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Properties of Complex Variables

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Detailed step by step calculations of the attached questions regarding complex variables including the domain, limits and continuity of complex functions.

1. For each of the functions below, describe the domain of definition, and write each function in the form f(z) = u(x,y) +iv(x,y)

1) f(z) = z^2 / (z+z)
2) f(z) = z^3
3) f(z) = |z| + z

2. Suppose that f(z) = x^2 - y^2 - 2y + i(2x-2xy), where z = x +iy. Use the expressions
x = z+z / 2 and y = z-z /2i

to write f(z) in terms of z, and simplify the result.

3. Sketch the following sets and determine which are domains. If the set is NOT a domain, briefly explain why not.

[see the attachment for the full problem]

4. Evaluate the following limit. Show steps. If the limit does not exist, explain why not.

5. Let [see the attachment for the full equation]

Determine the value w so that the function f is continuous at z=0

6. Let f(x_iy) = e^x (cos y + i sin y) for all x, y, R. Prove that f is continuous on the complex plain.

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Step-by-step solutions that find the domain, limit and continuity of complex variables.

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