Mathematical Methods for Physics and Engineering - Matematica.NET

3Complex numbers andhyperbolic functionsThis chapter is concerned with the representation and manipulation of complexnumbers. Complex numbers pervade this book, underscoring their wide applicationin the mathematics of the physical sciences. The application of complexnumbers to the description of physical systems is left until later chapters andonly the basic tools are presented here.3.1 The need for complex numbersAlthough complex numbers occur in many branches of mathematics, they arisemost directly out of solving polynomial equations. We examine a specific quadraticequation as an example.Consider the quadratic equationEquation (3.1) has two solutions, z 1 and z 2 , such thatz 2 − 4z +5=0. (3.1)(z − z 1 )(z − z 2 )=0. (3.2)Using the familiar formula for the roots of a quadratic equation, (1.4), thesolutions z 1 and z 2 , written in brief as z 1,2 ,arez 1,2 = 4 ± √ (−4) 2 − 4(1 × 5)√2−4=2±2 . (3.3)Both solutions contain the square root of a negative number. However, it is nottrue to say that there are no solutions to the quadratic equation. The fundamentaltheorem of algebra states that a quadratic equation will always have two solutionsand these are in fact given by (3.3). The second term on the RHS of (3.3) iscalled an imaginary term since it contains the square root of a negative number;83