lab5.html

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lab5.mws --- Trigonometric Integrals

> restart;
with( plots ):

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Auxiliary Plotting Commands -- execute, do not modify!

>    FourierAnim := proc(f,n)
  local a, b, Fcos, Fsin, Pf, Psincos;
  a := (f,n) -> int( f*sin(n*x), x=-Pi..Pi )/Pi:
  b := (f,n) -> int( f*cos(n*x), x=-Pi..Pi )/Pi:
  Fsin := (f,n) -> add( a(f,m)*sin(m*x), m=1..n ):
  Fcos := (f,n) -> b(f,0)/2 + add( b(f,m)*cos(m*x), m=1..n ):
  Psincos := (f,n) -> plot( Fsin(f,n)+Fcos(f,n), x=-3*Pi..3*Pi, color=blue ):
  Pf := plot( f, x=-Pi..Pi ):
  return display( [seq(
                    display( [Psincos(f,m),Pf], title=sprintf("n=%a",m) ),
                    m=1..n )],
                  insequence=true ):
end proc:

Lab Overview

This lab is intended to replace a need to lecture on Section 8.2.

Our first example will be to compute the definite integrals integrals .

We will conclude the demonstration with the Fourier sine and cosine coefficients of a function. These computations will be illustrated with an example based on the "square wave function" defined to be -1 on ( , 0 ) and 1 on ( 0, ) and extended periodically with periodic .

The questions at the end of this lab require you to apply the two examples to slightly different situations. In particular, Questions 4, and 5 refer to the "sawtooth function". This is the function that is defined to be the periodic extension of the linear function on [ , ]. The topics in this lab are the basis for a later lab on Fourier Series.

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Example 1 -

The functions , where = 1, 2, 3, ..., have the property that . In this example we will verify this claim.

Step 1: Develop Intuition with Graphical Animations

We begin by looking at the graph of the products for all combination of = 1, 2, .., 6 and = 1, 2, .., 6.

> sm := sin(m*x);
sn := sin(n*x);

>    N := 6:
for m from 1 to N do
  q := [seq(
         plot( sm*sn, x=-Pi..Pi, view=[-Pi..Pi,-1..1],
               title=sprintf("Plot of %a*%a", sm, sn) ),
        n=1..N )]:
  P||m := display( q, insequence=true );
end do:
unassign( 'm', 'n' );

The above loop creates an animation, Pn , for each = 1, 2, .., 6. Now, we display these animations. In each animation, try to determine if the "area" under the curve is positive, negative, or zero. Hint: Look for symmetries.

> P1;

> P2;

> P3;

> P4;

> P5;

> P6;

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Do you see any patterns to these results?
How are the plots different when from what you see when ?

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Step 2: Analytic Evaluation with Maple

Next, we attempt to evaluate these definite integrals. To get started we use , = 1, 2, 3 (please increase this to 10, and do not be concerned with the display of the 10x10 matrix). This time the results will be displayed in a matrix. To facilitate this, define a Maple function that accepts the values of and and returns the definite integral .

> N := 3:
F := unapply( Int( sm*sn, x=-Pi..Pi ), (m,n) ):
M := Matrix( N, N, F, shape=symmetric );

Then ask Maple to evaluate the integrals

> M = map( value, M );

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Step 3: Explicit Evaluation of the Definite Integrals

How did Maple do this? The answer involves using some trigonometric identities for products of sine functions.

> Smn := sm*sn:
q1 := combine( Smn, trig ):
Smn = q1;

When each side of this equation is integrated with respect to :

> q2 := Int( q1, x ):
q3 := value( q2 ):
Int( Smn, x ) = q2;
` ` = q3;

Observe that this formula does not make sense when (or , but that is not an issue at this time). Because the sine function has value 0 for all integer multiples of (and because and are integers when and are integers), it is pretty clear that the value of the antiderivative at is 0 and the value at is 0:

> eval( q3, x=Pi ) assuming integer;

> eval( q3, x=-Pi ) assuming integer;

Hence, whenever and are integers with .

To study the case when note that the product of sine functions simplifies to a square:

> Smm := eval( Smn, n=m );

Here the appropriate trigonometric identity is

> q4 := combine( Smm, trig ):
Smm = q4;

Once again, this expression is easily integrated with respect to :

> q5 := Int( q4, x ):
q6 := value( q5 ):
Int( Smm, x ) = q5;
` ` = q6 + C;

When this antiderivative is evaluated at and , the results are

> q7 := eval( q6, x=Pi ) assuming integer:
q7;

and

> q8 := eval( q6, x=-Pi ) assuming integer:
q8;

respectively, Therefore,

> Int( Smm, x=-Pi..Pi ) = q7 - q8;

This verifies the result obtained automatically by Maple in the previous section and the general result:

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Example 2 - Fourier sine and cosine coefficients for the sign function

Let f be a function defined on the interval ( , ). The Fourier sine and cosine coefficients of f are defined to be

, for = 1, 2, 3, ....

, for = 0, 1, 2, ....

To illustrate this definition, let f be the ``sign function'':

In Maple, this function is implemented with the signum command.

> plot( signum(x), x=-Pi..Pi, discont=true, title="The sign function" );

Do you see the isolated point at the origin in the above plot?

