Is this the hessian bilinear form or $1/2$ of it?

Question

Look at:

the author writes "full hessian", but it's clearly $\frac12\operatorname{hess}r$. Or not?

Other question: why does the author mean by "real harmonic polynomial"? And why it should be equal to the real part of the sum he written?!

Any hint will be appreciated! Many thanks!

EDIT Here's a more complete part of my book:

Answer 1

I would assume that $\partial_z\partial_zr=\partial_\overline{z}\partial_\overline{z}r$, in which case summing over repeated indices will give us twice the Hessian, so we take $1/2$ to get the correct value. Also by "real harmonic" I believe the first two sums (summed together) give a harmonic function with values in $\Bbb R$, i.e. a real harmonic function. Note that this supports my first assumption :)