4CRegions

Regions in the Complex Plane

A region in the complex plane is described by a complex inequation.

Example

math \Big \lbrace z: 0 \leqslant Arg(z) < \dfrac {\pi}{4}, \; z \in C \Big\rbrace math

Recall that when graphing regions,
 * a solid line is used to represent £ or ³
 * a broken line is used to represent

Either or
 * the region required can be shaded in,
 * the region __not__ required can be shaded

provided it is clearly indicated somewhere on the graph where the required region is.

Generally, shading the region __not__ required turns out to be more useful for regions of combined graphs.

Example

Sketch math \lbrace z: |z| <3 \rbrace math

This would give a circle
 * centred at the origin
 * with radius = 3.
 * Drawn with a broken line
 * Region required is __inside__ the circle.

If unsure about which section of a graph is the required region,
 * select a simple point which is either clearly inside or clearly outside of the shape
 * see if it meets the inequality requirement.

eg
 * select **z = 1 + 0i**
 * |1 + 0i| gives1 **< 3**,
 * so **1 + 0i** __is__ within required region.

Combining Curves or Regions

When two curves or regions are combined using the " È " (Union) symbol.
 * __All__ points from the two relations are included.

When two curves or regions are combined using the " Ç " (Intersection) symbol.
 * Only the points that belong to both relations are included. (ie the overlap between the two)

Example

math \big\lbrace z: \; \left| z+2 \right| =2 \big\rbrace \cup \big\lbrace z: \; \left| z-2 \right| =2 \big\rbrace math

The Union of the two circles means __all__ the points in __both__ circles are included.

Example

math \big\lbrace z: \; \left| z-2i \right| \leqslant 4 \big\rbrace \cap \big\lbrace z: \; Im(z) >2 \big\rbrace math

To find the intersection of the two regions,
 * shade the areas NOT required by each relation.
 * The remaining clear area will be the region required.

In the diagram to the right,
 * I have shaded everything under the Im(z) = 2 line
 * with a diagonal sloping up to the right.


 * Everything outside the circle
 * is shaded with a diagonal sloping down to the right.


 * Some areas have both types of shading.

The clear area is therefore the region required.

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