SolitaryRoad.com

Website owner:  James Miller


[ Home ] [ Up ] [ Info ] [ Mail ]

Properties and Fourier transforms of even and odd functions. Hermitian function. Complex conjugates.



Def. Even function. A function f such that f(-x) = f(x) for all x in the domain of f.


Examples. x2, cos x


Def. Odd function. A function f such that f(-x) = -f(x) for all x in the domain of f.


Examples. x3 and sin x since (-x)3 = -x3 and sin(-x) = - sin x



The concepts of evenness and oddness are defined for both real-valued and complex-valued functions of a real or complex variable. In the case of real-valued functions of a real variable, the graph of f(-x) is the graph obtained by reflecting the graph of f(x) about the y axis. The graph of an even function is symmetric with respect to the y-axis and the graph of an odd function is symmetric with respect to the origin (it is unchanged when reflected about both the x-axis and y-axis).



Theorem 1. Any function f(x) can be expressed uniquely as a sum of an even function E(x) and an odd function O(x)


            f(x) = E(x) + O(x)


where


            E(x) = ½[f(x) + f(-x)]

            O(x) = ½[f(x) - f(-x)]


Proof


 

Properties of even and odd functions



Properties involving addition and subtraction

   ● The sum of two even functions is even, and any constant multiple of an even function is even.

   ● The sum of two odd functions is odd, and any constant multiple of an odd function is odd.

   ● The difference between two odd functions is odd.

   ● The difference between two even functions is even.

   ● The sum of an even and odd function is neither even nor odd, unless one of the functions is equal to zero over the given domain.

 

Properties involving multiplication and division

  ● The product of two even functions is an even function.

  ● The product of two odd functions is an even function.

  ● The product of an even function and an odd function is an odd function.

  ● The quotient of two even functions is an even function.

  ● The quotient of two odd functions is an even function.

  ● The quotient of an even function and an odd function is an odd function.

 

Calculus properties

  ● The derivative of an even function is odd.

  ● The derivative of an odd function is even.

  ● The integral of an odd function from -A to +A is zero (where A is finite, and the function has no vertical asymptotes between -A and A).

  ● The integral of an even function from -A to +A is twice the integral from 0 to +A (where A is finite, and the function has no vertical asymptotes between -A and A. This also holds true when A is infinite, but only if the integral converges).

 

Series properties

  ● The Maclaurin series of an even function includes only even powers.

  ● The Maclaurin series of an odd function includes only odd powers.

  ● The Fourier series of a periodic even function includes only cosine terms.

  ● The Fourier series of a periodic odd function includes only sine terms.

 

 

Algebraic properties

Any linear combination of even functions is even, and the even functions form a vector space over the reals. Similarly, any linear combination of odd functions is odd, and the odd functions also form a vector space over the reals. The vector space of all real-valued functions is the direct sum of the subspaces of even and odd functions. In other words, every function f(x) can be written uniquely as the sum of an even function and an odd function.

 

 

Sine - Cosine form of Fourier Transform. By definition, the Fourier transform of the function f (t) is F(ω) as given by the formula

 

ole.gif

 

Using the fact that for any real number x,

 

            e-ix = cos x - i sin x

 

1) becomes

 

ole1.gif

 

or

 

ole2.gif

 

 

 

Fourier transforms of even and odd functions

 

Let us consider the function

 

4)        f (x) = e(x) + o(x)

 

where e(x) is an even function of x, o(x) is an odd function of x, and e(x) and o(x) are in general

ole3.gif

complex. The Fourier cosine transform of e(x) is

 

ole4.gif

 

and the Fourier sine transform of o(x) is

 

ole5.gif

 

and the Fourier transform of f (x) = e(x) + o(x) is

 

ole6.gif

 

The function f (x) is a complex-valued function of a real variable x. This function can be depicted using a three-dimensional Cartesian coordinate system with one axis labeled “x”, another axis labeled “Real”, and a third axis labeled “Imaginary” as shown in Fig. 1. For each value x = xi , f (x) = ai + bi i and the real and imaginary components of f (x) are plotted as a function of x. There are a number of possibilities for the function f (x). It can be real and even, real and odd, imaginary and even, imaginary and odd, complex and even, complex and odd, etc. Each of these possibilities gives rise to a characteristic Fourier transform as shown in the following table.

 

Function f (x)                                           Fourier transform

_______________________________________________________________

Real and even                                                 Real and even

Real and odd                                                  Imaginary and odd

Imaginary and even                                        Imaginary and even

Imaginary and odd                                         Real and odd

Complex and even                                          Complex and even

Complex and odd                                           Complex and odd

Real and asymmetrical                                   Complex and asymmetrical

Imaginary and asymmetrical                          Complex and asymmetrical

Real even and imaginary odd                         Real

Real odd and imaginary even                         Imaginary

Even                                                               Even

Odd                                                                 Odd

 

These different transform results are depicted in Fig. 1 and are summarized in the following diagram:

 

ole7.gif

                                                                                                                                                

 

 

Hermitian functions

 

Def. Hermitian function. A function defined by the property f (x) = f *(-x), where the asterisk denotes the complex conjugate.

 

A property of a Hermitian function is that its real part is even and its imaginary part is odd. See Fig. 2.

 

Theorem 2. If a function has the property that f (x) = f *(-x) then its real part is even and its imaginary part is odd.

 

ole8.gif

Proof. Let f (x) be written as the sum of an even and odd function:

 

f (x) = E(x) + O(x)

f (x) = E + O + iE + iO

 

Then

f (-x) = E - O + iE - iO 

 

and

 

f *(-x) = E - O - iE + iO

 

If we now require that f (x) = f *(-x) then necessarily O = 0 and E = 0. Thus f (x) = E + iO.

 

Complex conjugates. The Fourier transform of the complex conjugate of a function f (x) is F*(-s) i.e. it is the reflection of the conjugate of the transform. Special cases are as follows:

 

Function f (x)                                           Fourier transform

_______________________________________________________________

real                                                                  F(s)

imaginary                                                       -F(s)

even                                                                F*(s)

odd                                                                  -F*(s)

 

 

References

  Hsu. Fourier Analysis

  Bracewell. The Fourier Transform and Its Applications

  James, James. Mathematics Dictionary

  Spiegal. Laplace Transforms. (Schaum) Chap. 6

  Wikipedia



More from SolitaryRoad.com:

The Way of Truth and Life

God's message to the world

Jesus Christ and His Teachings

Words of Wisdom

Way of enlightenment, wisdom, and understanding

Way of true Christianity

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Who will go to heaven?

The superior person

On faith and works

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

The teaching is:

On modern intellectualism

On Homosexuality

On Self-sufficient Country Living, Homesteading

Principles for Living Life

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

America has lost her way

The really big sins

Theory on the Formation of Character

Moral Perversion

You are what you eat

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

These things go together

Television

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

The True Christian

What is true Christianity?

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Walking a solitary road

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

On responding to wrongs

Real Christian Faith

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

Sizing up people

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-discipline

Self-control, self-restraint, self-discipline basic to so much in life

We are our habits

What creates moral character?


[ Home ] [ Up ] [ Info ] [ Mail ]