Prove that $A \smallsetminus (A \smallsetminus B) = A \cap B$

$A$ and $B$ are any sets, prove that $A \smallsetminus (A \smallsetminus B) = A \cap B.$ This formula makes sense when represented on a Venn diagram, but I am having trouble with proving it mathematically.

I have tried letting $x$ be an element of $A$ and continue from there, but it doesn't seem to work out as a valid proof anyways.

Could anyone please point me in the right direction?

Many thanks.

• Please show us your work: starting with $x \in A\setminus(A\setminus B)$, or the other way around, starting with $x \in (A\cap B)$. Please don't claim you tried ABC, unless you also show us you effort using ABC. Would you like to get an answer that says only: "Yup, that's right, start by letting $x \in A\setminus (A\setminus B)$... – Namaste Jan 27 '18 at 18:42

Let $x \in A \setminus (A \setminus B).$

Then $x$ is an element such that $x \in A$ and $x \notin A \setminus B$. But if $x \notin A\setminus B$, with some additional work, we realize this implies that $x \in A$ and $x \in B$. So $x \in A \cap B$.

Vice-versa: let $x \in A \cap B$ so $x \in A$ and $x \in B$. This implies that $x \notin A \setminus B$. But given that $x \in A$ and $x \notin A \setminus B$ this implies that $x \in A \setminus (A \setminus B)$.

• I would like this answer if you hadn't skipped from "$x \in A$ and $x \notin A\setminus B$, but if $x\in A \setminus B$" to "we realize that $x \in A$ and $x\in B$." I added the words ("with some additional work") in between them. I would like this answer more if you explained that $x \in A$ and $x \notin (A\setminus B)$, then $x \in A$ and, it is not the case that ($x \in A$ and $x \notin B$.) By Demorgans, we get that $(x \in A \land \lnot x\in A),$ or $(x \in A \land x\in B)$. The first disjunct is always false (no element can be in a set, and not be in a set. – Namaste Jan 27 '18 at 19:13
• ...That leaves us to conclude, that "we realize this implies that $x\in A$ and $x\in B$. – Namaste Jan 27 '18 at 19:13

Notice that \begin{eqnarray*} x\in A\setminus(A\setminus B) &\Leftrightarrow& (x\in A)\wedge \neg(x\in A\setminus B)\\ &\Leftrightarrow & (x\in A)\wedge \neg((x\in A)\wedge \neg (x\in B))\\ &\Leftrightarrow & (x\in A)\wedge ((x\notin A)\lor(x\in B))\\ &\Leftrightarrow & ((x\in A)\wedge (x\notin A))\lor ((x\in A)\wedge(x\in B))\\ &\Leftrightarrow & (x\in A)\wedge (x\in B)\\ &\Leftrightarrow & x\in A\cap B. \end{eqnarray*}

• Yes, when in doubt go back to the logic – Tom Collinge Jan 27 '18 at 18:28
• I like this, because the equivalence makes the proof bi-directional, and doesn't skip important steps between $x \in A) \land$\lnot (x \in A\setminus B$and its equivalent,$x\in A \land x \in B$Those are likely the steps on which the asker got tripped up, I suspect. Nice answer. – Namaste Jan 27 '18 at 18:51 • I like this, because the equivalence makes the proof bi-directional, and doesn't skip important steps between$x \in A \land \lnot (x \in A\setminus B$and its equivalent,$x\in A \land x \in B.$Those are likely the steps on which the asker got tripped up, I suspect. Nice answer. – Namaste Jan 27 '18 at 19:01 • @amWhy: Thanks. Indeed, often mistakes are made in these basic logical steps. I've seen many people making mistakes using the notation$x\notin B$. I think it's safer to write$\neg(x\in B)$, indeed, if$B$is a more complicated expression, we still now how to negate the logical expression appearing inside the brackets. This basic logical way of thinking is user friendly and not very prone to errors. – Mathematician 42 Jan 27 '18 at 20:43 Did you know that $$A \smallsetminus B = A \cap \overline{B} \;\;?$$$\begin{align} A \smallsetminus (A \smallsetminus B) &= A \smallsetminus (A \cap \overline{B}) \\ \\ &= A \cap \overline{ (A \cap \overline{B})}\\ \\ &= A \cap (\overline{A} \cup B)\\ \\ &= (A \cap \overline{A} )\cup (A \cap B) \\ \\ &= A \cap B \end{align}$• cmon, why this get downvoted, it is correct. hmm, maybe you want to precise that$\bar B=B^{\complement}$and not the closure in this context. – zwim Jan 27 '18 at 18:55 • @zwim: Not sure why this got downvoted, however this proof assumes knowledge about standard set operations which you should be able to prove directly as well. You might end up running in circles and not proving anything. – Mathematician 42 Jan 27 '18 at 18:58 Let$x\in A\cap B$. Then$x\in A$and$x\in B$. Then$x\not \in A\setminus B$, so$x\in A\setminus (A\setminus B)$. Conversely, if$x\in A\setminus (A\setminus B)$, then$x\in A$and$x\not \in (A\setminus B)$. So$x\in A$and$x\in B$. That is$x\in A\cap B\$...

