Complex Numbers - Forms and Parts

  1. Hi, I have a complex number and understand that the rectangular form of the number is represented by

    s = σ + jω, where σ is the real part and jω is imaginary.

    I am having trouble locating them in the number below:

    I know that "2" is a real number, and the numerator is imaginary along with j*2*pi*k. Since the numerator is dividing both the elements at the bottom, does this number have a real and imaginary part? (This is where I am a little confused).

    My guess would be that σ = 2 and the rest is imaginary.

    If I could figure out what parts are real and imaginary, I can go on to find the rectangular form and the polar form.

  2. jcsd
  3. Euler's relation e^ix = cos(x) + i*sin(x) vastly simplifies the exponential in the numerator. Cos and sin have period 2pi so the numerator is -1. Then multiply numerator and denominator by the conjugate of the denominator (assuming k is real). That leaves you with a real number in the denominator and a complex number in the numerator whose real and imaginary parts can be readily evaluated.

    By the way, multiplying numerator and denominator by the conjugate of the denominator is the standard thing to do with this kind of problem.
    Last edited: Feb 11, 2012
  4. HallsofIvy

    HallsofIvy 41,267
    Staff Emeritus
    Science Advisor

    No, the numerator is NOT imaginary. In fact it is real- it is [itex]e^{j3\pi}= cos(3\pi)+ jsin(3\pi)= -1[/itex].
    Last edited: Feb 12, 2012
  5. Just try a look into and enter

    Exp[3 Pi I] / (2 + 2 Pi k I)

    what is the Mathematica version of your formula
  6. Deveno

    Deveno 906
    Science Advisor

    as others have suggested, evaluate the numerator at the specific angle (3pi), and then multiply the resulting fraction by:

    [tex]\frac{2 - j2\pi k}{2 - j2\pi k} (= 1)[/tex]

    to make the denominator real.
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