Form of Kronecker Delta Not Recognized in Mathematica

[DocZaius]

kron[m_,n_]:=\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(2\)]\(\(Sin[
\*FractionBox[\(n\ \[Pi]\ x\), \(2\)]]\ Sin[
\*FractionBox[\(m\ \[Pi]\ x\), \(2\)]]\) \[DifferentialD]x\)\)

This is the integral over x of sin(n pi x / 2) times sin(m pi x / 2) from 0 to 2. This is one way to define the Kronecker Delta function. Given m and n as integers and not both zero, if m = n then the output is 1, if m ≠ n the output is 0.

Why then is it that when I type:

Simplify[kron[m, n],
Element[m, Integers] && Element[n, Integers]]

that I get zero?

If I do kron[1, 1] I get 1! How can Mathematica say a function is 0 over the integers when I just put in two integers and got a non-zero ouput?

It is certainly not identical to zero right?

 

[djelovin]

When you force Mathematica to simplify expression it is assumed that m and n are not equal. This:

Assuming[{Element[m, Integers], Element[n, Integers], m == n}, kron[m, n]]
Assuming[{Element[m, Integers], Element[n, Integers], m != n}, kron[m, n]]

gives 1 and 0 respectively!

 

[DocZaius]

Thank you for looking into it! I didn't know Mathematica assumed m and n are not equal. Why is Mathematica making that assumption? Given two integers m and n, the possibility that they are equal is always there.

Rather worrisome Mathematica assumption that I will keep in mind from now on! I will have to handhold Mathematica by separately asking it the = and ≠ cases.

 

[djelovin]

When you call Simplify[expression] Mathematica assumes that all symbols in expression have unique values, otherwise not!

 

[alphy]

As a scientist, it is important to understand that Mathematica is a powerful mathematical software program, but it is not infallible. In this case, it seems that the simplification rule for the Kronecker Delta function may not be properly recognized by Mathematica. This could be due to a variety of reasons, such as a bug in the program or a limitation in its algorithms. It is also possible that the simplification rule for the Kronecker Delta function in Mathematica is different from the one that you have defined in your code.

It is important to carefully check the results and outputs of any software program, especially when dealing with complex mathematical functions. In this case, it seems that the simplification rule for the Kronecker Delta function may not be accurately represented in Mathematica. It is always a good idea to cross-check your results with other sources and to report any discrepancies to the developers of the software. As scientists, we must always critically evaluate our tools and methods to ensure accurate and reliable results.