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This is very similar to other questions people have asked (I've Googled extensively), but not identical. I have been trying to use integration by parts to generate an explicit formula in terms of $x$ for the indefinite integral

$$\int \text{sinc}^{2 k-1}(2 \pi x) \, \mathrm dx$$

There's a lot of guidance on finding definite integrals for lower powers of $\text{sinc}$, and I found a general expression for the definite integral from $-\infty$ to $\infty$ for any positive integer power $k$ here... But none of these are quite what I'm looking for, and it turns out they don't help me.

I have tried splitting $\text{sinc}^{2 k-1}(2 \pi x)$ into parts many different ways. Aside from the obvious numerator-denominator split, I've tried using the complex exponential for $\sin$, I've tried converting $\sin^2$ to an expression in $\cos$... But after each iteration of integration by parts, I end up with a new integral that seems more complex than the first. I've tunneled down $4$ iterations in each case, and the functions just get more and more unwieldy.

There has to be a way because some web pages reference old books with a solution in them. But I don't have access to a library.

Could someone please talk me through how to handle this? (Or at least provide a solution!)

mrtaurho
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Richard Burke-Ward
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    For any $k$, a CAS provide an explicit solution but really messy. For example, for $k=3$, $6144 \pi ^5 x^4,I_3$ is $$80 \pi ^4 x^4 \text{Si}(2 \pi x)-3240 \pi ^4 x^4 \text{Si}(6 \pi x)+5000 \pi ^4 x^4 \text{Si}(10 \pi x)+40 \pi ^3 x^3 \cos (2 \pi x)-540 \pi ^3 x^3 \cos (6 \pi x)+500 \pi ^3 x^3 \cos (10 \pi x)+20 \pi ^2 x^2 \sin (2 \pi x)-90 \pi ^2 x^2 \sin (6 \pi x)+50 \pi ^2 x^2 \sin (10 \pi x)-30 \sin (2 \pi x)+15 \sin (6 \pi x)-3 \sin (10 \pi x)-20 \pi x \cos (2 \pi x)+30 \pi x \cos (6 \pi x)-10 \pi x \cos (10 \pi x)$$ – Claude Leibovici Feb 04 '19 at 11:00
  • Thanks @Claude. But I was kind of hoping that there might be a closed form solution, perhaps like the solution for the definite integral found here... – Richard Burke-Ward Feb 04 '19 at 12:53
  • As I wrote, the problem is to get a general formula for $I_k$. I guess that for any specific value of $k$, we can get an explicit result. – Claude Leibovici Feb 05 '19 at 04:13

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Apologies (and thanks) to all who have read or responded. It seems my Googling was nowhere nears thorough as I had thought! I always take Wikipedia with a pinch of salt, but I read here that these integral cannot be expressed in terms of elementary functions.

But I'd be thrilled if someone could prove Wikipedia wrong!

Richard Burke-Ward
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