Derivative of Constant/Complex

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Theorem

Let $f_c \left({z}\right)$ be the constant function on an open domain $D \in \C$, where $c \in \C$.


Then:

$\forall z \in D : f_c' \left({z}\right) = 0$


Proof

The function $f_c: D \to \C$ is defined as:

$\forall z \in D: f_c \left({z}\right) = c$

Thus:

\(\ds f_c' \left({z}\right)\) \(=\) \(\ds \lim_{h \mathop \to 0} \frac {f_c \left({z + h}\right) - f_c \left({z}\right)} h\) Definition of Complex Differentiation
\(\ds \) \(=\) \(\ds \lim_{h \mathop \to 0} \frac {c - c} h\)
\(\ds \) \(=\) \(\ds \lim_{h \mathop \to 0} \frac 0 h\)
\(\ds \) \(=\) \(\ds 0\)

$\blacksquare$