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- Jun 20, 2014

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Here is this week's POTW:

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Let $F : (0, \infty)\times (0,\infty) \to \Bbb R$ be given by

$$F(\alpha, \beta) = \int_0^\infty \frac{\cos(\alpha x)}{x^4 + \beta^4}\, dx$$ Show that $$\frac{F(\alpha,\beta)}{F(\beta,\alpha)} = \frac{\alpha^3}{\beta^3}$$ as long as there is no positive integer $n$ such that $\alpha = \dfrac{(4n-1)\pi\sqrt{2}}{4\beta}$.

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Remember to read the POTW submission guidelines to find out how to submit your answers!

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Let $F : (0, \infty)\times (0,\infty) \to \Bbb R$ be given by

$$F(\alpha, \beta) = \int_0^\infty \frac{\cos(\alpha x)}{x^4 + \beta^4}\, dx$$ Show that $$\frac{F(\alpha,\beta)}{F(\beta,\alpha)} = \frac{\alpha^3}{\beta^3}$$ as long as there is no positive integer $n$ such that $\alpha = \dfrac{(4n-1)\pi\sqrt{2}}{4\beta}$.

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Remember to read the POTW submission guidelines to find out how to submit your answers!

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