Math Revisited: Complex Algebra

One graphical method of representing complex numbers. For example, here C = 4 + 3i


Back in 2010 and earlier this year we examined multiple features of complex analysis, including the nature of complex numbers, complex functions, simple manipulations -  for example,  getting a complex number into polar form, as well as complex roots (e.g. Ö2i), the Cauchy -Riemann equations and residue calculus.  A basic area includes complex algebra, or solving algebraic problems using the principles of complex numbers.  Below are a number of selected problems to challenge math -inclined readers looking for more than Instagram stuff or Trump's idiotic tweets:

1)   Let f(z) = ln r + i(q)  where r = êz ê and q = Arg(z)

Find f(1)

2)   Find f(2i -3) for f(z) = (z + 3)²(z – 5i)² 

3)   Find f(2i) for f(z) =  - 3z²


4)   Let f(z) = e ^(-3z)

Find the real and imaginary parts of the function f(z)

5)  Find all solutions for cos (z) =  5

6) Solve for z if:  sin z =  i sinh  1

7)  If   ln 1  = ln 1 +  ln 0i

Solve for z   

8)   Find: f(-3i) for f(z) = (z + 2 – 3i) ¸ (z + 4 – i)

9)  Find f(1+i) for: f(z) = 1/ (z² + 1)


10)   Solve: (z + 1)³ = z³