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Evaluating complex multiplication?

  1. Oct 26, 2003 #1
    How might one comprehend the product of complex numbers

    N
    [pi](an+ibn)=C
    n=1

    such as by representation with vectors in the complex plane, or algebraic simplification? Specifically, I would like to know the value Re(C).
     
  2. jcsd
  3. Oct 26, 2003 #2
    Hi

    Any complex number a+ib can be represented as R(& theta). Where R is the magnitude and & theta is the phase. So convert the complex number into R(& theta) form and multiply. The R parts multiply while the angle parts add up.

    e.g. R(& theta)* P(& Theta) = R*P(& theta + & Theta)

    Got it???


    Sridhar
     
  4. Oct 26, 2003 #3
    A bit rough, sridhar, but helpful in jogging my memory.

    Can you or another be more mathematical in regard to the transform involved?

    Is it tan-1(b/a)=[the] and r=(a2+b2)1/2?
     
  5. Oct 26, 2003 #4

    Hurkyl

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    Staff Emeritus
    Science Advisor
    Gold Member

    Right...ish

    if (a + bi) = r exp(iθ), then it is true that

    r = (a^2 + b^2)^(1/2)
    and
    tan θ = b/a

    But you have to make sure that θ is in the correct quadrant. (iow you might have to add π).
     
  6. Oct 26, 2003 #5
    Hurkyl,

    Is there a simplifying (exact) identity for the arithmetic series

    N
    [sum]tan-1(bn/an)
    n=1

    and for the geometric series

    N
    [pi](an2+bn2)1/2
    n=1

    or, more importantly, for my original statement concerning Re(C)?
     
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