Removable Singularity/Examples/Sine of z over z

Example of Removable Singularity

Let $f: \C \setminus \set 0 \to \C$ be the complex function defined as:

$\map f z = \dfrac {\sin z} z$

Then $f$ has a removable singularity at the point $z = 0$.

Proof

We are given that $f: \C \setminus \set 0 \to \C$ is the complex function:

$\map f z = \dfrac {\sin z} z$

By Limit of Sine of X over X at Zero:

$\ds \lim_{z \mathop \to 0} \frac {\sin z} z = 1$

So $f$ is holomorphic

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Hence $f$ has a removable singularity at the point $z = 0$.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): singular point (singularity): 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): singular point (singularity): 1.