## Mathematics of Periodicity 1

### Functions

Function: procedure which, for any set of inputs in some range, assigns a corresponding output

#### Some Non-Periodic Functions (of 1 Variable)

• f(x) = 5

• f(x) = x

• f(x) = x^2 ("x squared")

#### Some Periodic Functions

• Continuous periodic functions
• sin (x)

• cox (x)

• Discontinuous periodic functions
• Functions with varying periods or amplitudes
• cos (x^2)

• (cos (x)) / x

#### Phase

• Phase: in a periodic process, the point to which the repeating process has advanced considered in relation to an assumed starting point
• Phase vs. time: phase ignores absolute time because it cares only about position within the cycle, independent of which cycle it is and how long the period is
• Phase angle: a measure of phase, usually in radians (0 to 2) or a proportion of a cycle (0 to 1)
• In phase: describing two or more periodic processes whose phases match; that is, the phases of maximum or minimum amplitude are simultaneous
• Functions in phase with each other
• sin(x), .5 sin(x)

• cos(x), cos(2x)

• Functions out of phase with each other
• sin(x), cos(x)

• sin(x), sin(x+)