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., $s<0$?

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 
Senior Member Joined: Jun 2019 From: USA Posts: 310 Thanks: 162  
October 11th, 2019, 05:01 AM  #5  
Senior Member Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98  Quote:
So $\displaystyle s<0$. Last edited by idontknow; October 11th, 2019 at 05:05 AM.  
October 11th, 2019, 05:36 AM  #6 
Senior Member Joined: Jun 2019 From: USA Posts: 310 Thanks: 162 
That was essentially my argument from the beginning. In your second post, you said, 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? 

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convergence, integral 
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