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2008-05-18T19:11:51Z

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Vipul: New page: {{particular entire function}} ==Definition== The '''cosine function''' is a holomorphic function from $\mathbb{C}$ to $\mathbb{C}$ (i.e., an entire function). ...

2008-04-27T21:19:10Z

New page: {{particular entire function}} ==Definition== The '''cosine function''' is a holomorphic function from <math>\mathbb{C}</math> to <math>\mathbb{C}</math> (i.e., an entire function). ...

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{{particular entire function}}

==Definition==

The '''cosine function''' is a holomorphic function from <math>\mathbb{C}</math> to <math>\mathbb{C}</math> (i.e., an [[entire function]]). It is defined as:

<math>\cos z := \sum_{n=0}^\infty (-1)^n \frac{z^2n}{(2n)!}</math>

Alternatively, it can be defined as:

<math>\cos z := \frac{\exp(iz) + \exp(-iz)}{2}</math>

For <math>z \in \R</math>, it coincides with the [[real cosine function]].

The cosine function does not extend to a [[meromorphic function]] on the [[Riemann sphere]]: it has an [[essential singularity]] at <math>\infty</math>.

Cosine function - Revision history

Vipul: 1 revision

Vipul: New page: {{particular entire function}} ==Definition== The '''cosine function''' is a holomorphic function from \mathbb{C} to \mathbb{C} (i.e., an entire function). ...

Vipul: New page: {{particular entire function}} ==Definition== The '''cosine function''' is a holomorphic function from $\mathbb{C}$ to $\mathbb{C}$ (i.e., an entire function). ...