• ditto of @Skills. – Namaste Jan 27 '18 at 18:36


# Math Help - sequences problem

1. ## sequences problem

A hardware supplier has designed a series of six plastic containers with lids where each container (after the first) can be placed into the next larger one for storage purposes. The containers are rectangular boxes, and the dimensions of the largest one are 120 cm by 60 cm by 50 cm. The dimension of each container is decreased by 10 percent with respect to the next larger one.

Determine the volume of each container, and the volume of all the containers. Write your answers in litres (one litre = 1000 cm3).

Is it 10% from each dimension or from the volume? Hard to figure out from the problem.

If from the each dimension then:
V 1= 50x 60x 120 = 360000 cm3 = 360 litres
V2=45 x 54 x 108 = 262440 cm3 = 262.4 liltres
V 3 = 40.5 x 48.6 x 97.2 = 191.3litres
V4 = 36.45 x 43.74 x 87.48 =139.4 litres
V5 = 32.81 x 39.37 x 78.74 = 101.7 litres
V6 = 29.61 x 35.47 x 70.94 =74.5 litres

I need some feedback.

2. I expect since it says "the dimension of each container is decreased by 10% with respect to the larger one" I would say it's the dimensions and not the volume that is decreased by 10%.

3. Originally Posted by terminator
A hardware supplier has designed a series of six plastic containers with lids where each container (after the first) can be placed into the next larger one for storage purposes. The containers are rectangular boxes, and the dimensions of the largest one are 120 cm by 60 cm by 50 cm. The dimension of each container is decreased by 10 percent with respect to the next larger one.

Determine the volume of each container, and the volume of all the containers. Write your answers in litres (one litre = 1000 cm3).

Is it 10% from each dimension or from the volume? Hard to figure out from the problem.
Why? The problems says "The dimension of each container is decreased by 10 percent". That tells you precisely which is intended

If from the each dimension then:
V 1= 50x 60x 120 = 360000 cm3 = 360 litres
V2=45 x 54 x 108 = 262440 cm3 = 262.4 liltres
V 3 = 40.5 x 48.6 x 97.2 = 191.3litres
V4 = 36.45 x 43.74 x 87.48 =139.4 litres
V5 = 32.81 x 39.37 x 78.74 = 101.7 litres
V6 = 29.61 x 35.47 x 70.94 =74.5 litres

I need some feedback.

4. Hello, terminator!

A hardware supplier has designed a series of six plastic containers with lids
. . where each container (after the first) can be placed into the next larger one.

The containers are rectangular boxes.
. . The dimensions of the largest one are 120 cm by 60 cm by 50 cm.

The dimensions of each container are decreased by 10 percent
. . with respect to the next larger one.

(a) Determine the volume of each container.
(b) Determine the volume of all the containers.
Write your answers in litres (one litre = 1000 cubic cm).

The original box has dimensions $L,\,W,\,H$
. . Its volume is: . $L\!\cdot\!W\!\cdot\!H\text{ cm}^3$

The next box has dimensions: $0.9L,\,0.9W,\,0.9H$
. . Its volume is: . $(0.9L)(0.9W)(0.9H) \:=\:0.729(LWH)$

That is, each box is 0.729 of the volime of the next larger box.

