Grassmann numbers

Question

I am reading Zee's Quantum Field Theory in a Nutshell and am having some questions about Grassmann numbers. Let $x,y$ be Grassmann numbers. I think I have two relations:

$$e^{x+y}=1+x+y\tag{1}$$ and $$e^{x+y}=e^xe^y=(1+x)(1+y)=1+x+y+xy,\tag{2}$$

Which is wrong and why? In fact, my question originates from the derivation of the formula:

$$\int dx\int dy\,e^{yAx}=\det A\tag{3}$$

where $x=(x_1,x_2,\dots,x_N),y=(y_1,y_2,\dots,y_N)$ both are $N$ Grassmann numbers and $A$ is an antisymmetric $N$ by $N$ matrix. For, if $(1)$ holds, it seems that $$\int dx\int dy\,e^{yAx}=\int dx\int dy\exp\left(\sum_{ij}y_iA_{ij}x_j\right)=\int dx\int dy\left(1+\sum_{ij}y_iA_{ij}x_j\right)=0\,,\tag{4}$$ which is, of course, wrong. So, where is the problem?

Answer 1

The power series is the definition of exponentiation.

$e^{x+y}=e^xe^y$ is a property of that power series that can be derived for the special case when $x$ and $y$ commute.

Answer 2

It seems that the problem is here. For Grassmann numbers $x,y,a,b,c,d$, we have $$e^{x+y}=1+x+y$$but$$e^{ab+cd}=1+ab+cd+\frac{1}{2}(2abcd)\neq 1+ab+cd$$ Therefore, $(1)$ is correct but the derivation in $(4)$ is wrong.