For $ z \in \mathbb{C}, t \in \mathbb{R}, \\f : \mathbb{C} \times \mathbb{R} \to \mathbb{C}, \\a : \mathbb{C} \times \mathbb{R} \to \mathbb{R}, \\b : \mathbb{C} \times \mathbb{R} \to \mathbb{R}$
And given that $\frac{\partial f(z,t)}{\partial t} = a(z,t) +ib(z,t)$, under what conditions will the following equations be true?
$\frac{\partial f^R(z,t)}{\partial t} = a(z,t) \\ \frac{\partial f^I(z,t)}{\partial t} = b(z,t) $
where $f^R : \mathbb{C} \times \mathbb{R} \to \mathbb{R}$ and $f^I : \mathbb{C} \times \mathbb{R} \to \mathbb{R}$ such that $f(z,t) = f^R(z,t)+if^I(z,t)$.