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I am trying to evaluate the definite integral 1 to obtain a closed-form solution, with z being the integration variable, and the other parameters are real positive constants.

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The integral can be solved with the aid of Taylor series and variable transform (I guess), however, I am trying to avoid using infinite series solutions. I also tried integration by parts as in 2, dv can easily yield the gamma function, but the difficulties come with the blue part (u).

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Can anyone advise on a solution? Special functions are very welcomed.

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