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Complex Numbers Problems Part 1
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This is split into parts because MathJaX takes a lot of time to render if there 
is lots of Math on the same page.

Find the square root of following complex numbers:

1. :math:`7+8i`
2. :math:`3+4i`
3. :math:`a^2-b^2+2abi`
4. :math:`7-25\sqrt{-2}`
5. :math:`\sqrt[4]{-81}`
6. Find the square root of

   .. math::
      \frac{x^2}{y^2}+\frac{y^2}{x^2}+\frac{1}{2i}\left(\frac{x}{y}+\frac{y}{x}
      \right) + \frac{31}{16}

7. Find the square root of

   .. math::
      \frac{x^2}{y^2}+\frac{y^2}{x^2}-\frac{1}{i}\left(\frac{x}{y}-\frac{y}{x}
      \right) - \frac{9}{4}

8. Find the square root of

   .. math::
      x^2+\frac{1}{x^2}+4i\left(x-\frac{1}{x}\right)-6

9. Find :math:`\sqrt{2+3\sqrt{-5}}+\sqrt{2-3\sqrt{-5}}`

10. Find :math:`\sqrt{i}\sqrt{-i}`

    Simplify following in the form of :math:`A+iB`

11. :math:`i^{n+80}+i^{n+50}`

12. :math:`\left(i^{17}+\frac{1}{i^{15}}\right)^3`

13. :math:`\frac{(1+i)^2}{2+3i}`

14. :math:`\left(\frac{1}{1+i} + \frac{1}{1-i}\right)\frac{7+8i}{7-8i}`

15. :math:`\frac{(1+i)^{4n+7}}{(1-i)^{4n-1}}`

16. :math:`\frac{1}{1-cos\theta + 2isin\theta}`

17. :math:`\frac{(cosx+isinx)(cosy+isiny)}{(cotu+i)(i+tanv)}`

18. Find the complex number :math:`z` such that :math:`z^2 + |z|=0`

19. Show that for :math:`z\in C, |z|=0` if and only if :math:`z=0`

20. If :math:`z_1` and :math:`z_2` are :math:`1-i` and :math:`2+7i` find
    :math:`Im\left(\frac{z_1z_2}{\overline{z_1}}\right)`

    Find :math:`x` and :math:`y` if

21. :math:`(x+5i)-(3-iy)=7+8i`

22. :math:`\frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y+i}{3-i}=i`

23. If :math:`|z|=1` then prove that :math:`\frac{z-1}{z+1}` is purely
    imaginary.

24. If :math:`|z-i|<1` then prove that :math:`|z+12-6i|<14`

25. If :math:`|z+6|=|2z+3|` then prove that :math:`|z|=3`

26. If :math:`\sqrt{a-ib}=x-iy` prove that :math:`\sqrt{a+ib}=x+iy`

27. Show that the equation

    .. math::
       \frac{A^2}{x-a}+\frac{B^2}{x-b}+ ... + \frac{H^2}{x-h} = x+l

    where :math:`A, B, ..., H; a, b, ..., h \text{ and } l` are real; cannot have
    imaginary roots.

28. Show that a unimodular complex number, not purely real can always be
    expressed as :math:`\frac{c+i}{c-i}` for some real :math:`c`.

29. If the expression

    .. math::
       \frac{sin\frac{x}{2}+cos\frac{x}{2}-itanx}{1+2isin\frac{x}{2}}

    is real, then find all the possible set of values for :math:`x`

30. Find the conjugate, modulus and argument of :math:`\sqrt{3}+2i`

31. Put :math:`\frac{a+ib}{x-iy}` in polar form.

    For any two complex numbers :math:`z_1` and :math:`z_2` prove that

32. :math:`|z_1+z_2|^2+|z_2-z_2|^2 = 2(|z_1|^2 + |z_2|^2)`

33. :math:`|z_1+z_2|^2=z_1^2+z_2^2+2Re(z_1\overline{z_2}) =
    z_1^2+z_2^2+2Re(\overline{z_1}z_2)`

34. If :math:`z_1=1 \text{ and } |z_2|=1` then prove that

    .. math::
       |z_1+z_2|= \left|\frac{1}{z_1}+\frac{1}{z_2}\right|

35. If :math:`|z-2|=2|z-1|` then show that :math:`|z|^2=\frac{4}{3}Re(z)`

36. If :math:`\sqrt[3]{a+ib}=x+iy` then prove that :math:`\frac{a}{x} +
    \frac{b}{y} = 4(x^2 - y^2)`

37. If :math:`x+iy=\sqrt{\frac{a+ib}{c+id}}` then prove that
    :math:`(x^2+y^2)^2= \frac{a^2+b^2}{c^2+d^2}`

38. If :math:`\frac{3}{2+cos\theta+isin\theta}=a+ib` then prove that
    :math:`a^2+b^2=4a-3`

39. If :math:`|2z-1|=|z-2|` then prove that :math:`|z|=1`

40. If :math:`x` is real and :math:`\frac{1-ix}{1+ix}=m+in` then prove that
    :math:`m^2+n^2=1`

41. If :math:`m+in=\frac{x+iy}{x-iy}, \text{ where }  x,y,m,n` are real and
    :math:`x+iy\ne 0` and :math:`m+in\neq 0` then prove that :math:`m^2 + n^2 = 1`

42. If :math:`\left(1+i\frac{x}{a}\right) \left(1+i\frac{x}{c}\right)
    \left(1+i\frac{x}{c}\right) ... = A+iB` then prove that
    :math:`\left(1+\frac{x^2}{a^2}\right) \left(1+\frac{x^2}{b^2}\right)
    \left(1+\frac{x^2}{c^2}\right) ... = A^2+B^2`

43. Let :math:`z_1` and :math:`z_2` be complex numbers such that :math:`z_1\ne
    z_2` and :math:`|z_1|=|z_2|.` If :math:`z_1` has positive real part and
    :math:`z_2` has negative imaginary part then prove that
    :math:`\frac{z_1+z_2}{z_1-z_2}` is either zero or purely imaginary.

44. For complex numbers :math:`z_1=x_1+iy_1` and :math:`z_2=x_2+iy_2` the
    notation of :math:`z_1\cap z_2` if :math:`x_1\le x_2` and :math:`y_1\le y_2.`
    Show that for all complex numbers :math:`z` with :math:`1\cap z` we have
    :math:`\frac{1-z}{1+z}\cap 0.`

45. If :math:`a>0, z|z|+az+1=0,` show that :math:`z` is a negative real number.

46. Find the range of real number :math:`\alpha` for which the equation
    :math:`z+\alpha|z-1|+2i=0; z=x+iy` has a solution. Also, find the solution.

47. For every real number :math:`a\ge 0,` find all the complex numbers
    satisfying the equation :math:`a|z|-4az+1+ia=0`

48. Show that :math:`(x^2+y^2)^5=(x^5-10x^3y^2+5xy^4)^2+(5x^4y-10x^2y^3+y^5)^2`

49. Express :math:`(x^2+a^2)(x^2+b^2)(x^2+c^2)` as sum of two squares.

50. If :math:`(1+x)^n=a_0+a_1x+a_2x^2+ ... +a_nx^n,` prove that
    :math:`2^n=(a_0-a_2+a_4- ...)^2 + (a_1-a_3+a_5- ...)^2`