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## Piecewise-Defined Functions and Periodic Functions

important note: f ( x) = - f ( - x) → anti-symmetric → cos-trans are zero! ∫ 0 2 π f ( x) cos ( k x) d x = 0 b k = 1 π ∫ 0 2 π f ( x) sin ( k x) d x = 1 π ∫ 0 π 1 sin ( k x) d x + 1 π ∫ π 2 π ( - 1) sin ( k x) d x = 1 π ( - cos ( k x) k | 0 π - - cos ( k x) k | π 2 π) = 1 π

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## Periodic Function

Definition 1. A function f: D → R, D ⊆ R is said to be periodic if there exists a positive real number ω such that f ( t) = f ( t + ω) for all t ∈ D. The number ω is called the period of f.

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## Periodic functions

A function f (x) is said to be periodic, if there exists a positive real number T such that f (x+T) = f (x). The smallest value of T is called the period of the function. Note: If the value of T is

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## List of periodic functions

Example 1: Find the period of the periodic function Sin (4x + 5). Solution: The given function is Sin (4x + 5) The period of Sinx is 2π., and the period of Sin (4x + 5) is 2π/4 = π/2. Therefore, the ## Example: Periodic step function

Periodic Function Formula. A function f is said to be periodic if, for some non-zero constant P, it is the case that: f ( x + P) = f ( x) For all values of x in the

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