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Form of Kronecker Delta Not Recognized in Mathematica

  1. Apr 18, 2012 #1
    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?
     

    Attached Files:

    Last edited: Apr 18, 2012
  2. jcsd
  3. Apr 21, 2012 #2
    When you force Mathematica to simplify expression it is assumed that m and n are not equal. This:

    Code (Text):
    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!
     
  4. Apr 21, 2012 #3
    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.
     
  5. Apr 21, 2012 #4
    When you call Simplify[expression] Mathematica assumes that all symbols in expression have unique values, otherwise not!
     
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