Problem Set III 1.- Use Laplace's method to obtain an asymtpotic expansion valid for x → ∞ of the complementary error function 2 ∞ erfc(x) = -t2 e dt 2 x2 -2tx -t² = e e dt, and compare your result with that obtained in class by means of integration by parts. (Note the factor 2/√ used here which is not included in the convention used in class.) Compare the asymp- totic results with the exact values erfc(2) 0.004677735... and erfc(4) = 0.00000 00154 173.... Is == the asymptotic expansion convergent? [Hints: Note that the function -2t (which is part of the exponent in the exponential e -2tx in the second integral above) has its maximum at an endpoint of the integration interval. Expand exp (−t²) in a power series around the origin.]