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I failed to evaluate the integral of $\frac{e^{ax}}{x\sinh(bx)}$ with respect to $x$ from negative infinite to positive infinite, What techniques can I use to evaluate the integrals of such kind for meixner levy process for the purpose of numerical analysis

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could you perhaps provide some more context ? Why do you need to integrate the function mentioned above in order to model the process ? –  Probilitator Apr 8 at 13:18

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The integral diverges, hence numerical integration will not yield a meaningful result.

Indeed, using the Taylor expansion of the hyperbolic sine one has that $$x \sinh (bx)= x\frac{e^{bx}-e^{-bx}}{2}=bx^2+\frac{b^3x^4}{6}+...$$ Therefore, at the vicinity of $x=0$ $$\frac{e^{ax}}{x \sinh (bx)}\sim\frac{1}{bx^2}$$ which is a non-integrable singularity even in terms of the Cauchy principal value.

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I must humbly concede that you are correct –  Probilitator Apr 8 at 13:11

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