Sine Function is Absolutely Convergent/Complex Case

Theorem

The complex sine function $\sin: \C \to \C$ is absolutely convergent.

Proof 1

The definition of the complex sine function is:

$\ds \forall z \in \C: \sin z = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}$

By definition of absolutely convergent complex series, we must show that the power series

$\ds \sum_{n \mathop = 0}^\infty \size {\paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!} }$

is convergent.

We have

\(\ds \ds \sum_{n \mathop = 0}^\infty \size {\paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!} }\)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \frac {\size z^{2 n + 1} } {\paren {2 n + 1}!}\)

Definition of Complex Modulus This article, or a section of it, needs explaining. In particular: If we need this step explained, then it needs to be broken out into a series of steps that illustrates that the complex modulus is absolutely multiplicative, or whatever, and of course that $\size {\paren {-1}^n} = 1$. That is, if we are going to bother doing that here. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

\(\ds \)

\(\le\)

\(\ds \sum_{n \mathop = 0}^\infty \paren{ \frac {\size z^{2 n + 1} } {\paren {2 n + 1}!} + \frac {\size z^{2 n } } {\paren {2 n }!} }\)

Squeeze Theorem for Complex Sequences

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \frac {\size z^n} {n!}\)

changing indices

\(\ds \)

\(=\)

\(\ds \exp \size z\)

Taylor Series Expansion for Exponential Function

The result follows from Squeeze Theorem for Complex Sequences.

$\blacksquare$

Proof 2

Radius of Convergence of Power Series Expansion for Sine Function shows that the radius of convergence of the complex sine function is infinite.

Then Existence of Radius of Convergence of Complex Power Series shows that the complex sine function is absolutely convergent.

$\blacksquare$

Sources

• 2001: Christian Berg: Kompleks funktionsteori: $\S 1.5$