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 roots, 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
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