Software question

I am looking for a software package that can solve simple random equations like f(x,y) as an "input" and plot the 3-D results as the "output" using x, y and jx. My object is to view both real and imaginary parts of the graph on one x, y and jx 3-D coordinate system while using y as the independent variable. Complex math softwa ...continues

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 Variables: Properties of Moduli

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

Complex Variables : De Moivre's Theorem and Rectangular Coordinates

5. Use de Moivre’s formula to derive the following trigonometric identities. (a) cos 3θ = cos3 θ – 3cos θּsin2 θ (b) sin 3θ = 3cos2 θּsin θ – sin3 θ 6. By writing the individual factors on the left in exponential form, performing the needed operations, and finally changi ...continues

Complex Variables : Lagrange's Trigonometric Identity

10. Establish the identity 1 + z + z2 + ּּּּ +zn = (1 + zn+1) / (1 – z) (z ≠ 1) and then use it to derive Lagrange’s trigonometric identity: 1 + cos θ + cos 2θ + ּּּ + cos nθ = ½ + (sin [(2n + 1)θ / 2]) / 2sin(θ/2) (0 < θ ...continues

Proof : Nth Roots - Show that if c is any nth root of unity other than unity itself, then 1 + c + c^2 + ּּּּ + c^(n-1) = 0.

7. Show that if c is any nth root of unity other than unity itself, then 1 + c + c^2 + ּּּּ + c^(n-1) = 0.

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.

Simplify a Complex Expression

Suppose that f (z) = x2 – y2 – 2y + iּ(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.