In the book Advanced Mathematical Methods for Scientists and Engineers by Bender and Orszag (question 6.50) we are asked to compute the asymptotic expansion of $\int_{0}^{1}\sqrt{t}e^{ixt}dt$ fully. We are told the answer is


My problem is that the answer I keep computing is the following:


I obtain this by using the original expansion up to first order given in the book and then proceeding by using the hint given in the book which says:


The expansion started in the book says that:


From here I use the hint and perform a contour integration to get the term $\frac{i\sqrt{\pi}}{2x^{\frac{3}{2}}}e^{i\frac{\pi}{4}}$ and end up with:




Finally I repeatedly use integration by parts to try an obtain the answer. I have obviously made a mistake somewhere. If someone could point it out to me I would be very grateful. Thank you in advance.

  • 2
    $\begingroup$ I haven't tried computing the series myself by numerically it appears that your answer yields a much better approximation to the integral than the one given by the book. I suspect there is a typo in the book's answer. $\endgroup$ – Antonio Vargas Oct 29 '14 at 20:59
  • $\begingroup$ @AntonioVargas Thank you for your response! $\endgroup$ – user99163 Oct 30 '14 at 3:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.