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How can I evaluate this integral (without complex analysis)?

$\int_{-\infty}^\infty\sinh [x(1-b)] \exp(iax) dx\qquad a, b\in \mathbb R$

Thanks.

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    @AleksVlasev $\large\int_{-\infty}^{\infty}{\rm e}^{{\rm i}ax}\,{\rm d}x = 2\pi\,\delta\left(a\right)$ ---> [See this link](http://en.wikipedia.org/wiki/Dirac_delta_function)2014-01-28

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This is the Fourier transform of $\sinh$, which can be broken down as a sum of exponentials. Because the Fourier transform is a linear operator, your integral is a sum of Fourier transforms of exponentials, that is a sum of Lorentzian functions (as it's shown for instance here).