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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

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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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