The first box has volume: . $120\cdot60\cdot50 \:=\:360,\!000\text{ cm}^3$

. . That is: . $V_1 \:=\:360\text{ liters.}$

(a) Therefore, the $\,n^{th}$ box has volume: . $V_n \;=\;360(0.729)^{n-1}$

The sum of the volumes of all the boxes is:

. . $S \;=\;360 + 360(0.729) + 360(0.729^2) + 369(0.729^3) + \hdots$

. . $S \;=\;360\underbrace{\bigg[1 + 0.729 + 0.729^2 + 0.729^3 + \hdots\bigg]}_{\text{a geometric series}}$

The sum of the geometric series is: . $\dfrac{1}{1 - 0.729} \:=\:\dfrac{1}{0.271}$

(b) Therefore: . $S \;=\;360\left(\frac{1}{0.271}\right) \;\approx\;1328.4\text{ liters}$


The Gambler’s Fallacy

Probability seems simple enough to many people that it can fool them into wrong conclusions. We have had many questions that involve the “Gambler’s Fallacy”, both from people who naively assume it without thinking, and from some who defend it using technical ideas like the Law of Large Numbers.

The Gambler’s Fallacy

Here is a question from 2003 that is a good introduction to the idea:

Gambler's FallacyA current co-worker and I are in a friendly disagreement about the probability of selecting the winning number in any lottery, say Pick 5. He states that he would rather bet the same set of five numbers every time for x period of time, but I insist that the probability is the same if you randomly select any set five numbers for the same period of time. The only assumption we make here is betting one set of numbers on any given day. Who is correct?

I tried explaining to him that the probability of betting on day one is the same for both of us. On day two it is the same. On day three it is the same, etc. Therefore the sum of the cumulative probabilities will be the same for both of us.

Doctor Wallace first examined the appropriate calculation, before moving on to the likely underlying error:

You are correct. If you have the computer randomly select a different set of 5 numbers to bet on every day, and your friend selects the same set of numbers to bet on every day, then you both have exactly the same probability of winning.

Tell your friend to think of the lottery as drawing with tickets instead of balls. If the lottery had a choice of, say, 49 numbers, then imagine a very large hat containing 1 ticket for every possible combination of 5 numbers.  1, 2, 3, 4, 5;  1, 2, 3, 4, 6;  etc.

On the drawing day, ONE ticket is pulled from the hat. It is equally likely to be any of the C(49,5) tickets in the hat. (There would be 1,906,884 tickets in the hat in this case.)

Since both you and your friend have only ONE ticket in the hat, you both have the same chance of winning.

On the next drawing day for the lottery, ALL the tickets are replaced.  Each lottery draw is an event independent of the others. That is to say, the probability of any combination winning today has absolutely NO effect on the probability of that or any other combination winning tomorrow. Each and every draw is totally independent of the others.

That perspective makes it “obvious”: if two people each have one “ticket”, they have the same probability, whether or not it is the same one that was taken last time, since the “ticket” is chosen randomly each time without regard to the past.

So why is the other opinion tempting?

The reason your friend believes that he has a better chance of winning with the same set of numbers is probably due to something called the "gambler's fallacy." This idea is that the longer the lottery goes without your friend's "special" set of numbers coming up, the more likely it is to come up in the future. The same fallacy is believed by a lot of people about slot machines in gambling casinos. They hunt for which slot hasn't paid in a while, thinking that that slot is more likely to pay out. But, as the name says, this is a fallacy; pure nonsense. A pull of the slot machine's handle, like the lottery draw, is completely independent of previous pulls. The slot machine has no memory of what has come before, and neither has the lottery. You might play a slot machine for 2 weeks without hitting the big jackpot, and someone else can walk in and hit it in the first 5 minutes of play.  People wrongly attribute that to "it was ready to pay out." In reality, it's just luck.  That's why they call it gambling.  :)

The same thing comes up in math classes:

This used to be a "trick" question on old math tests:

"You flip a fair coin 20 times in a row and it comes up heads every single time. You flip the coin one more time. What is the probability of tails on this last flip?"

