**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)

**F****ind f(2i -3) for f(z) = (z + 3)**^{2}(z – 5i)^{2}_{}

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_{}

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**Find f(2i) for f(z) = - 3z**

^{2}_{}

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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)**

_{}^{}

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

^{2}+ 1)**10)**

**Solve: (z + 1)**

^{3}= z^{3}_{}

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