Map

A way of associating unique objects to every point in a given Set. So a map from $A\mapsto B$ is an object $f$ such that for every $a\in A$, there is a unique object $f(a)\in B$. The terms Function and Mapping are synonymous with map.

The following table gives several common types of complex maps.

Mapping	Formula	Domain
Inversion	$f(z) = {1\over z}$
Magnification	$f(z) = az$	$a \in \Bbb{R} \not = 0$
Magnification+Rotation	$f(z) = az$	$a\in\Bbb{C} \not = 0$
Möbius Transformation	$f(z) = {az+b\over cz+d}$	$a, b, c, d \in \Bbb{C}$
Rotation	$f(z) = e^{i\theta}z$	$\theta\in\Bbb{R}$
Translation	$f(z) = z+a$	$a \in \Bbb{C}$

See also 2x mod 1 Map, Arnold's Cat Map, Baker's Map, Boundary Map, Conformal Map, Function, Gauss Map, Gingerbreadman Map, Harmonic Map, Hénon Map, Identity Map, Inclusion Map, Kaplan-Yorke Map, Logistic Map, Mandelbrot Set, Map Projection, Pullback Map, Quadratic Map, Tangent Map, Tent Map, Transformation, Zaslavskii Map

References

Arfken, G. ``Mapping.'' §6.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 384-392, 1985.

Lee, X. ``Transformation of the Plane.'' http://www.best.com/~xah/MathGraphicsGallery_dir/Transform2DPlot_dir/transform2DPlot.html.