Most people will respond that the chance of tails is now very high.
(Ask your friend and see what he says.)  However, the true answer is that the probability is 1/2.  It's 1/2 on EVERY flip, no matter what results came before.  Like the slot machine and the lottery, the coin has no memory.

That type of question is still valuable. It tests an important idea that students need to think about.

For more about picking the same number in a lottery, see

Lottery Strategy and Odds of Winning

For a very nice refutation of the Gambler’s Fallacy in coin tossing, see

What Makes Events Independent?

The Law of Large Numbers

That question was based on a naive approach to gambling. The next, from 2000, is based on probability theory. (The questions were asked in imperfect English, which I will restate as I understand it, correcting a misinterpretation in the archived version. Doctor TWE got it right.)

Law of Large Numbers and the Gambler's FallacyIf we throw three dice at a time, three times altogether, is the result the same as if we throw nine dice one time?

Do we have the same probability to get a given number in the two cases?
Where and how different is it?

Doctor TWE answered, covering several possible interpretations of “result”:

The sum, the average, and the probabilities of getting a particular value on one die or more aren't affected by whether the dice are rolled one at a time, in groups of 3, or all 9 at once. These are called "independent events," and the order in which they happen doesn't affect the outcome.

Elvino replied,

So, I have 42.12% probability to get at least one six (or other number), hen I throw three dice at once. If I throw the same dice, in the same way, again and again I will always have the same probability!

But, there is a law that says, the more I throw, the more probability I have to obtain a certain number. How does this law of probabilities work?

Checking his calculation, and thereby confirming what he meant, I get the probability of rolling at least one six on three dice to be the complement of rolling no sixes: $$\displaystyle 1 – \frac{5^3}{6^3} = 42.1%$$. Next time I will discuss this further.

Doctor TWE confirmed his calculation, then explained what this law means, and does not mean:

There is something called the Law of Large Numbers (or the Law of Averages) which states that if you repeat a random experiment, such as tossing a coin or rolling a die, a very large number of times, your outcomes should on average be equal to (or very close to) the theoretical average.

Suppose we roll three dice and get no 6's, then roll them again and still get no 6's, then roll them a third time and STILL get no 6's. (This is the equivalent of rolling nine dice at once and getting no 6's, as we discussed in the last e-mail; there's only a 19.38% chance of this happening.) The Law of Large Numbers says that if we roll them 500 more times, we should get at least one 6 (in the 3 dice) about 212 times out of the 503 rolls (.4213 * 503 = 211.9).

This is *not* because the probability increases in later rolls, but rather, over the next 500 rolls, there's a chance that we'll get a "hot streak," where we might roll at least one 6 on three or more consecutive rolls. In the long run (and that's the key - we're talking about a VERY long run), it will average out.

There is also something called the Gambler's Fallacy, which is the mistaken belief that the probability on the next roll changes because a particular outcome is "due." In the example above, the probability of rolling at least one 6 in the next roll of the three dice (after three rolls with no 6's) is still 42.13%. A (non-mathematician) gambler might think that the dice are "due," that in order to get the long-term average back up to 42%, the probability of the next roll getting at least one 6 must be higher than 42%. This is wrong, and hence it's called "the Gambler's Fallacy."

The important thing here is that things will “average out” in the long run, so that you get at least one 6 in 42.1% of the rolls, but not because at any one time there is a greater chance, to make up the average — it happens only because there is a very long time to do so. It is not because past events have any effect on future events, pulling them into line with the “averages”.

For more on the Law of Large Numbers, see

The Law of Large Numbers

Digging deeper

In 2008, we had the following detailed question about that from Konstantinos:

Do Prior Outcomes Affect Probabilities of Future Ones?What I want to know is if in matters of luck, such as games of dice, or lotteries, or flipping coins, the future outcomes have any relativity with past results.  What I mean in each case is:  If I flip a coin and get tails, don't I have bigger or smaller possibilities to get heads on the next roll?  I mean I know that the possibility is always 1/2 but since I have already thrown the coin 5 times and rolled 5 tails there aren't possibilities that in the next throws there will be heads?  The same question goes for dice rolls, and for lotto numbers.  If a number has come more times than others, isn't it possible that for a limited amount of coming times, numbers that haven't come yet, will start showing more?

