# Can the delta function be even or odd?

Say I have the function,

$$\delta\left(\tau-\frac{T}{2}\right)\mathrm d\tau$$ where T = $$\frac{2π}{ω}$$

Is this even or odd? Or neither?

Reason I ask is to find out whether I can cancel some cosine and sine terms for a convolution computation.

• The expression looks funny. $\partial$ ? Feb 15, 2020 at 22:21
• I have no idea what your bizarre notation means
– MPW
Feb 15, 2020 at 22:22
• Are you trying to write $\delta (t -\frac{T}{2})\mathrm dt$? Either way, this doesn't seem to denote any function. Feb 15, 2020 at 22:23
• @Macuser That looks like a differential, not a function. Are you simply asking whether or not the Dirac delta function is even or odd? Feb 15, 2020 at 22:27
• @Macuser Do you know the definitions of even and odd functions? Have you checked whether or not they apply? Please add this context to your question with an edit Feb 15, 2020 at 22:29

$$\delta(x)$$ is an even function which peaks at $$x=0$$ and $$\delta(-x)=\delta(x)$$. But $$f(t)=\delta(t-T/2)$$ peaks at $$t=T/2$$ and it is symmetric about $$t=T/2$$ as $$f(T-t)=\delta(T-t-T/2)=\delta(T/2-t)=\delta(t-T/2)=f(t)$$