agm variants of ordinary trigonometric functions

The ordinary sine and cosine are averages of exponentials. Here, I play around with replacing the arithmetic average with the arithmetic geometric mean

Let \mathrm{cas}(z)=\mathrm{agm}(e^{iz},e^{-iz})

And since the $\mathrm{agm}(x,y)$ may be represented in terms of a complete elliptic integral of the first kind $K(m)$:

K(m)=\int_0^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-m\sin^2(\theta)}}

\mathrm{agm}(x,y) = \frac{\pi}{4} (x + y) \; / \; K\!\left[\left( \frac{x - y}{x + y} \right)^2 \right]

Therefore
\mathrm{cas}(z) = \mathrm{agm}(e^{iz},e^{-iz}) = \frac{\pi}{4} (e^{iz} + e^{-iz}) \; / \; K\!\left[\left( \frac{e^{iz} - e^{-iz}}{e^{iz} + e^{-iz}} \right)^2 \right]

Therefore:
\mathrm{cas}(z) = \frac{\pi\cos(z)}{2K(-tan^{2}(z))}

Domain colored, with mpmath, it sort of looks like this:

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s