} e^{- i x\xi} f(x)\ dx = \int^\infty_0 e^{-i x\xi}f(x)\ dx + \int^0_{-\infty} e^{- i x\xi}f(x)\ dx\\ =&\ 2\int^\infty_0 \cos(x\xi) f(x)\ dx \end{align} which means we are trying to find an even function $f$ such that \begin{align} \int^\infty_0 \cos(x\xi) f(x)\ dx = \sqrt{\frac{\pi}{2}} \hat f(\xi) = \frac{e^{-\xi}\sin \xi}{\xi} \end{align} for all $\xi>0$ or equivalently \begin{align} \hat f(\xi) = \sqrt{\frac{2}{\pi}}\frac{e^{-\xi}\sin \xi}{\xi} \end{align} for $\xi>0$. If we extend symmetric $\hat f(\xi)$ about $\xi=0$, then we are interested in finding $f$ such that \begin{align} \hat f(\xi) = \sqrt{\frac{2}{\pi}}\frac{e^{-|\xi|}\sin \xi}{\xi}= \sqrt{\frac{2}{\pi}} e^{-|\xi|}\cdot \frac{\sin \xi}{\xi} \end{align} which means \begin{align} f(x) = \sqrt{\frac{2}{\pi}}(e^{-|\xi|})^\vee \ast (\operatorname{sinc}(\xi))^\vee(x). \end{align} Since we have following inversion \begin{align} (\operatorname{sinc}(\xi))^\vee(x) = \chi_{[-1, 1]}(x) \end{align} and \begin{align} (e^{-|\xi|})^\vee(x) = \sqrt{\frac{2}{\pi}} \frac{1}{1+x^2} \end{align} which means \begin{align} f(x) =&\ \frac{2}{\pi}\int^\infty_{-\infty} \frac{1}{1+(x-y)^2} \chi_{[-1, 1]}(y)\ dy =\frac{2}{\pi}\int^1_{-1} \frac{1}{1+(x-y)^2}\ dy \\ =&\ \frac{2}{\pi} [\arctan(x+1)-\arctan(x-1)]. \end{align}

Remark: The constant $2/\pi$ might be off.

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  • $\begingroup$ (+1) Well done. It is probably worth noticing that $$\arctan(x+1)-\arctan(x-1) = \arctan\frac{2}{x^2} = \frac{\pi}{2}-\arctan\frac{x^2}{2}.$$ $\endgroup$ – Jack D'Aurizio Oct 17 '16 at 18:43
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Since

$$\int_0^{\infty} e^{-x}\cos(\alpha x)\ dx = \frac{1}{\alpha^2+1}$$

we can take

$$f(x) = e^{-\alpha-x}\left(\alpha+\frac{1}{\alpha}\right)\sin(\alpha)$$

which works.

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    $\begingroup$ Are you sure about this? Just because you plugged $\alpha$ into your definition of $f$, doesn't mean this is the variable of the transform, so it doesn't work. More explicilty: $\hat{f}(k) =\int f(x) \cos(k x) \mathrm{d}x$ won't give the correct answer as a function of $k$. $\endgroup$ – Ranc Oct 17 '16 at 15:19
  • $\begingroup$ By the equation we have for $f$, we see that the integral of $f(x)\cos(ax)$ at $a=0$ (take the limit $a\rightarrow0$) is $1$. But according to your solution, integration of your $f$ at $a=0$ gives $0$. $\endgroup$ – Ranc Oct 17 '16 at 15:36
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The answer is the inverse Fourier cosine transform: \begin{align} f(x) & = \frac{2}{\pi}\int_{0}^{\infty}e^{-\alpha}\frac{\sin\alpha}{\alpha}\cos(\alpha x)d\alpha, \\ f'(x)& =-\frac{2}{\pi}\int_{0}^{\infty}e^{-\alpha}\sin(\alpha)\sin(\alpha x)d\alpha \\ &=\frac{1}{\pi}\int_{0}^{\infty}e^{-\alpha}\{\cos(\alpha(x+1))-\cos(\alpha(x-1))\}d\alpha \\ &=\Re\frac{1}{\pi}\int_{0}^{\infty}e^{-\alpha(1+i(x+1))}-e^{-\alpha(1+i(x-1))}d\alpha \\ &=\Re\frac{1}{\pi}\left[\frac{1}{1+i(x+1)}-\frac{1}{1+i(x-1)}\right] \\ &=\frac{1}{\pi}\left[\frac{1}{1+(x+1)^2}-\frac{1}{1+(x-1)^2}\right] \\ f(x) &= \frac{1}{\pi}\{\tan^{-1}(1+x)-\tan^{-1}(x-1)\}+C. \end{align} Evaluating the original expression at $x=0$ gives $C$: $$ \frac{2}{\pi}\int_{0}^{\infty}e^{-\alpha}\frac{\sin\alpha}{\alpha}d\alpha = \frac{2}{\pi}2\tan^{-1}(1)+C = 1+C. $$

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