## 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:

http://www.wolframalpha.com/input/?i=define+inverse+Laplace+transform&a=*DPClash.MathWorldE.inverse+Laplace+transform-_*LaplaceTransform-

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

http://www.wolframalpha.com/input/?i=inverse+Laplace+transform+arctan%283%2Fs%29+&a=*C.s-_*Variable-

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

Please let us know if you have any other questions.

Best wishes,

Elif
The Wolfram|Alpha Team
www.wolframalpha.com