April 11, 1996    Math 312,       Quiz 9


    

  1. Let K(x,t) = e^(-x^2/4t)/sqrt(4 pi t).

      

    • Show that K_t = K_xx, for all t > 0.

    • Show that the integral of K(x,t) over the entire line equals one
for any t > 0.

    • Show that the limit as t tends to zero from above of the integral
from epsilon to infinity of K(x,t), where epsilon is any positive
number equals zero.

    

  2. Let  K(x,y) = (1/pi)(y/(x^2+y^2).

      

    • Show that K_xx + K_yy = 0, for all y > 0.

    • Show that the integral of K(x,y) over the entire line equals one
for any y > 0.

    • Show that the limit as y tends to zero from above of the integral
from epsilon to infinity of K(x,y), where epsilon is any positive
number equals zero.



   

Postscript
Postscript