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 October 9th, 2019, 12:12 PM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98 Integral convergence For which values of s the integral converges ? $\displaystyle \int_{0}^{\infty}e^{sx}\cos(sx)dx \; , s\in \mathbb{R}$. Last edited by idontknow; October 9th, 2019 at 12:18 PM. October 9th, 2019, 03:14 PM #2 Senior Member   Joined: Jun 2019 From: USA Posts: 310 Thanks: 162 Cosine is on the order of 1. Wouldn't it be the same values for which integral of e^sx converges, i.e., $s<0$? Thanks from idontknow and SDK October 10th, 2019, 12:04 AM #3 Senior Member   Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98 Integral converges only if $\displaystyle |e^{sx}\cos(sx)|\leq e^{sx}$. Now the RHS must converge or $\displaystyle s<0$ . $\displaystyle e^{-sx}|_{0}^{\infty }$ converges then the integral converges . Last edited by idontknow; October 10th, 2019 at 12:14 AM. October 10th, 2019, 04:45 AM   #4
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Quote:
 Originally Posted by idontknow $\displaystyle |e^{sx}\cos(sx)|\leq e^{sx}$
Find me a value of s or x where this inequality is false. October 11th, 2019, 05:01 AM   #5
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 Originally Posted by DarnItJimImAnEngineer Find me a value of s or x where this inequality is false.
$\displaystyle \int_{D} e^{sx}cos(sx) < \int_{D} e^{sx} =\frac{e^{sx}}{s} \displaystyle |_{0}^{\infty}$ diverges , so it's own reciprocal expression converges , which is $\displaystyle se^{-sx}=s(e^{x})^{-s}$.
So $\displaystyle s<0$.

Last edited by idontknow; October 11th, 2019 at 05:05 AM. October 11th, 2019, 05:36 AM   #6
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That was essentially my argument from the beginning.
In your second post, you said,
Quote:
 Originally Posted by idontknow Integral converges only if $\displaystyle |e^{sx}\cos(sx)|\leq e^{sx}$.
and I asked you to give me an example where this would not be true. It's true for all combinations of real s and x, isn't it? Tags convergence, integral 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 noobinmath Calculus 2 February 27th, 2015 10:59 AM jams Complex Analysis 1 March 5th, 2012 04:02 PM dunn Real Analysis 5 January 15th, 2012 12:46 PM fed2black Calculus 1 April 12th, 2010 02:26 PM lime Calculus 2 February 15th, 2010 10:04 PM

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