What I find most confusing is that the relation between probabilities and past possibilities, and the outcome in real life.  How can a coin come tails 5 or six times in a row when the possibility is always 1/2?  Are probabilities only theoretical?

I tried noting down results of different trials of luck (dice, past lotteries, coins) but in the end they don't seem to make true to any theory I have heard.  I would like to see the magic of numbers in real life and on this subject and how it works as to prove a theory.  Do I have to toss the coin 10,000 times?  And if I do will I see heads coming in a row after 10 subsequent tails?

I responded to this one:

What you are suggesting is called the Gambler's Fallacy--the WRONG idea that future results of a random process are affected by past results, as if probability deliberately made things balance out.  The law of large numbers says that it WILL balance out eventually; but that does not happen by changing the probabilities in the short term. The long-term balance just swamps the short-term imbalance.

If you think about it, what could cause a coin to start landing heads up more often after a string of tails?  There is no possible physical cause for this; the coin has no memory of what it, or any other coin, previously did.  And probability theory does not make things happen without a physical cause; it just describes the results.

If I tossed a coin and it landed tails 5 times, I would just recognize that that is a perfectly possible (and not TOO unlikely) thing to happen.  If I got tails 100 times, I would NOT expect the next to be heads; I would inspect the coin to make sure it actually has a heads side!  An unlikely string of outcomes not only does not mean that the opposite outcome is more likely now; it makes it LESS likely, because it suggests statistically that the basic probability may not be what I was originally assuming.

In response to the question about probabilities being merely theoretical, I explained what probability is and isn’t:

Probabilities are theoretical, but have experimental results, IN THE LONG RUN.  The law of large numbers says that, if you repeat an experiment ENOUGH TIMES, the proportion of times an event occurs will PROBABLY be CLOSE to the predicted theoretical probability.  Probability can't tell you what will happen the next time, but it does predict what is likely to happen on the average over the next, say, million times.  If you started out with ten tails in a row, you will not necessarily get ten heads in a row at any point, or even more heads than tails in the next few tosses; you will just get enough heads in the next million to keep it balanced.

Note all the capitalized words, emphasizing the vagueness of this statement. A formal statement of this theorem will define those more clearly, but they will then all become statements of probability (my “probably”) and limits (my “enough” and “close”). See, for example, the formal statements of the Law of Large Numbers in the Wikipedia page (my emphases in the last paragraph):

The weak law of large numbers (also called Khinchin’s law) states that the sample average converges in probability towards the expected value $$\displaystyle \overline {X}_{n}\overset{P}{\rightarrow} \mu \ {\textrm {when}}\ n\to \infty .$$.

That is, for any positive number ε, $$\displaystyle \lim _{n\to \infty }\Pr \!\left(\,|{\overline {X}}_{n}-\mu |>\varepsilon \,\right)=0$$.

Interpreting this result, the weak law states that for any nonzero margin specified, no matter how small, with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value; that is, within the margin.

What it does not say is that over, say, 100 trials (or 10,000 trials), we can be sure that the average will be exactly what we expect.

I continued:

In particular, if you were to throw five coins at a time (to make it easier than throwing the same coin five times in a row), and do that 1000 times, you would expect that you would get 5 tails about 1/32 of those times (since the probability of all five being tails is 1/2^5). That's about 31 times!  So it's not at all unreasonable to expect that it will occur once in a while.

On the other hand, the probability of getting 100 tails in a row is 1/2^100, or 1/1,267,650,600,228,229,401,496,703,205,376, which makes it very unlikely that it has ever happened in the history of the world, though it could!
Again, probability can't tell you what WILL happen, specifically; it is all about unpredictable events.  But if you tossed a coin 1000 sets of 10 times, on the average one of those is likely to yield 10 tails. (The probability is 1/2^10 = 1/1024.)  The probability of some ten in a row out of 10,000 tosses is a little bigger, but that's harder to calculate.

