The integral \(\int_0^\infty e^{-ax^2+bx} dx\) does not have a closed-form solution and is classified as an error function (erf) integral, requiring numerical methods for evaluation. In contrast, the integral \(\int_{-\infty}^\infty e^{-ax^2+bx} dx\) has a closed-form solution expressed as \(\sqrt{\frac{\pi}{a}}\). The discussion emphasizes the distinction between closed-form and numerical solutions, noting that while erf functions are closed forms, they still necessitate numerical computation for specific values.
PREREQUISITESMathematicians, physicists, and engineers involved in advanced calculus, particularly those working with integrals involving complex variables and Gaussian functions.
\int_{-\infty}^\infty e^{-ax^2} dx={\sqrt{\frac{\pi}{a}}}}Yegor said:\int_0^\infty e^{-ax^2+bx} dx, a and b may be complex.
Does exist any formula for this integral?
Or for \int_{-\infty}^\infty e^{-ax^2+bx} dx?
It is true that the first will involve erf, while the second will have nicer form. Yet erf is a closed form. Also closed form verses numerical solution is kind of silly any way. log(2) is a closed form, but if you want a number you have to "do it numerically". The issue has more do do with how many function one want to define tabulate and use. The distinction between an answer erf(1) and one of sin(1) is mostly historical.mathman said:ax2-bx=a(x-b/2a)2+(b/2)2/a
Your first integral then becomes an erf integral, which can only be done numerically. The second integral has a closed form solution - integral of Gaussian.