How do I find the image $y=x$ under mapping $w=z^2$ ? Also, how do I graph the image?
How do I find the image $y=x$ under mapping $w=z^2$?
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calculus
real-analysis
matrices
complex-analysis
complex-numbers
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1$u+iv=(x+ix)^2=2ix^2$ – 2017-02-18
1 Answers
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Exacly like myGlasses suggests in his comment. Parametrize the domain as: $\{z\in\mathbb{C}:y=x\}$ as $\{z\in \mathbb{C}:z=x+xi\}$ and calculate the image as: $u+iv=(x+xi)^2=2ix^2$.
This will place your line on the $y$ axis. Because the image is an even and non-negative function of $x$, you will get a stretched wrap around 0. Here's some Maple code to see the map:
restart; w := proc (z) options operator, arrow; z^2 end proc; with(plots); complexplot({x+Ix, w(x+Ix)}, x = -1 .. 1, scaling = constrained);