Regression to the mean

In an unarchived question from 2014, Adam brought in another idea:

If I flip a coin 10 times, the most likely outcome is that I will flip a total of 5 heads and a total of 5 tails.  If each round of coin flipping (one round being 10 flips, in my example) is independent of previous rounds, then the probability of flipping a total of 5 heads and a total of 5 tails never changes.  However, the concept of "regression to the mean" implies that "rare" events are likely to be followed by less rare events and vice versa.  So, if I flip 10 heads in a row in round 1 (a rare event), the odds of flipping a total of 5 heads and 5 tails in round 2 are greater than if I'd flipped 5 heads and 5 tails in round 1.  What is the correct way to see this, do the odds change or not?

Coin flipping rounds are independent of one another, which implies that the probabilities never change. Regression to the mean states that rare events are likely to be followed by less rare events, implying that the probability of even random events does change.

I replied:

The odds don't change. You are NOT more likely to flip 5 heads if you previously flipped 10.

Regression to the mean only says that the most likely event on ANY toss will be about 5 heads, so if you got 10 on one toss, you are more likely to get something closer to 5 on the next toss, simply because something closer to 5 is ALWAYS more likely than 10. If you got 5 on the first toss, you are as likely to get 5 again as ever; but any deviation that does occur will be away from the mean, because there is no direction to go except away!

It is an entirely wrong interpretation of the phenomenon to think that, having tossed 10 heads, you are more likely to toss 5 heads than under other circumstances. It's just that the event that is always more likely will be less extreme than that first toss.

