Don't understand how this summation of complex numbers returns a single value

Question

I have the equation that gets a height for a vector $ x $ over time $t$: $$h(\mathbf x,t)=\sum_{\mathbf k}\tilde h(\mathbf k,t)\exp(i\mathbf k\cdot\mathbf x)$$ Function $\tilde{h}$ is defined as: $$\tilde{h} (\mathbf k,t)=\tilde h_0(\mathbf k)\exp(i\omega(k)t)+\tilde h_0^*(-\mathbf k)\exp(-i\omega(k)t)$$ Where: $$\tilde h_0(\mathbf k)=\frac1{\sqrt2}(\xi_r+i\xi_i)\sqrt{P_h(\mathbf k)}$$ But I do not understand how this returns a single height value, since $\tilde h$ is the sum of a complex number and its conjugate, how then does this sum to a single height value for $x,t$?

Answer 1

Consider any complex number $z = a+ib$, with $a,b\in\mathbb R$, and its complex conjugate $\bar{z} = a-ib$. Then

$$z+\bar{z} = a+ib+a-ib = 2a$$

which is a real number.