Chapter 01: Complex Numbers


Notes of the book Mathematical Method written by S.M. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN.

A complex number is an element $(x,y)$ of the set $$ \mathbb{R}^2=\{(x,y): x,y \in \mathbb{R}\} $$ obeying the following rules of addition and multiplication.

For $z_1=(x_1,y_1)$, $z_2=(x_2,y_2)$, we put

  1. $z_1+z_2= (x_1+x_2, y_1+y_2)$
  2. $z_1 z_2 = (x_1 x_2 - y_1 y_2, x_1 y_2+y_1 x_2)$

The set $\mathbb{R}^2$ with operation defined above is denoted by $\mathbb{C}$.

Contents and summary

  • Complex numbers
  • Properties of complex numbers
  • The Argand's diagram
  • De Moivre's theorem
  • Roots of the complex numbers
  • Basic elementary functions
  • Logarithmic functions
  • Inverse hyperbolic functions
  • Inverse trigonometric functions
  • Complex power
  • Summation of series

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