Terrifying question about polynomial in analysis

[eileen6a]

the textbook says that:
"a non-constant analytic polynomial cannot be real-valued, for then both the partial derivative with respect to x and y would be real and the cauchyriemann equation cannot be satisfied."
why??there's no explanation in the book and this sentence is written as an example.
help.

 

[jackmell]

Well let's see what we got:

$$\begin{aligned} f(z)&=P_n(z)=u(x,y)+iv(x,y) \\ &=u(x,y)+i0\\ &\neq k \end{aligned}$$

So then:

$$\begin{aligned} \frac{\partial u}{\partial x}&=g(x,y) &\quad \frac{\partial u}{\partial y}&=h(x,y)\\ \frac{\partial v}{\partial x}&=0 &\quad \frac{\partial v}{\partial y}&=0 \end{aligned}$$

. . . so if f(z) is not equal to k (a constant) . . . then what?

 

[eileen6a]

thanks you must be a genius