(problem 1) Sketch each the following:
\begin{align*} i) & \; |z+1-i| = 2 \\ ii) & \; |z-1| + |z+i| = 4 \\ iii) & \; Re(z) \geq 2 \\ iv) & \; \gamma (t) = (t-1) + it^2, 0\leq t \leq 2 \\ v) & \; |i- z| < 3 \\ vi) & \; |z-i| - |z+i| = 1 \\ vii) & \; \gamma (t) = 2\cos t + 2i\sin t , \pi /4 \leq t \leq 3\pi /2 \\ viii) & \; 1 < |z -2i| < 4 \\ ix) & \; |z| > 1 \\ x) & \; \gamma (t) = \cos t |\cos t| + i\sin t |\sin t|, 0 \leq t \leq 2\pi \end{align*}
We learn to recognize and sketch special sets in the complex plane.
1 Circles and Disks
2 Lines and Half Planes
example 4 The set of points satisfying the equation where is a horizontal line and
the set of points satisfying the equation where is a vertical line. We can also express
these lines as and with the understanding that . Moreover, we can write to
represent a line in the complex plane with a slope of and intersecting the imaginary
axis at .
example 5 The set of points satisfying the inequality where is a half plane. If this
set is called the lower half plane (LHP).
3 Ellipses and Hyperbolas
4 Other Curves
If and are differentiable functions of the real variable and if and do not both vanish simultaneously, then is a smooth curve. Furthermore, if , but for all , then is called a simple closed curve.
example 8 The arc of a circle of radius centered at that goes from angle to angle
traversed in the counter-clockwise direction is given by
example 9 A straight line segment from the point to the point is given by This is
also commonly written as:
Jordan Closed Curve Theorem The complement of any simple closed curve can be
partitioned into two mutually exclusive domains, and , in such a way that is
bounded, is unbounded, and is the boundary for both and . In addition is the
entire complex plane. The domain is called the interior of , and the domain is called
the exterior of .
5 Problems
Here is a video solution to problem 1viii:
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Here is a video solution to problem 1x:
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