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[BOŞ_SES] Hello,
We viewed our previous example a box function.
It was a constant function.
Everywhere there were constant but different constants.
As a reminder, reset a divide between zero four,
a split between four to one half,
with a zero value in between with half function
We calculated in various ways and various terms
we observed that the convergence of numbers.
Here's a little more difficult by similar ease
but integral function more difficult, the more we want to see an example.
Again we see that four option separately calculate a hunch
We will try to improve.
The closer that we will see how each and every one how to develop convergence.
Accounts will absorb a little more.
This is our function; reset a divide between the two x equals y merger,
Among the half with a zero value.
We can do this function in four different ways.
One of them opened the series between the reset function.
It is a period here.
And when we move out of this function is moved translationally.
This function also resets a second representation
We extend the same function but in this symmetrical.
Each minus minus merger between zero x
The value of the function in the value of x in the same happening,
Or, according to the y-axis on the right and left from the same, such symmetry.
The third type of reset function with an all-in again
we are the same, but less stretched between the antisymmetric combined to zero.
Antisymmetric mean, of course, minus x
The value of x in the opposite sign of the function.
As you can see, this time we move them in two out of the region
a function to undergo screenings by taking the outside area, carry forward periods.
Here again, two-period, but this time antisymmetric function
carry the function to be obtained,
but reset to give the same result between all of them.
In general the same as those in the first notation with complex functions
A display as sine and kosinüsl Fourier series,
in the same situation, but there will be complicated but ak coefficients, such as the UK
Instead it will be a combination of two different factors it compiled the complex coefficient.
Our calculations also do not want to enter the same detail.
General notation zero, be a means
There were two periods divided by a denominator p sorry, had p in the denominator.
That means a split in the denominator for the two functions would still box
As we saw in the next two numbers.
It will take the hit merger integration, we find a zero.
We will hit the kosinüsl the integration,
We'll find whites, we find the shit hit the sinus.
Each integral reset halfway between the x, with half a
It will fall between one of the bottom half of the second portion to be zero.
So come out half resets all integrals.
It is not difficult to calculate the integral.
Especially the first one is very easy, of course integral x x squared divided by two a split in two
We account a split comes eight.
When you're doing here with this partial integration is so integral method
where the product, we feel like x times cosine of UV base.
So here you have the base of derivatives.
Once you have this integral u and v minus derivative multiply.
u will have a base because we get x.
Here only the first cosine,
This integration will be done to see to remain in sinus in the second.
Now we find when we calculate these integrals flow here,
We find in the UK.
One small detail; If you notice in the denominator k ak
In the UK there are only a short square.
Therefore, this short course has grown faster frame
As for the flow will go out and divide more quickly.
For example, if you receive a number found by dividing k on the front face,
share those numbers in a range, whereas a number of the UK's happening to him split.
That made the development of these accounts with those details.
Again, we see little flow coefficients as a function of the previous box
in the range of numbers it is obtained by dividing the k force.
This dimension also has its own in the number of trippings.
That is equal to three to five such a happening.
b five, b is equal to three.
Carrying this number minus zero, minus zero,
We are moving in a negative zero that way.
A minus zero, minus zero, we're going in a way.
In a sine minus one,
a negative one, we are moving as a negative one.
They are more simple expressions and turns of the Derley
We'll get the graphics.
Cosine shown me just how to extend symmetrically.
Again we get multiplication.
A significant difference; here are just short of the previous two had widely.
We are doing this is by far the multiplication.
Integral because again be made to partial integrals x times cosine
that's when the next second partial derivative of integrals x
I could just step back and to remain integral to his sinuses.
Here we find the coefficients here,
If you notice, there's the short term and the short term in the denominator are both square.
This is natural because it was going to just see the previous kbps.
k checkered was going well with the flow.
So in terms of details, but do not show them.
When we made the antisymmetric only a short extension again.
sinus x this time around.
Only a series of sinuses.
Only it consisted of a previous cosine.
Still Hosaka from making some account of these details, you can do calculations.
It also would be good but more important is to be able to internalize and follow the flow.
This series is obtained after to see the numbers.
I still complex denominator in the exponential representation
There are a number of two short periods because reset here
Because it is a function of the period between turns.
Therefore, a split pin in the denominator
also comes the top two we divided into two terms.
This again made an integral part of integrals.
Those who want a fast Fourier transform coefficient to income eligible.
They can be made.
We see the convergence of the results obtained as follows: here
simply cosine and sine series, we look at the overall convergence.
Do not be boring because he will come to roughly the same thing on the öbürkü.
There were only two terms.
See approached the end, this function will converge.
y equals x and zero after that.
See struggling to converge less.
Wherein n is four, i.e. four polynomial series.
Down it converged up a little better,
When we received eight terms thoroughly converging.
Upon receipt of these sixteen and thirty-two term top
The show also made the top in terms of convergence.
One black, one red as you can see how both together
close to both function quite nicely yakınsatıy.
Hundreds term is done on computer
if you take it too close and not spend too much time even more beautiful.
Here, too, it seems coefficients.
Blue aka coefficients, we have seen before.
They went by quickly flashing for short frame,
after a while you do not see the blue coefficient.
see, while this red coefficients.
They are quite slow flashing k'yl Go to the denominator.
There is also a box of these functions and the coefficients
We see here also that the function was composition.
Box function that carries an importance in this regard.
Here it seems the convergence of three different views.
Fourier series in the x column
general sinus, which kosinüsl terms.
The second column is just cosine,
The third column in the convergence of only sine.
Almost all of them in the same way as you can see the number of terms taken quite a few
approaching, converging, there's little difference.
There were four terms.
But the four were taken, but there also are four four cosine sine series in general.
There's only four cosine, sine there's only four.
There are minor details.
This is a zero, that zero here and the antisymmetric
scratch, passing very close to zero, is not exactly zero.
But the derivative is zero here and so many ten
When you get up in a beautiful way down to six and thirty-two in closer.
A similar approach in the sine.
So as you can see initially a little roughly in the area between Reset
though still a reminder of the approach there is a series of behavior.
By increasing the number of terms in a pretty good approximation.
Whether you want to find even better if you get two hundred and fifty-six it would not be a challenge.
Easily obtain much better convergence.
Now I want to pause here.
So far, a number of functions with artificial calculations
We tried to see it.
After vibration of a machine part,
As the vibrations of a sound recording of a molecule
We'll see functions and Fourier series.
Attila AşkarProf. Dr.
