**Background**

I am including this information about real higher order derivatives because it does not seem to be common knowledge. I also include a review of the complex Hessian.

If $f:\mathbb{R}^n \to \mathbb{R}^k$ is a function, then its derivative $Df:\mathbb{R}^n \to \textrm{Hom}(\mathbb{R}^n,\mathbb{R}^k)$. $Df(\mathbf{p}):\mathbb{R}^n \to \mathbb{R}^k$ is a linear map between tangent spaces for each point $\mathbf{p} \in \mathbb{R}^n$. $Df(\mathbf{p})$ is defined as the unique linear map so that

$$ \lim_{\vec{h} \to \vec{0}} \frac{\left| f(\mathbf{p}+\vec{h})-f(\mathbf{p})-Df(\mathbf{p})(\vec{h}) \right|}{|\vec{h}|} = 0 $$

Endowing $\textrm{Hom}(\mathbb{R}^n,\mathbb{R})$ with any norm you desire (the operator norm for example), we find that we can differentiate $Df$ to obtain a map $D^2f:\mathbb{R}^n \to \textrm{Hom}(\mathbb{R}^n,\textrm{Hom}(\mathbb{R}^n,\mathbb{R}^k))$.

Fundamentally, $D^2f$ allows us to approximate changes in the derivative:

$$D^2f(\mathbf{p})(\vec{h}_1)(\vec{h}_2) \approx Df_{\mathbf{p}+\vec{h}_2}(\vec{h_1}) -Df_\mathbf{p}(\vec{h_1})$$ We can reinterpret this as $D^2f(\mathbf{p}): \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^k$ being a bilinear map. In the special case $k=1$, $D^2f(\mathbf{p})$ is a bilinear form.

The second derivative as a bilinear form is given by the formula

$$ D^2f = \sum_{j,k=1}^n \frac{\partial^2 f}{\partial x_j \partial x_k} dx_j \otimes dx_k $$

and can be represented by a matrix called the "Hessian matrix" of $f$. Famously it turns out that this bilinear form is symmetric, and so is its corresponding matrix.

All of this generalizes easily to higher dimensional derivatives: the $D^kf(\mathbf{p})$ can be thought of as a $k$ linear form on $\mathbb{R}^n$ which measures changes in $D^{k-1}f$. The formula is just

$$ D^kf(\mathbf{p}) = \sum_{|I|=k} \frac{\partial^k f}{\partial x_I} dx_I $$

where I am using multi index notation.

For functions $f:\mathbb{C}^n \to \mathbb{C}$ the so called "Complex Hessian" of $f$ is incredibly important. It is given by the formula

$$\mathcal{H} = \sum_{j,k=1}^n \frac{\partial^2 f}{\partial z_j \partial \overline{z}_k} dz_j \otimes d\overline{z}_k$$

It turns out that another way to write this is as $H(v,w) = i\partial \overline{\partial} f(v,J(w))$ where $\partial$ and $\overline{\partial}$ are the Dolbeault operators and $J:\mathbb{C}^n \to \mathbb{C}^n$ is the complex structure map $J(z_1,z_2,...,z_n) = (iz_1,iz_2,...,iz_n)$.

**Questions**

Does anyone have any intuitive reasons for the extreme importance of the complex Hessian in several complex variables? It is really totally fundamental to the subject, and I use it all the time, but I still do not have a completely "natural" justification for it.

We can consider a function $f:\mathbb{C}^n \to \mathbb{C}$ as a complex valued function on $\mathbb{R}^{2n}$, and use the discussion above to generate a notion of a higher order derivative tensor $D^kf$, but this completely ignores the complex structure. Is there a natural generalization of the Complex hessian to higher order complex derivatives? Could this be related to the notion of "finite type" in several complex variables?