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The integral: $$S(x)=\int_0^x H^{(1)}_{k+1}(\eta)H^{(2)}_{k-1}(\eta)d\eta$$ can be expressed as a combination of Hypergeometric functions and trigonometric functions. I have some difficulty to calculate the previous integral $S(x)$ defined between $0$ and $+\infty$. Can someone give me any hint? Thanks

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Consider the case of half-integer $k$. Hankel functions can then be expressed in elementary functions. The integral diverges at the lower integration bound. For example, for $k=1/2$, $$H_{3/2}^{(1)}(\eta) H_{-1/2}^{(2)}(\eta) = -\frac{2}{\pi} \frac{\eta+i}{\eta^2}$$ –  Sasha Sep 13 '12 at 16:13

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