Solving Integral for All n≥2 | Evans PDE's (Page 48)

[NanoMath]

In the book from Evans on PDE's (page 48) I came across this integral. Here r > 0 and [itex] \delta [/itex] is an arbitrarily small number.
Could you give me some hint on how to solve this integral for all integers [itex] n\geq2 [/itex], i.e why does it go to zero as t approaches zero from the right side.

 

[eys_physics]

Try to a change of variable of the form $$x=r/\sqrt{16t}$$.

 

[mathman]

Let [tex]K(t)=e^{\frac{-\delta^2}{16t}}[/tex]. Put K as a multiplier outside the integral sign, and 1/K inside. The integral is bounded as t ->0+, while the coefficient -> 0. As for the integral itself, it looks like it will include the error function as a term.

 

[NanoMath]

Thank you for replies. Actually I only needed to check that the whole expression goes to zero and multiplying and dividing through by K(t) solves it.