 My Math Forum Generalized Fourier Integral and Steepest descent path, Saddle point near the endpoin

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 July 14th, 2019, 08:08 PM #1 Newbie   Joined: Jul 2019 From: Tokyo Posts: 1 Thanks: 0 Generalized Fourier Integral and Steepest descent path, Saddle point near the endpoin I am looking forward to solving the integration in the following equation with the assumption that $ka$ is very large \begin{align} H = 2jka\int_{-\pi/2}^{\pi/2}\cos{(\varphi-\phi)}e^{jka[\cos{\varphi}+\cos{(\varphi-\phi)}]}\ d\varphi \end{align} I used the steepest descent path method to handle the integration, then from the saddle point $\varphi_s = \frac{\phi}{2}$, I found the saddle point contribution to the integral. However, the endpoints also contribute to the integral and they are estimated by asymptotic expansion by integration by part technique. The endpoints contribution is as following: \begin{align} H^{endpoint}=\frac{-4j\sin{\phi}}{1+\cos{\phi}}\sin{[ka\sin{\phi}]} \end{align} The asymptotic result is the summation of saddle point contribution and the endpoints contribution. When I check the results and compared this asymptotic approximation by the numerical integration, I found something wrong with the asymptotic approximation when $\phi\rightarrow \pi$ (the saddle point goes to the endpoints). Somehow the results have been blowed up when $\phi\rightarrow \pi$. The comparation between two methods is shown as the below figure. Could someone help me how to find the exact asymptotic result when $\phi\rightarrow \pi$. [enter image description here] : Thank you so much for your attention! Last edited by skipjack; July 14th, 2019 at 09:06 PM. July 15th, 2019, 11:04 AM   #2
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I'm not going to pretend to know the solution, but...
Quote:
 Originally Posted by gemmy94 \begin{align} H^{endpoint}=\frac{-4j\sin{\phi}}{1+\cos{\phi}}\sin{[ka\sin{\phi}]} \end{align} Somehow the results have been blowed up when $\phi\rightarrow \pi$.
That doesn't seem surprising. Tags descent, endpoin, fourier, generalized, integral, path, point, saddle, steepest Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post UyQTran Geometry 0 March 14th, 2019 08:48 AM safyras Differential Equations 0 April 7th, 2014 01:54 AM aneuryzma Calculus 1 November 19th, 2009 03:30 PM Louis Algebra 2 April 21st, 2008 12:59 PM germanaries Real Analysis 0 March 18th, 2008 01:28 PM

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