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Both Laplace results were correct May 9, 2012

Posted by hackandfab in Uncategorized.

Hooray! my result was correct and it’s the first time I write a proof to someone else outside university.

From: The Wolfram|Alpha Team via RT <info@wolframalpha.com>
Date: Wed, May 9, 2012 at 5:30 PM
Subject: [W|A #302928] W|A Feedback

Dear Iker,

Both your result and the result given by Wolfram|Alpha are correct. There are various ways to write the result. Our result “alpha(laplace transform sin(3t)/t)”, and your result agree for positive s: alpha(arctan(3/s) – (pi/2 – arctan(s/3)))

Also, in the complex plane the two expressions are the same for Re(s)>0

plot3d |arctan(3/(x + i y)) – (pi/2 – arctan((x + i y)/3))|

Now, for the inverse Laplace transform one integrates along a contour
that goes parallel to the imaginary axis and is in the right-half plane:


As the two functions agree there, they both give the same inverse Laplace transform:


inverse Laplace transform pi/2-arctan(s/3)

Please let us know if you have any other questions.

Best wishes,

The Wolfram|Alpha Team


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