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Basic periodic functions

Not many of the functions that we encounter are periodic. There are few functions, which are periodic by their very definition. We are, so far, familiar with following periodic functions in this course :

  • Constant function, (c)
  • Trigonometric functions, (sinx, cosx, tanx etc.)
  • Fraction part function, {x}

Six trigonometric functions are most commonly used periodic functions. They are used in various combination to generate other periodic functions. In general, we might not determine periodicity of each function by definition. It is more convenient to know periods of standard functions like that of six trigonometric functions, their integral exponents and certain other standard forms/ functions. Once, we know periods of standard functions, we use different rules, properties and results of periodic functions to determine periods of other functions, which are formed as composition or combination of standard periodic functions.

Constant function

For constant function to be periodic function,

f x + T = f(x)

By definition of constant function,

f x + T = f(x) = c

Clearly, constant function meets the requirement of a periodic function, but there is no definite, fixed or least period. The relation of periodicity, here, holds for any change in x. We, therefore, conclude that constant function is a periodic function without period.

Trigonometric functions

Graphs of trigonometric functions (as described in the module titled trigonometric function) clearly show that periods of sinx, cosx, cosecx and secx are “2π” and that of tanx and cotx are “π”. Here, we shall mathematically determine periods of few of these trigonometric functions, using definition of period.

Sine function

For sinx to be periodic function,

sin x + T = x

x + T = n π + - 1 n x ; n Z

The term - 1 n evaluates to 1 if n is an even integer. In that case,

x + T = n π + x

Clearly, T = nπ, where n is an even integer. The least positive value of “T” i.e. period of the function is :

T = 2 π

Cosine function

For cosx to be periodic function,

cos x + T = cos x

x + T = 2 n π ± x ; n Z

Either,

x + T = 2 n π + x

T = 2 n π

or,

x + T = 2 n π x

T = 2 n π 2 x

First set of values is independent of “x”. Hence,

T = 2 n π ; n Z

The least positive value of “T” i.e. period of the function is :

T = 2 π

Tangent function

For tanx to be periodic function,

tan ( x + T ) = tan x x + T = n π + x ; n Z

Clearly, T = nπ; n∈Z. The least positive value of “T” i.e. period of the function is :

T = π

Fraction part function (fpf)

Fraction part function (FPF) is related to real number "x" and greatest integer function (GIF) as { x } = x [ x ] . We have seen that greatest integer function returns the integer which is either equal to “x” or less than “x”. For understanding the nature of function, let us compute few function values as here :

--------------------------------- x [x]x – [x] ---------------------------------1 1 0 1.25 1 0.251.5 1 0.5 1.75 1 0.752 2 0 2.25 2 0.252.5 2 0.5 2.75 2 0.753 3 0 3.25 3 0.253.5 3 0.5 3.75 3 0.754 4 0 ---------------------------------

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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cm
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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what is inorganic
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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