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Signals and Systems by Oktay Alkin

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Appendix A

Complex Numbers and Euler’s Formula

A.1 Introduction

Complex numbers allow us to solve equations for which no real solution can be found. Consider, for example, the equation

x2+9=0(A.1)

which cannot be satisfied for any real number. By introducing an imaginary unit1 j=-1 so that j2 = −1, Eqn. (A.1) can be solved to yield

x=j3

Similarly, the equation

(x+2)2+9=0(A.2)

has the solutions

x=-2j3(A.3)

A general complex number is in the form

x=a+jb(A.4)

where a and b are real numbers. The values a and b are referred to as the real part and imaginary part of the complex number x respectively. Following notation is used for real and imaginary parts of a complex number:

a=Re{ x }b=Im{ x }

For two complex numbers to be equal, both their real ...

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