The discussion centers on a statement from Hinch's perturbation theory book regarding the Gaussian integral. Specifically, it addresses the inequality \(\int_z^{\infty}\dfrac{d e^{-t^2}}{t^9}<\dfrac{1}{z^9}\int_z^{\infty}d e^{-t^2}\). The conclusion drawn is that the function \(1/t^9\) for \(t\) in the interval \((z, \infty)\) is consistently less than \(1/z^9\), which justifies the inequality. A graphical representation is suggested as a method to visualize this relationship.
PREREQUISITESStudents and researchers in applied mathematics, particularly those studying perturbation theory and Gaussian integrals, will benefit from this discussion.
liyz06 said:Homework Statement
I'm reading Hinch's perturbation theory book, and there's a statement in the derivation:
...\int_z^{\infty}\dfrac{d e^{-t^2}}{t^9}<\dfrac{1}{z^9}\int_z^{\infty}d e^{-t^2}...
Why is that true?
Homework Equations
The Attempt at a Solution
Dick said:Because 1/t^9 for t in (z,infinity) is less than 1/z^9. Draw a graph.