ABSTRACT
A function f(x) is said to be periodic if f ( x + T ) = f ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq2242.tif"/> for all values of x, where T is some positive number. T is the interval between two successive repetitions and is called the period of the function f(x). For example, y = sin x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq2243.tif"/> is periodic in x with period 2π since sin x = sin ( x + 2 π ) = sin ( x + 4 π) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq2244.tif"/> , and so on. Similarly, y = cos x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq2245.tif"/> is a periodic function with period 2π since cos x = cos ( x + 2 π ) = cos ( x + 4 π ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq2246.tif"/> , and so on. In general, if y = sin ωt https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq2247.tif"/> or y = cos ωt https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq2248.tif"/> then the period of the waveform is 2π/ω. The function shown in Figure 63.1 is also periodic of period 2π and is defined by: https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/fig63_1.tif"/> f ( x ) = { − 1 , when − π ≤ x ≤ 0 1 , when 0 ≤ x ≤ π https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq2249.tif"/>
