Tuesday, December 10, 2013

Formulas of complex numbers


  • Complex conjugate number

    z=a+ib

    \overline{z} = a - ib.\,

  • Absolute value of complex number

    \textstyle r=|z|=\sqrt{x^2+y^2}.\,

  • Polar form

    z = r(\cos \varphi + i\sin \varphi ).\,

  • Euler's formula

    e^{ix} = \cos x + i\sin x\,

  • De Moivre's formula

    (\cos x + i \sin x)^n = \cos (nx) + i \sin (nx).\,

  • n-th roots of a complex number

    \sqrt[n]{z} = \sqrt[n]r \left( \cos \left(\frac{\varphi+2k\pi}{n}\right) + i \sin \left(\frac{\varphi+2k\pi}{n}\right)\right)
    for any integer k satisfying 0 ≤ k ≤ n − 1

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