[?] Simplify Conjugate[DiracDelta[x]]?

Questions
,
1

Assuming x is Real, how to get Mathematica to recognize the truth of this equation?

In[513]:= FullSimplify[Conjugate[DiracDelta[x]] == DiracDelta[x], 
 x \[Element] Reals]

Out[513]= Conjugate[DiracDelta[x]] == DiracDelta[x]
2

Try ComplexExpand.

3

Thanks.

4

When x=0, DiracDelta[x] is not considered a number:

In[8]:= DiracDelta[0] // NumberQ

Out[8]= False

Mathematica simplifies as you expect when x!=0:

Simplify[Conjugate[DiracDelta[x]] == DiracDelta[x], 
 x \[Element] Reals && x != 0]
5

Thank you. That is helpful.