In the middle of an engineering problem (fatigue modeling), I sought Gradshteyn and Ryzhik to solve the following: \(\int \limits_0^a x^\alpha e^{\beta x - x^2}~dx\) It looks deceptively simple and two nights of work on this didn't yield any useful results. Is it only solvable by Mathematica? One of my coworkers found a solution with hypergeometric series but, is there a simpler solution? Alpha = real Beta = real a = positive real
I don't see why you think that this would be simple. You have a product, so the first thing that comes to mind is substitution. So lets use \(x = u + \frac{\beta}{2}\) to get \(u = x - \frac{\beta}{2} , \; du = dx, \; -u^2 = -\frac{\beta^2}{4} + \beta x - x^2\). So we have \(\int \limits_0^a x^\alpha e^{\beta x - x^2} \, dx = e^{\frac{\beta^2}{4}} \int \limits_{- \frac{\beta}{2}}^{a- \frac{\beta}{2}} \left( u + \frac{\beta}{2} \right)^\alpha e^{- u^2} \, du\) Now by using the generalized binomial theorem and assuming \(\alpha\) is not a negative integer and \(0 < 2 a < \left| \beta \right|\) we have: \( \left( u + \frac{\beta}{2} \right)^\alpha = \sum \limits_{k=0}^{\infty} \frac{ \beta^{\alpha -k} u^k }{k! 2^{\alpha -k} } \prod \limits_{j=0}^{k-1} (\alpha -j) = \sum \limits_{k=0}^{\infty} { \alpha \choose k } \frac{ \beta^{\alpha -k} u^k }{2^{\alpha -k}} \) So \(\int \limits_0^a x^\alpha e^{\beta x - x^2} \, dx \\ = e^{\frac{\beta^2}{4}} \int \limits_{- \frac{\beta}{2}}^{a- \frac{\beta}{2}} \left( u + \frac{\beta}{2} \right)^\alpha e^{- u^2} \, du \\ = e^{\frac{\beta^2}{4}} \sum \limits_{k=0}^{\infty} { \alpha \choose k } \frac{ \beta^{\alpha -k} }{2^{\alpha -k}} \int \limits_{- \frac{\beta}{2}}^{a- \frac{\beta}{2}} u^k e^{- u^2} \, du \) And these integrals we can do with the incomplete Gamma function: \(\int u^k e^{- u^2} \, du = - \frac{1}{2} \Gamma \left( \frac{1 + k }{2} , \, u^2 \right) + C \) Looks nice, but it's not. We get some extra minus signs in there for the \(u^2\) term and it gets worse if a is too large. Let's test for \(a = \frac{1}{10}, \alpha = 3, \beta = 1\) \(\int \limits_0^{\frac{1}{10}} x^3 e^{x - x^2} \, dx \\ = e^{\frac{1}{4}} \sum \limits_{k=0}^{3} { 3 \choose k } 2^{k-3} (-1)^k \int \limits_{- \frac{5}{10}}^{- \frac{4}{10}} u^k e^{- u^2} \, du \\ = \frac{e^{\frac{1}{4}} }{16} \left( \left( \Gamma \left( \frac{1}{2} , \, \frac{16}{100} \right) - \Gamma \left( \frac{1}{2} , \, \frac{25}{100} \right) \right) - 6 \left( \Gamma \left( 1 , \, \frac{16}{100} \right) - \Gamma \left( 1 , \, \frac{25}{100} \right) \right) + 12 \left( \Gamma \left( \frac{3}{2} , \, \frac{16}{100} \right) - \Gamma \left( \frac{3}{2} , \, \frac{25}{100} \right) \right) - 8 \left( \Gamma \left( 2 , \, \frac{16}{100} \right) - \Gamma \left( 2 , \, \frac{25}{100} \right) \right) \right) \) 0.000026904852296752657572155912554654375603197730055...
