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Homework Help: Complex variables(1)

  1. Sep 23, 2010 #1
    1. The problem statement, all variables and given/known data
    I'm beginning my studies in complex variables and have some questions...

    Q1) We know that x2=9 => x=+/- √9 = +/- 3.
    Suppose z^2 = w where z and w are COMPLEX numbers, then is it still true to say that
    z = +/- √w ? Why or why not?

    Q2) "Let az2 + bz + c =0, where a,b,c are COMPLEX numbers, a≠0.
    Then the usual quadratic formula still holds."

    My concern is with the √(b2-4ac) part. How can we find √(b2-4ac) when b2-4ac is a COMPLEX number?
    For example, what does √(-1+4i) mean on its own and how can we find it? I know there is a general procedure(using polar form and angles) to solve for the nth root of a complex number (z^n=w), but I still don't understand what √(-1+4i) means on its own.
    Even for real numbers, there is a difference between solving x2=9 and finding √9, right? So is there any difference between finding √(-1+4i) and solving z2=-1+4i for z using polar form and angles?

    2. Relevant equations
    Complex variables

    3. The attempt at a solution
    As shown above.

    I hope someone can explain these. Any help is much appreciated!
     
    Last edited: Sep 23, 2010
  2. jcsd
  3. Sep 23, 2010 #2

    Office_Shredder

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    z = +/- √w loses meaning because there is no distinguished choice for √w. Normally we define √x to be the positive square root of x, but there's no such thing as positive and negative in the complex numbers. However what we CAN say is that if a2=w and b2=w, then a2-b2=0, and factoring (a-b)(a+b)=0. So a=b or a=-b

    √(-1+4i) is just going to mean a complex number which, when squared, gives -1+4i. It doesn't specify which one, but for the quadratic formula it doesn't matter which one you pick, because they both work. When writing [tex] \pm[/tex] in the quadratic formula for real numbers, what you really mean is that you can use either square root of the number and you will get a root. The same holds for complex numbers as well
     
  4. Sep 24, 2010 #3
    Q2) z= [-b +/- sqrt(b^2 - 4ac)] / 2a

    Say b^2 - 4ac turns out to be -1+4i
    Let w^2 = -1+4i. Use polar form and angles to solve for w, we get 2 solutions, call them w1 and w2.

    And you mean we can pick either one of w1, w2 as √(-1+4i), right?
    Say if I pick w1, then the solutions will be
    z=[-b +/- w1] / 2a

    Alternatively, if I pick w2, then the solutions will be
    z=[-b +/- w2] / 2a which will be the SAME as the above, right? Why are they the same?

    Thanks for explaining!!
     
  5. Sep 24, 2010 #4

    ehild

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    How are w1 and w2 related? Compare their polar forms.

    ehild
     
  6. Sep 25, 2010 #5
    I think w1 = -w2, but is this always true? Why or why not?
     
  7. Sep 25, 2010 #6

    ehild

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    w1 and w2 can be written also in polar form:

    as w=|w1|=|w2|,

    and -1=e

    w1=w eîφ and w2=-w1=w ei(φ+Π).

    Use w1 and w2 in the quadratic form instead of ±.



    ehild
     
    Last edited: Sep 25, 2010
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