Suppose we consider a point dipole i.e. an ideal dipole. At the origin the potential:  $$ \phi(\mathbf r) = \frac{1}{4\pi\epsilon_0} \frac{\mathbf p\cdot\hat{\mathbf r}}{r^2}$$  is undefined due to the $1/r^2$ term in the potential, but electric field:  $$ \mathbf E = \frac{1}{4\pi\epsilon_0}\frac{3(\mathbf p\cdot\hat{\mathbf r})\hat{\mathbf r} - \mathbf p}{r^3} - \frac{1}{3\epsilon_0}\mathbf p\delta^3(\mathbf r) $$  remains well defined. How can we calculate electric field at the origin by equation $\mathbf E=-\nabla\phi$?
I am a bit lost how one has deduced the formula for electric field with electric dipole because of some inconsistency between different sources. The  contains a delta function in the formula, but  gives only the formula without the delta function,  $$\bar{E} = -\nabla \phi = \frac{-p}{4\pi \epsilon_{0} r^{3}} ( \hat{e}_{z}-3\hat{u}_{r}(\hat{u}_{r} \cdot \hat{u}_{z}) ) $$  Is it intentionally left out?  My notes mention that the dipole is with a circular thing $P=Q d$ where $d$ is distance and $Q$ is charge.
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