Vtyjkyuk

Context: A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 9.1, 9.3 (asked about here)

Question

From (here): How do we have that $$lim_z rightarrow z_o f(z) in mathbbC textfor removable singularity z_0 tag1$$ under the characterisation that $f$ has a removable singularity if $$lim_z rightarrow z_o (z-z_0) f(z) = 0 tag2$$?

Update: Attempt is now moved to answer.

Some intuition:

$(2) Leftarrow (1)$

• If $L in mathbbC$, then $lim_z rightarrow z_o (z-z_0) f(z) = lim_z rightarrow z_o (z-z_0) lim_z rightarrow z_o f(z) = (0)(L) = 0$

$(2) Rightarrow (1)$

• If $L = infty$, then $lim_z rightarrow z_o (z-z_0) f(z) = 0 cdot infty$, indeterminate.




• If $L$ dne, then $lim_z rightarrow z_o (z-z_0) f(z)$ dne.

Upon reflection, this actually seems like a simple complex extension of a simple elementary real analysis proof.

Pf $(2) iff (1)$:

Firstly, let $lim_z rightarrow z_o f(z) =: L$, a letter which could be a finite complex number, $infty$ or 'dne'.

Now for the forward implication:

$(2) Leftarrow (1)$:

• Given: $forall varepsilon_1 > 0, exists delta_1 > 0: |f(z)-L|<varepsilon_1 textwhenever 0<|z-z_0|<delta_1$




• TS: $forall varepsilon > 0, exists delta > 0: |(z-z_0)f(z)|<varepsilon textwhenever 0<|z-z_0|<delta$




• Pf: Let $varepsilon > 0$. $$|(z-z_0)f(z)| = |(z-z_0)[f(z)-L+L]|$$

$$= |(z-z_0)| |f(z)-L+L| le |z-z_0| (|f(z)-L|+|L|)$$

$$stackrel(*)le |z-z_0| varepsilon_1+|L| < varepsilon textwhenever 0 < |z-z_0| < delta := minfracvarepsilonvarepsilon_1+,delta_1$$

(*) $textwhenever 0 < |z-z_0| < delta_1$

QED $(2) Leftarrow (1)$

Now for the backward implication:

$(2) Rightarrow (1)$

• Given: $forall varepsilon > 0, exists delta > 0: |(z-z_0)f(z)|< varepsilon textwhenever 0<|z-z_0|<delta$




• TS: $forall varepsilon_1 > 0, exists delta_1 > 0: |f(z)-L|< varepsilon_1 textwhenever 0<|z-z_0|<delta_1$




• Pf: Let $varepsilon_1 > 0$. $$|f(z)-L| = |frac(z-z_0)[f(z)]z-z_0-L| le |frac(z-z_0)[f(z)]z-z_0| + |L|$$

$$stackrel(**)le |fracvarepsilonz-z_0| + |L| le varepsilon_1 textwhenever 0<|z-z_0|<delta_1 := mindelta,fracvarepsilonL$$

(**) $textwhenever 0 < |z-z_0| < delta$

QED $(2) Rightarrow (1)$

Now both the forward and backward implications have been proven.

QED $(2) iff (1)$