Tetraviews of Elementary Functions

function

Sin

Cos

Exp

SubscriptBox[\(φ\), \(x, y\)]

SubscriptBox[\(φ\), \(x, u\)]

SubscriptBox[\(φ\), \(x, v\)]

SubscriptBox[\(φ\), \(y, u\)]

SubscriptBox[\(φ\), \(y, v\)]

SubscriptBox[\(φ\), \(u, v\)]

The natural living space of a function

zf(z)

with

z

being a complex variable is the two-dimensional complex space with coordinates

{z,f(z)}

or the four-dimensional real space with coordinates

{x=Re(z),y=Im(z),u=Re(f(z)),v=Im(f(z))}

. This Demonstration shows three-dimensional projections from the four-dimensional space. By carrying out rotations in the four-dimensional space, different views of the function

f(z)

can be obtained (including views of the inverse function

-1

f

(z)

).