Limits of Real and Imaginary Parts
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Theorem
Let $f: D \to \C$ be a complex function, where $D \subseteq \C$.
Let $z_0 \in D$ be a complex number.
Suppose $f$ is continuous at $z_0$.
Then:
- $(1): \quad \ds \lim_{z \mathop \to z_0} \map \Re {\map f z} = \map \Re {\lim_{z \mathop \to z_0} \map f z}$
- $(2): \quad \ds \lim_{z \mathop \to z_0} \map \Im {\map f z} = \map \Im {\lim_{z \mathop \to z_0} \map f z}$
where:
- $\map \Re {\map f z}$ denotes the real part of $\map f z$
- $\map \Im {\map f z}$ denotes the imaginary part of $\map f z$.
Proof
- $\forall \epsilon > 0: \exists \delta > 0: \cmod {z - z_0} < \delta \implies \cmod {\map f z - \map f {z_0} } < \epsilon$
Given $\epsilon > 0$, we find $\delta > 0$ so for all $z \in \C$ with $\cmod {z - z_0} < \delta$:
\(\ds \epsilon\) | \(>\) | \(\ds \cmod {\map f z - \map f {z_0} }\) | ||||||||||||
\(\ds \) | \(\ge\) | \(\ds \cmod {\map \Re {\map f z - \map f {z_0} } }\) | Modulus Larger than Real Part | |||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {\map \Re {\map f z} - \map \Re {\map f {z_0} } }\) | Addition of Real and Imaginary Parts |
It follows that:
- $\forall \epsilon > 0: \exists \delta > 0: \cmod {z - z_0} < \delta \implies \cmod {\map \Re {\map f z} - \map \Re {\map f {z_0} } } < \epsilon$
Then equation $(1)$ is proven by:
\(\ds \lim_{z \mathop \to z_0} \map \Re {\map f z}\) | \(=\) | \(\ds \map \Re {\map f {z_0} }\) | Definition of Limit of Complex Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {\lim_{z \mathop \to z_0} \map f z}\) | Definition of Continuous Complex Function |
The proof for equation $(2)$ with imaginary parts follows when $\Re$ is replaced by $\Im$ in the equations above.
$\blacksquare$