Taylor series of complex multivariate function I am looking for a reference or some literature on Taylor series of complex multivariate functions. I found material for complex functions and material for multivariate functions, but not for both.
Is there an expression for at least the 2 or 3 first terms of the Taylor expansion of a function $f: \mathbb{C}^n \rightarrow \mathbb{C}$?
 A: I'll take a stab at it blending the two concepts together. 
In a single complex variable we have $f(z)=a(x,y)+ib(x,y)$ where $z=x+iy$. 
Suppose we have two complex variables. 
Then I'd guess: $f(z_1,z_2)=a(x_1,x_2,y_1,y_2)+ib(x_1,x_2,y_1,y_2)$
Before we worry about a Taylor series, we need some condition for testing analysity. 
In the single complex variable case, we have some conditions. We want the derivative tweaking just the real component to give us the derivative tweaking just the imaginary component. 
So: $\frac{a(x+\Delta x,y)+ib(x+\Delta x, y)-a(x,y)-ib(x,y)}{\Delta x}=\frac{a(x,y+\Delta y) + i b(x, y+ \Delta y)-a(x,y)-ib(x,y)}{i\Delta y}$
Then we equat real and imaginary parts: 
$\frac{\partial a}{\partial x}+i\frac{\partial b}{\partial x}=(-i)(\frac{\partial a}{\partial y}+i\frac{\partial b}{\partial y})$
So we get : 
$\frac{\partial a}{\partial x}=\frac{\partial b}{\partial y}$
$\frac{\partial b}{\partial x}=-\frac{\partial a}{\partial y}$
By similar reasoning, I think we need:
$\frac{\partial a}{\partial x_1}=\frac{\partial b}{\partial y_1}$
$\frac{\partial b}{\partial x_1}=-\frac{\partial a}{\partial y_1}$
$\frac{\partial a}{\partial x_2}=\frac{\partial b}{\partial y_2}$
$\frac{\partial b}{\partial x_2}=-\frac{\partial a}{\partial y_2}$
$\frac{\partial a}{\partial x_1}=\frac{\partial a}{\partial x_2}$
$\frac{\partial b}{\partial x_1}=\frac{\partial b}{\partial x_2}$
$\frac{\partial b}{\partial y_1}=\frac{\partial b}{\partial y_2}$
$\frac{\partial a}{\partial y_1}=\frac{\partial a}{\partial y_2}$
If those criteria are satisfied, then I think we can proceed with a Taylor series. 
$f(z_1,z_2)=f(x_{10},x_{20},y_{10}, y_{20})+\frac{\partial a}{\partial x_1}\Delta x_1 +i\frac{\partial b}{\partial x_1}\Delta x_1+\frac{\partial a}{\partial x_2}\Delta x_2+ \frac{\partial b}{\partial y_1}\Delta y_1 - i \frac{\partial a}{\partial y_1}+... + \frac{\partial^2 a}{\partial x_1^2}\Delta x_1^2+... $
I believe you proceed in the usual way for multivariable functions just keeping in mind which terms need the imaginary unit. An x derivative of the real part of the function and  a y derivative of the imaginary part do not need the imaginary units. The others do. 
