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