Wrong result for keyword fourier transform?

[Cornel]

Hello,

Input:

where 

is the step function:

 

Rewriting the above result:

And calculating fourier transform by definition for the above input, results:

 (which is the correct answer)

So, where is the mistake?

The 2 above results should not have been equal?

Mathcad Prime 8 file attached.

[LucMeekes]

The mistake is in Prime.

If Mathcad 11 may be the judge:

Chances are this bug has been solved in Prime 9, otherwise/anyway you can/should report this bug to PTC.

Success!
Luc

P.S. this is what Prime 7 produces:

[Werner_E]

Prime is a work in (sloooow) progress - maybe some day it will get it right.

Once more its clear - when it comes to symbolic math, Mathcad 11 with Maple rules!

[Werner_E]

BTW, here is Wolfram Alpha

EDIT:

Ahh! Got it!

I was aware of different scaling/normalization in the definitions of the fourier transform, but not of something like this "oscillatory factor"!

As far as I could see there is no definition of the fourier transform mentioned in the Mathcad and Prime help.

Different "oscillatory factors" may be an explanation for Primes results !?

[Cornel · Accepted solution]

@Werner_E @-MFra- @ttokoro 

I think I discovered where the difference comes from:

So, fourier transform in keyboard has this definition for calculation of fourier transform:

and not

The difference comes from that - sign, as Werner points out in Wolfram Alpha.

Is this definition used by fourier transform from keyword correct?

From what I know the fourier transform is defined in this way:

So, including that - sign.

[Cornel]

Rewriting the above expression:

[Werner_E]

---

From what I know the fourier transform is defined in this way:

---

I am not an expert, but basically this is also the only definition I am aware of.

That was also the reason I was confused about of Wolfram Alphas ""oscillatory factor".
Perhaps Wolframs explanation can help to understand: https://mathworld.wolfram.com/FourierTransform.html

[-MFra-]

A few years ago a list of all the "defective" calculations of MATHCAD 15 was published in the MATHCAD community. In Prime some have been corrected, but I see that others remain...