# Why are these expressions with shifted delta functions not equal?

I have the following equation

Exp[a-t]*DiracDelta[t-a] == DiracDelta[t-a]


which cannot be evaluated. However, if I plug in 0 for a:

In[1] = Exp[t]*DiracDelta[t] == DiracDelta[t]
Out[1] = True


This does not seem to work with values other than 0.

Of course, the equality with delta functions may be mathematically tricky, but in this case Exp[a-t] == 1 for a == t, thus the first expression should yield True (in my opinion...). How can I get to this value?

• Strangely it works if you do Simplify[Exp[a - t]*DiracDelta[t - a] == DiracDelta[t - a], t - a == k] where the second argument to Simplify is an assumption. – flinty Aug 6 '20 at 17:26
• Set a to any constant value and it works. It works for a algebraic if you integrate each side, which is what delta functions are for after all. – Bill Watts Aug 7 '20 at 7:09
• I am surprised it works at all for non integrals, because the formula you are using setting a to t is the formula for integrating delta functions. – Bill Watts Aug 7 '20 at 7:16
• The equation you try to check without integration makes no sense! If it would be true you could conclude Exp[t-a]=?=1 !?! – Ulrich Neumann Aug 7 '20 at 7:27