Even though the complex numbers do not exist in nature directly, they are very useful in physics and engineering applications:
1. In the theory of complex functions, there are pairs of functions called the conjugate harmonic functions. They are useful in finding solutions of the Laplace equation in two dimensions.
2. The method of analytic continuation is a very important tool in finding solutions of differential equations and evaluating definite integrals.
3. Infinite series, infinite products, asymptotic solutions, and stability calculations are other areas, where the complex techniques are very helpful.
4. Even though the complex techniques are very useful in certain problems of physics and engineering, which are essentially defined in the real domain, complex numbers appear as an essential part of the physical theory in quantum mechanics.
11.1 Complex Algebra
A complex number, where can be defined by giving a pair of real numbers representing the real and the imaginary parts, respectively. A convenient way to represent complex numbers is to introduce the complex -plane (Figure 11.1), where a point is shown as
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where 
can be defined by giving a pair of real numbers 
representing the real and the imaginary parts, respectively. A convenient way to represent complex numbers is to introduce the complex -plane (Figure 11.1), where a point is shown as