Complex maps

This applet demonstrates a number of complex maps w = f(z). By default the identity map f(z) = z 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. The "Clear" button will clean the grids.

To get the most out of this applet, try the functions one by one (I recommend starting with the top map and moving downwards), and play with each function until you understand exactly what it does, and why it does it. (Some of the later maps are less intuitive, but do try to make the effort to understand them).

More notes and things to try appear below.

General notes:

With each map, ask yourself the following questions:

Is shape preserved by this map? If you draw a square (for instance) on the domain, is a square drawn on the range?

Is orientation preserved by this map? If you draw a clockwise arrow on the domain, is the arrow on the range clockwise or anticlockwise?

Are angles preserved by this map? If you intersect two lines at right angles (for instance) on the domain, do the two curves on the range also intersect at right angles?

Are there any singularities of the map, i.e. points in the domain where the map becomes discontinuous or undefined? One usually has very erratic behaviour if one tries to approach a singularity.

Trace some grid lines on the domain (e.g. real axis, imaginary axis, Re(z)=1). What are the images of these lines (i.e. what gets drawn on the right-hand side of the grid)? Do they intersect at right angles?

Some notes on individual functions:

f(z) = z
This map is called the identity map, and is extremely boring.

f(z) = z+i
This is a translation upwards by one unit.

f(z) = 2z
This is a dilation by 2. Note that the right-hand grid is not large enough to contain the entire range of this function.

f(z) = -z
This is a dilation by -1, a.k.a. a reflection through the origin. Is orientation preserved?

f(z) = iz
This is a clockwise rotation by pi/2. How would one make an anticlockwise rotation by pi/2?

f(z) = (1+i)z
This is a dilation combined with a rotation. What is the dilation factor of this map, and how much does it rotate things?

f(z) = zbar
This is the conjugation map (zbar is a lousy ASCII way of writing z with an overline on top of it). How would you describe it graphically? Does it preserve orientation?

f(z) = z^2
This is the squaring map; verify that the square of -2 is 4, the square of i is -1, etc. Note that the magnitude gets squared, while the phase gets doubled. If you draw a straight line on the domain, what do you get on the range? (hint: it is a conic section). If you trace around the origin once anticlockwise on the domain, how many times do you trace around the origin on the range? Is orientation preserved? Are angles preserved?

f(z) = Re(z)
This function returns the real part of z. You'll notice one very distinctive thing about this function: it has a very limited range! Clearly this function does not preserve shape or angle.

f(z) = x + 2iy
This function keeps the real part x unchanged, but doubles the imaginary part y. Are angles preserved by this map (e.g. does a 45 degree angle on the left grid correspond to a 45 degree angle on the right)?

f(z) = 1/z
This is the inversion map, which is the standard example of a Möbius transformation. Verify that the reciprocal of 2 is 1/2, the reciprocal of -i is i, etc. Note that magnitude is reciprocated, while the phase gets flipped. You may have to stick fairly close to the origin to see anything at all appear on the range. If you draw a straight line on the domain, what do you get on the range? If you draw a circle on the domain which passes through the origin, what do you get on the range? Try to draw the circle as carefully as possible. What about circles that don't go through the origin? If you trace around the origin once anticlockwise on the domain, what happens on the range? What about if you trace around someplace other than the origin? What happens as you get very close to the origin? Are angles preserved?

f(z) = 9/zbar
This map represents reflection through the circle {z: |z|=3}. How is it related to the inversion map f(z) = 1/z? What is the image of a straight line under this map? What happens near the origin? Is orientation preserved for circles away from the origin? What about circles containing the origin? Trace out some grid lines for this function.

f(z) = exp(z). If you move z to the right, then exp(z) gets larger; if you move z to the left, it gets smaller. That's not surprising; it works the same way with the real exponential. But what happens if you move z up or down? Does this map preserve shape? size? angles? orientation? What happens if z is imaginary? Is the map one-to-one? Do you think exp(z) can ever reach zero exactly?

	Is shape preserved by this map? If you draw a square (for instance) on the domain, is a square drawn on the range?
	Is orientation preserved by this map? If you draw a clockwise arrow on the domain, is the arrow on the range clockwise or anticlockwise?
	Are angles preserved by this map? If you intersect two lines at right angles (for instance) on the domain, do the two curves on the range also intersect at right angles?
	Are there any singularities of the map, i.e. points in the domain where the map becomes discontinuous or undefined? One usually has very erratic behaviour if one tries to approach a singularity.
	Trace some grid lines on the domain (e.g. real axis, imaginary axis, `Re(z)=1`). What are the images of these lines (i.e. what gets drawn on the right-hand side of the grid)? Do they intersect at right angles?

	`f(z) = z` This map is called the identity map, and is extremely boring.
	`f(z) = z+i` This is a translation upwards by one unit.
	`f(z) = 2z` This is a dilation by 2. Note that the right-hand grid is not large enough to contain the entire range of this function.
	`f(z) = -z` This is a dilation by -1, a.k.a. a reflection through the origin. Is orientation preserved?
	`f(z) = iz` This is a clockwise rotation by `pi/2`. How would one make an anticlockwise rotation by `pi/2`?
	`f(z) = (1+i)z` This is a dilation combined with a rotation. What is the dilation factor of this map, and how much does it rotate things?
	`f(z) = zbar` This is the conjugation map (`zbar` is a lousy ASCII way of writing `z` with an overline on top of it). How would you describe it graphically? Does it preserve orientation?
	`f(z) = z^2` This is the squaring map; verify that the square of -2 is 4, the square of `i` is -1, etc. Note that the magnitude gets squared, while the phase gets doubled. If you draw a straight line on the domain, what do you get on the range? (hint: it is a conic section). If you trace around the origin once anticlockwise on the domain, how many times do you trace around the origin on the range? Is orientation preserved? Are angles preserved?
	`f(z) = Re(z)` This function returns the real part of `z`. You'll notice one very distinctive thing about this function: it has a very limited range! Clearly this function does not preserve shape or angle.
	`f(z) = x + 2iy` This function keeps the real part `x` unchanged, but doubles the imaginary part `y`. Are angles preserved by this map (e.g. does a 45 degree angle on the left grid correspond to a 45 degree angle on the right)?
	`f(z) = 1/z` This is the inversion map, which is the standard example of a Möbius transformation. Verify that the reciprocal of 2 is 1/2, the reciprocal of `-i` is `i`, etc. Note that magnitude is reciprocated, while the phase gets flipped. You may have to stick fairly close to the origin to see anything at all appear on the range. If you draw a straight line on the domain, what do you get on the range? If you draw a circle on the domain which passes through the origin, what do you get on the range? Try to draw the circle as carefully as possible. What about circles that don't go through the origin? If you trace around the origin once anticlockwise on the domain, what happens on the range? What about if you trace around someplace other than the origin? What happens as you get very close to the origin? Are angles preserved?
	`f(z) = 9/zbar` This map represents reflection through the circle `{z: \|z\|=3}`. How is it related to the inversion map `f(z) = 1/z`? What is the image of a straight line under this map? What happens near the origin? Is orientation preserved for circles away from the origin? What about circles containing the origin? Trace out some grid lines for this function.
	`f(z) = exp(z)`. If you move `z` to the right, then `exp(z)` gets larger; if you move `z` to the left, it gets smaller. That's not surprising; it works the same way with the real exponential. But what happens if you move `z` up or down? Does this map preserve shape? size? angles? orientation? What happens if `z` is imaginary? Is the map one-to-one? Do you think `exp(z)` can ever reach zero exactly?