http://en.wikipedia.org/wiki/Regression_toward_the_mean

Past Events and ProbabilityBatting Averages


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The Fourier transform of this image exhibits an "infinite" series of harmonics or higher order terms, although these do not actually go out to infinity due to the finite resolution of the original image. So, responding to your comment, a 1 kHz square wave doest not include a. Since sines and cosines (and in turn, imaginary exponentials) form an orthogonal set1, this se-ries converges for any moderately well-behaved function f(x). Often in solid state physics we need to work with functions that are periodic. The square wave is a special case of a pulse wave which allows arbitrary durations at minimum and maximum amplitudes. In other words, Fourier series can be used to express a function in terms of the frequencies (harmonics) it is composed of. A few examples are square waves, saw-tooth waves, and triangular pulses. Example: Compute the Fourier series of f(t), where f(t) is the square wave …. In general, given a repeating waveform , we can evaluate its Fourier series coefficients by directly evaluating the Fourier transform: but doing this directly for sawtooth and parabolic waves …. They are helpful in their ability to imitate many different types of waves: x-ray, heat, light, and sound. The complex Fourier expansion coeﬃcients are cn= 4 π 1 n nodd 0 neven 0 2 4 6 8 10 0. In case of a string which is struck so that say at x=a only the string has a …. Fourier Series using LabVIEW Fourier Series using LabVIEW Student-developed LabVIEW VI The student is then asked to approximate a square wave in both the time and frequency domain using a summed set of 5 sine waves The frequency and amplitude from the LabVIEW interface provide the coefficients of the Fourier Series …. ABSTRACT Fourier analysis and a computer were used to generate the Fourier series and coefficients for the transmission distribution of a square wave …. Complex Fourier Series Animation of the square wave The 4 upper rotating vectors correspond to the 4 lower formula components. From the real to the complex Fourier series Proposition The complex Fourier coe cients of f(x) = a0 2 + X1 n=1 an cosnx + bn sinnx are cn = an ibn 2; c n = an + ibn 2: M. Fourier Series for a … Odd 3: Complex Fourier Series - Imperial College London It can be done by using a process called Fourier analysis. Draw a square wave of amplitude 1 and period 1 second whose trigonometric Fourier Series Representation consists of only cosine terms and has no DC component. m m! Again, we really need two such plots, one for the cosine series and another for the sine series. 3 1) Compute the Fourier series that corresponds to a square wave 1 - /4 /4 (0) 1. Square WaveFourier Series Examples And Solutions Square Wave Thank you enormously much for downloading fourier series examples and solutions square wave. Such a Fourier expansion provides an interpetation of the wave …. For comparison, let us find another Fourier series, namely the one for the periodic extension of g(x) = x, 0 x 1, sometimes designated x …. c) Write the first three nonzero terms in the Fourier expansion of f ( t). Jean Baptiste Joseph Fourier (21 March 1768 - 16 May 1830) Fourier series. He initialized Fourier series, Fourier transforms and their applications to problems of heat transfer and vibrations. 1) Compute the Fourier series that corresponds to a square wave. A Fourier series is nothing but the expansion of a periodic function f(x) with the terms of an infinite sum of sins and cosine values. The exact combination of harmonics will vary depending on the way the string is set in motion; e. Check the time t of the 10 rotations, t=24sec->T=2. Again, of course, you’re not going to get a perfect square wave with a finite number of Fourier terms in your series (in essence, it’s then not a …. 13 Applications of the Fourier transform 13. of Electrical Engineering, Southern Methodist University. The synthesis technique is also called additive synthesis. This chapter includes complex differential forms, geometric inequalities from one and several complex …. For instance, the square wave …. The ideas are classical and of transcendent beauty. An Introduction to Fourier Analysis Fourier Series, Partial Differential Equations and Fourier Transforms. As mentioned in the previous section, perhaps the most important set of orthonormal functions is the set of sines and cosines. s(t) = AN ODD SQUAREWAVE with DC. PDF Characterization of Signals Frequency Domain. Finding the Fourier series coefficients for the square wave sq T (t) is very simple. A Fourier series uses the relationship of orthogonality between the sine and cosine functions. , the output will always have a negative portion and positive portion. 1 The Real Form Fourier Series as follows: x(t) = a0 2 + X∞ n=1 an cosnω0t+bn sinnω0t (1) This is called a trigonometric series. In practice, the complex exponential Fourier series (5. (one period is T which is equal to 2PI) Looking at the figure it is clear that area bounded by the Square wave …. Plot one-sided, double-sided and normalized spectrum. continuation of part 1a - Introduction to Complex Fourier Series. The coefficients become small quickly for the triangle wave, but not for the square wave or the sawtooth. This means a square wave in the time domain, its Fourier …. Concept: The complex Exponential Fourier Series representation of a periodic signal x(t) with fundamental period To is given by, \$$x\\left(. In class, we mentioned how complex exponentials version of the Fourier Series can be used to represent a 2Tperiodic function f(t) as follows: can be used to represent a Fourier Series: f(t) = X1 n=1 c ne inˇ T t (1) The crux of Fourier Series analysis is to nd the c n values specifying how much the nth harmonic contributes to the function f(t). The synthesis of a complex wave …. 