Image of x+y=1 under f(z) = z^2

Thread starter complexnumber
Start date Jul 31, 2009
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SUMMARY

The discussion focuses on determining the image of the line defined by the equation x + y = 1 under the complex function f(z) = z^2. By substituting z = x + i(1 - x) into the function, the transformation yields u(x,y) = x^2 - y^2 and v(x,y) = 2xy. This results in a mapping of the line in the complex plane to a new set of coordinates defined by these equations, illustrating the behavior of the function on the specified line.

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Understanding of complex functions, specifically f(z) = z^2
Familiarity with complex number representation in the form z = x + iy
Knowledge of mapping in the complex plane
Basic algebraic manipulation of equations

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Explore the properties of complex functions, particularly polynomial mappings
Learn about the geometric interpretation of complex transformations
Investigate the implications of the Cauchy-Riemann equations on complex mappings
Study the effects of linear transformations on the complex plane

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Jul 31, 2009

complexnumber

For f(z) = z^2 find the image of x + y = 1

f(z) = z^2 = (x + iy)^2 = x^2 + 2ixy - y^2

u(x,y) = x^2 - y^2
v(x,y) = 2xy

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Jul 31, 2009

Mark44

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Here's how I would approach this problem. Every point on the line has coordinates (in the complex plane) of (x, 1 - x). A point on this line can be represented as z = x + i(1 - x). What does your function f do to these complex numbers?