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The Fourier sine coefficients for this function are defined in terms of the definite integral

> q1 := Int( signum(x)*sin(n*x), x=-Pi..Pi )/Pi:
a[n] = q1;

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Step 1: Develop Intuition with Graphical Animations

Sine Coefficients

First, we look at the graph of the products for = 1, 2, .., 10.

> sn := sin(n*x);

>    N := 10:
q := [seq(
        plot( signum(x)*sn, x=-Pi..Pi, view=[-Pi..Pi,-1..1],
              title=sprintf("Plot of %a", signum(x)*sn) ),
        n=1..N )]:
display( q, insequence=true );

Do you see a pattern to the area between the curve and the -axis in these pictures?
What can you say by symmetry?
What is different about the symmetry for frames with even and odd?
Are these functions even or odd (or neither)? (How does that help you to evaluate the integrals?)

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Cosine Coefficients

Repeating this step for the cosine coefficients for = 0, 1, 2, .., 10:

> cn := cos(n*x);

>    N := 10:
q := [seq(
        plot( signum(x)*cn, x=-Pi..Pi, view=[-Pi..Pi,-1..1], discont=true,
              title=sprintf("Plot of %a", signum(x)*cn) ),
        n=0..N )]:
display( q, insequence=true );

Do you see a pattern to the area between the curve and the -axis in these pictures?
What can you say by symmetry?
What is different about the symmetry for frames with even and odd?
Are these functions even or odd (or neither)?

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Step 2: Analytic Evaluation with Maple

Next, we attempt to evaluate the definite integrals for the sine and cosine coefficients.

Sine Coefficients

To get started we use = 1, 2, .., 10. This time the results will be displayed in a column vector. To facilitate this, define a Maple function that accepts a value for and returns the definite integral

> N := 10:
A := unapply( Int( signum(x)*sn, x=-Pi..Pi )/Pi, n ):
M := Matrix( N, 1, A );

To evaluate these integrals

> M = map( value, M );

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Cosine Coefficients

Repeating these steps for the cosine coefficients:

> N := 10:
B := unapply( Int( signum(x)*cn, x=-Pi..Pi )/Pi, n ):
M := Matrix( N, 1, B );

To evaluate these integrals

> M = map( value, M );

Here the pattern is pretty obvious!

But, we still have to compute . Replace the %? in the following command to create a valid Maple command to compute . Then, execute both commands (in order) and find the value of .

> B0 := %? ;

> value( B0 );

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To conclude, we have discovered that

, for all = 1, 2, 3, ....

, for all = 0, 1, 2, ....

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Step 3: Explicit Evaluation of the Definite Integrals

To evaluate these integrals by hand, we would divide the interval ( , ) into two pieces ( , 0 ) and ( 0, ).

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Sine Coefficients

> q2 := Int( signum(x)*sin(n*x), x=-Pi..Pi )/Pi:
A(n) = q2;
` ` = value( q2 );

When we tell Maple that n (and any other variables in this expression) are integers:

> value( q2 ) assuming integer;

To reduce this even further, look at the cases when n is odd and even separately:

> value( q2 ) assuming odd;

> value( q2 ) assuming even;

This is consistent with the results found in Steps 1 and 2.

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Cosine Coefficients

> q3 := Int( signum(x)*cos(n*x), x=-Pi..Pi )/Pi:
B(n) = q3;
`` = value( q3 );

This, too, is consistent with the results found in Steps 1 and 2.

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Discussion

Why do we care about Fourier sine and cosine coefficients. While this will be studied in greater detail in a later lab, here is a preview. Define the functions

for = 1, 2, ....

Note that more and more coefficients are used in as n increases. The following FourierAnim command creates a 12-frame animation in which each frame consists of the signum function on [ , ] and on [ , ].

> FourierAnim( signum(x), 12 );

Notice how the graphs of have period for all . Moreover, as increases, the graphs of provide better and better approximations to the signum function on [ , ].

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Lab Questions

1. Consider for integers and . What is the value of ?

2. Consider for integers and . What is the value of ?

Hint: Lab Questions 1 and 2

> cm := cos(m*x);
cn := cos(n*x);

Copy the Example 1 section (or selected Steps) here twice - once for Question 1 and once for Question 2. Use cm and cn defined above to replace appropriate occurrence of sm and sn in Example 1 and execute the appropriate commands.

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3. For the square wave, . Why is ?

4. The ``sawtooth function'' is , for <= <= , and f is periodic with period , i.e., for <= <= and an integer.

Find the Fourier sine coefficient for the sawtooth function. Report both the general formula and the first five Fourier sine coefficients., , = 1, 2, .., 5.

Hint: Lab Question 4

To implement the sawtooth function in Maple, you may write either f (after executing the following command) or, since all integration will be done on ( , ), simply type x whenever you need to reference the function f.

> f := x;

I recommend copying the Example 2 section (or selected Steps from Example 1) and pasting it following this hint. Make the changes in the copy of Example 2.

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5. Use the FourierAnim command to create an animation showing that the graphs of converge to the sawtooth function. What is the smallest value of for which the graph of has a maximum value larger than ? Write this function .