Formulas and Tables

Number Sets

Complex Numbers

Set of complex numbers: \(\mathbb{C}\)
Imaginary unit: \(i\)
Complex numbers: \(z\), \({z_1}\), \({z_2}\)
Real numbers: \(a\), \(b\), \(c\), \(d\)

Absolute value of a complex number: \(r\), \({r_1}\), \({r_2}\)
Argument of a complex number: \(\varphi\), \({\varphi_1},\) \({\varphi_2}\)
Whole numbers: \(k\)
Natural numbers: \(n\)

  1. Algebraic form of complex number \(z = a + bi\)
  2. Powers of the imaginary unit
Powers of the imaginary unit
  1. Complex plane
Complex plane
  1. Equality of complex numbers
    \(a + bi = c + di\), if \(a = c\) and \(b = d\)
  2. Addition of complex numbers
    \(\left( {a + bi} \right) + \left( {c + di} \right) =\) \( \left( {a + c} \right) + \left( {b + d} \right)i\)
  3. Subtraction of complex numbers
    \(\left( {a + bi} \right) – \left( {c + di} \right) =\) \( \left( {a – c} \right) + \left( {b – d} \right)i\)
  4. Multiplication of complex numbers
    \(\left( {a + bi} \right)\left( {c + di} \right) =\) \( \left( {ac – bd} \right) + \left( {ad + bc} \right)i\)
  5. Division of complex numbers
    \(\large\frac{{a + bi}}{{c + di}}\normalsize =\) \( {\large\frac{{ac + bd}}{{{c^2} + {d^2}}} + \frac{{bc – ad}}{{{c^2} + {d^2}}}\normalsize} i\)
  6. Conjugate of a complex number \(\overline {a + bi} = a – bi\)
  7. Modulus \(r\) and argument \(\varphi\) of a complex number
    \(z = a + bi,\) \(r = \sqrt {{a^2} + {b^2}} ,\) \(\varphi = \arctan \large\frac{b}{a}\normalsize\)
Modulus and argument of a complex number
  1. Polar form of complex number
    \(z = a + bi =\) \( r\left( {\cos \varphi + i\sin \varphi } \right)\)
  2. Product of complex numbers in polar form
    \({z_1} \cdot {z_2} = \) \({r_1}\left( {\cos {\varphi _1} + i\sin {\varphi _1}} \right) \cdot\) \( {r_2}\left( {\cos {\varphi _2} + i\sin {\varphi _2}} \right) =\) \( {r_1}{r_2}\big[ {\cos \left( {{\varphi _1} + {\varphi _2}} \right) }+\) \({ i\sin \left( {{\varphi _1} + {\varphi _2}} \right)} \big]\)
  3. Conjugate of a complex number in polar form
    \(\overline {r\left( {\cos \varphi + i\sin \varphi } \right)} =\) \( r\left[ {\cos \left( { – \varphi } \right) + i\sin \left( { – \varphi } \right)} \right]\)
  4. Inverse of a complex number in polar form
    \({\large\frac{1}{{r\left( {\cos \varphi + i\sin \varphi } \right)}}\normalsize} =\) \( {\large{\frac{1}{r}}\normalsize}\left[ {\cos \left( { – \varphi } \right) + i\sin \left( { – \varphi } \right)} \right]\)
  5. Division of complex numbers in polar form
    \({\large\frac{{{z_1}}}{{{z_2}}}\normalsize} =\) \( {\large{\frac{{{r_1}\left( {\cos {\varphi _1} + i\sin {\varphi _1}} \right)}}{{{r_2}\left( {\cos {\varphi _2} + i\sin {\varphi _2}} \right)}}}\normalsize} =\) \({\large\frac{{{r_1}}}{{{r_2}}}\normalsize}\big[ {\cos \left( {{\varphi _1} – {\varphi _2}} \right) }+\) \({ i\sin \left( {{\varphi _1} – {\varphi _2}} \right)} \big]\)
  6. Exponentiation of a complex number
    \({z^n} =\) \( {\left[ {r\left( {\cos \varphi + i\sin \varphi } \right)} \right]^n} =\) \( {r^n}\left[ {\cos \left( {n\varphi } \right) + i\sin \left( {n\varphi } \right)} \right]\)
  7. De Moivre’s formula \({\left( {\cos \varphi + i\sin \varphi } \right)^n} =\) \( \cos \left( {n\varphi } \right) + i\sin \left( {n\varphi } \right)\)
  8. Nth root of a complex number
    \(\sqrt[\large n\normalsize]{z} =\) \( \sqrt[\large n\normalsize]{{r\left( {\cos \varphi + i\sin \varphi } \right)}} =\) \( \sqrt[\large n\normalsize]{r}\Big( {\cos {\large\frac{{\varphi + 2\pi k}}{n}\normalsize} + i\sin \large\frac{{\varphi + 2\pi k}}{n}} \Big),\;\) \(k = 0,1,2, \ldots ,n – 1\)
  9. Euler’s formula \(\exp \left( {ia} \right) = \cos a + i\sin a\)
  10. Exponential form of complex number \(z = r\exp \left( {i\varphi } \right)\)
  11. \(\sin a = \large\frac{{\exp \left( {ai} \right) – \exp \left( { – ai} \right)}}{2}\)
  12. \(\cos a = \large\frac{{\exp \left( {ai} \right) + \exp \left( { – ai} \right)}}{2}\)
  13. \(\tan a = \large\frac{{\exp \left( {ai} \right) – \exp \left( { – ai} \right)}}{{\exp \left( {ai} \right) + \exp \left( { – ai} \right)}}\)
  14. \(\cot a = \large\frac{{\exp \left( {ai} \right) + \exp \left( { – ai} \right)}}{{\exp \left( {ai} \right) – \exp \left( { – ai} \right)}}\)

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