Why is the int (integral) of the dirac function sign/2 instead of the heaviside function?

In MATLAB, the result of taking the integral of the "dirac" function is sign/2.

In many resources online and in textbooks the integral of the dirac delta function is defined as the heavyside step function. One such example can be found at:

https://en.wikipedia.org/wiki/Dirac_delta_function

Why is there a discrepancy?

Answer 1

[MathWorks Support Team]

Int(f(x), x) in MATLAB does not explicitly include the +C (an arbitrary constant that must be included when integrating).

The output of the heavyside function is 0 if negative, 1 if positive. The output of the sign function is 1 if positive and -1 if negative. if you divide the sign function output range by two your possible results become [-1/2, 1/2].

Adding 1/2 to this set of possible answers produces [0, 1], identical outputs to the heavyside function. The 1/2 that we added is included in that implicit +C.

Please run the below command in the command window of installed MATLAB R2019a version to get release specific documentation on the implicit +C for the "dirac" function:

>> web(fullfile(docroot, 'symbolic/dirac.html'))

As we can see, in the context of integrals, the heavyside function and the sign function divided by 2 are equivalent when each has an arbitrary constant.  These are both equally valid solutions. 

Please follow the below link to search for the required information regarding the current release:

https://www.mathworks.com/help/