Derivative of Constant/Complex
< Derivative of Constant(Redirected from Complex Derivative of Constant)
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Theorem
Let $\map {f_c} z$ be the constant function on an open domain $D \in \C$, where $c \in \C$.
Then:
- $\forall z \in D : \map { {f_c}'} z = 0$
Proof
The function $f_c: D \to \C$ is defined as:
- $\forall z \in D: \map {f_c} z = c$
Thus:
\(\ds \map { {f_c}'} z\) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\map {f_c} {z + h} - \map {f_c} z} h\) | Definition of Derivative of Complex Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {c - c} h\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac 0 h\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
$\blacksquare$