# Complex map

Complex maps . By default the identity map is displayed, but other maps can be chosen. The left grid represents the z complex plane (the domain of f), and the right grid represents the w complex plane (the range of f). Moving the mouse around the z plane will cause a pointer to move in the w plane according to whatever complex map was selected. The spacings in the grids are one unit long; thus both grids have real part and imaginary part ranging from -5 to 5. Dragging the mouse around on the z plane will trace out a red curve on the z plane, as well as a corresponding curve on the w plane. To get the most out of this applet, try the functions one by one until you understand that each of them does, and why it does it.
Complex mapping is applying to the complex number a function that computes from it some (usually also complex) value. Hence complex function is not much different from the real function like . This article and two applets provide overview of the most important mappings.

Each complex mapping has the following features that may be different for different functions:

• Is shape preserved? Does the square maps to the square, would the written text remain readable after such a transform?
• Is orientation preserved? If you draw a clockwise arrow on the domain, is the arrow on the range also clockwise or it is instead anticlockwise?
• Are angles preserved? If you intersect two lines at right angles (for instance), do the two curves also intersect at right angles after the mapping is applied?
• Are there any singularities of the map? One usually has very erratic behaviour at or near singularity.
• What are the images of real (x, or Re(z)) and imaginary (y, or Im(z)) axis? Do they still intersect at right angles?

## Simple functions

We list below some of the most widely known complex mappings. The questions in this list can be easily answered by experimenting with the applet.

• f(z) = z*, the identity map, boring.
• f(z) = z+i, a shift upwards by one unit.
• f(z) = 2z, a dilation by 2. Note that the right-hand grid is not large enough to contain the entire range.
• f(z) = -z, a reflection through the origin. Is orientation preserved?
• f(z) = iz, a clockwise rotation by the right angle. How to rotate anticlockwise by the same angle?
• f(z) = (1+i)z, a dilation combined with a rotation. What is the dilation factor and how much are the things rotated?
• f(z) = zbar, the conjugation map (zbar is an ASCII way of writing z with an overline on top of it). How would you describe it graphically and does it preserve orientation?
• f(z) = z^2, the squaring. While the magnitude gets squared, the phase gets doubled. A straight line on the domain maps to the conic section. If you trace around the origin once anticlockwise, how many times the mapped value moves around its origin? Is orientation preserved? Are angles preserved?
• f(z) = Re(z), f(z) = Im(z) - picking only real or only imaginary part. The mapped point is always on the coordinate axis.
• f(z) = x + 2iy keeps the real part x unchanged, but doubles the imaginary part y. Are angles preserved?
• f(z) = 1/z, the inversion map, and also the standard example of a Möbius transformation. The magnitude is reciprocated, while the phase gets flipped. What is the mapping of the straight line (hold shift to draw it)? What is the mapping of a circle which passes through the origin? What about circle that does not go through the origin? If you trace around the origin anticlockwise, how does the mapped point move? What about if you trace around some place other than the origin? Are angles preserved?
• f(z) = 9/zbar, a reflection through the circle {z: |z|=3}. What is the mapping of a straight line? What happens near the origin? Is orientation preserved for circles that are away from the origin? For circles containing the origin? Trace out some grid lines.
• f(z) = exp(z) is more interesting than the exponent of the scalar value, as introducing the imaginary part (by moving up or down) forces the mapped point to circle around the origin, as defined by the Euler formula:

Moving from left to right shows exponential growth, same as with real values.

## Multi value functions

Functions may have several possible values. For instance, may return either x or -x. One of the values has been designated as the "principal" value of the function. Picking only principal value restricts the multi-valued function to a single-valued function. Such restricted function that is the branch of the original function. The branch that returns the principal value is the principal branch.

Mapping of functions which are multi-valued rather than single-valued. The principal value is marked by a red ball, and you can optionally choose to show the principal branch only. Most of branches have a discontinuity at the negative real axis. If you move the mouse across this line, the principal branch will suddenly switch from one value of the multiple-valued function to another. The negative real line is called a branch cut for the principal branch. Branch cuts are a necessary whenever one tries to prune a multiple-valued function into a single valued function.

• f(z) = |z|, the absolute value, repeats 'z' above the real axis but also creates a mirror image of z below the axis. This image is flipped not only vertically but also horizontally, as if passing a converging lens of some kind.
• f(z) = z^{1/2}, the square root, has two branches. The principal branch always stays in the upper half plane and another branch is a flipped reflection of the principal branch. Lines are mapped into conic sections. Dragging the mouse around the origin feels like rotating some geared mechanism: the mapped value also rotates in the same direction, but two times slower. Do perpendicular lines remain perpendicular?
• f(z) = z^{1/3}, the cube root, has three branches, rotated from each other by 120 degrees. The "gearing mechanism" now slows rotation three rather than two times.
• f(z) = z^{1/4} has the similarly rotated four branches, f(z) = z^{1/5} has five branches and so on. They similarly divide the plane into the equal-angle sectors (one per branch). One full revolution of the initial z around the origin map to the crossing of one such sector in the transform.
• f(z) = log(z) actually has the infinite number of values, but we only display three at a time. The values of the principal branch of the logarithm stay inside a certain strip (what is this strip?).

What happens to the logarithm if you move the mouse toward the origin, away from it, or rotate around the origin? What happens when you cross the branch cut?