Complex Variables: Associative and Commutative Laws for Multiplication

3. Use the associative and commutative laws for multiplication to show that: (z1z2)(z3z4) = (z1z3)(z2z4) 4. Prove that if z1z2z3 = 0, then at least one of the three factors is zero. *Please see attachment for proper citation of equations (they did not transfer over into the text box properly)

Complex Variables: Verify Inequality

3. Verify that (sqrt(2))ּ|z| ≥ |Re z| + |Im z|. Suggestion: Reduce this inequality to (|x| - |y|)2 ≥ 0. Sqrt(2) means square root of 2.

Complex Variables: Sketch Points Determined by Given Condition

4. In each case, sketch the set of points determined by the given condition: (a) |z -1 + i| = 1 (b) |z + i| ≤ 3 (c) |z -4i| ≥ 4

Complex Variable: Using Established Properties of Moduli ...

7. Use established properties of moduli to show that when |z3| ≠ |z4|, |z1 +z2| / |z3 + z4| ≤ ( |z1| + |z2|) / | |z3| - |z4| |.

500 level complex variable in undergraduate

The problems are from complex variable 500 level in undergraduate. Please specify your notation(if necessary) and explain clearly each step of your solution. Thank you very much.