10 Trigonometric Fourier series of another square wave. Note that the Fourier coefficients are complex numbers, even though the series in Equation [1], evaluated with the coefficients in Equation [4], result in a real function. Similarly, if G(x) is an odd function with Fourier coe cients a nfor n 0 and b n for n 1, then a n= 0 for all n 0, and a n= 2 L Z L 0 G(x)sin nˇx L dxfor all n 0(16) In particular, the fourier series of an even function only has cosine terms and the fourier series of an odd function only has sine terms. It is very easy to see that an vanishes if f is an odd function, while bn vanishes if f is even. Index Terms—Fourier series, periodic function, recursive. Based on what I read at this link: Any periodic . It is natural for complex numbers and negative frequencies to go hand-in-hand. Even Square Wave (Exponential Series) Consider, again, the pulse function. 3: Fourier Cosine and Sine Series, day 1 Trigonometric Fourier Series (Example 2) Complex fourier Series - Example Fourier Transform (Solved Problem 1) Fourier Analysis: Fourier Transform Exam Question ExampleFourier Series: Complex Version! Part 1 Fourier …. (a) Determine the complex exponential Fourier series of x (t). Young (translator), There are excellent discussions of Fourier series …. Since the time domain signal is periodic, the sine and cosine wave …. Write a computer program to calculate the exponential Fourier series of the half-wave rectified sinusoidal current of Fig. This function is a square wave; a plot shows the value 1 from x=p to x = 0 followed by the . 3 Square wave Analysis (breaking it up into sine waves). Summation of just five odd harmonics gives a fairly decent representation in Figure 15. Complex fourier Series - Example Fourier Transform (Solved Problem 1) Fourier Analysis: Fourier Transform Exam Question ExampleFourier Series: Complex …. One of the principles of Fourier analysis is that any imaginable waveform can be constructed out of a carefully chosen set of sinewave components, assembled in a particular way (the frequency -> time task). (Image will be uploaded soon) Laurent Series Yield Fourier Series (Fourier Theorem). The complex Fourier series Recall the Fourier series expansion of a square wave, triangle wave, and sawtooth wave that we looked at before. Schrödinger's Equation Up: Wave Mechanics Previous: Electron Diffraction Representation of Waves via Complex Numbers In mathematics, the symbol is conventionally used to represent the square …. So here's the final wave, listening to the final waveform. You can work out the 2D Fourier transform in the same way as you did earlier with the sinusoidal gratings. In fact, in many cases, the complex Fourier series is easier to obtain rather than the trigonometrical Fourier series In summary, the relationship between the complex and trigonometrical Fourier series …. To motivate this, return to the Fourier series…. Recall from Chapter 1 that a digital sound is simply a sequence of num- Fourier analysis with complex exponentials which will often result in complex square wave are now 0 and 1,contraryto−1 and 1 before. 5) The Laurent expansion on the unit circle has the same form as the complex Fourier series, which shows the equivalence between the two expansions. Fourier Series | Brilliant Math & Science Wiki. Also, as with Fourier Sine series…. Theorem 2 lim N→∞ sup 0≤x≤1 f −S N(f) = 0 holds for any continuous function g. Compute the Exponential Fourier Series for the square wave shown below assuming that. Louis, MO April 24, 2012 The Fourier series is a tool for solving partial differential equations. (We assume the reader is already at least somewhat familiar with these. Step 1: Obtain the Fourier series of F(t). magnitude of the square of the Fourier transform: SFEt {()}2 This is our measure of the frequency content of a light wave. Plot the function over a few periods, as well as a few truncations of the Fourier series. It is not a mathematical proof, and several terms are used loosely (particularly those in quotes). We can be confident we have the correct answer. %Fourier series of rectangular wave clc; close all; clear all; j=1; T=4; %Time period of square wave tau=1; %2tau= On time of the square wave w0=2*pi/T;. 3–81 Spectrum of a Sum of Cosine Signals spectrum cosines 3–82 3. I understand the bounds that he chooses and. If a square waveform of period T is defined by  \left\{ \begin{array}{l l} f(t)= 1 \text{ if } t= T/2 \end{array} \right. Series coefficients c n (d) Fig. Discrete Time Fourier Transforms The discrete-time Fourier transform or the Fourier transform of a discrete–time sequence x[n] is a representation of the sequence in terms of the complex exponential sequence. There's also the infamous Square wave …. Since this wave is periodic, its harmonic content can be found using Fourier series as follows: The Fourier coefficients are,. Answer The function is discontinuous at t = 0, and we expect the series to converge to a value half-way between the upper and lower values; zero in this case. Let f(x) = {1 if -pPDF Fourier analysis for vectors. 2 Complex conjugate sqrt(x) Square root log(x) Natural logarithm Suppose we want to enter a vector x consisting of points Sine Wave 0. Example 1 Find the Fourier sine coefficients bk of the square wave SW(x). Fourier Analysis: Fourier Series with Complex Exponentials n The Complex Fourier series can be written as: where: n Complex cn n *Complex conjugate * n Note: if x(t) is real, c-n = cn 32 Fourier Analysis: Fourier Series Line Spectra n n Line Spectra refers the plotting of discrete coefficients corresponding to their frequencies For a periodic. Start by forming a time vector running from 0 to . See below Once rectified, it is even , so you only need the cosine series. Importantly there is nothing special about the square …. Where To Download Fourier Series Examples And Solutions Square Wave Fourier Series Examples And Solutions Square Wave As recognized, adventure as well as experience practically lesson, amusement, as skillfully as arrangement can be gotten by just checking out a books fourier series examples and solutions square wave …. The 8-term Fourier series approximations of the square wave and the sawtooth wave: Mathematica code: f[t_] := SawtoothWave[t] T = 1; Fourier series decomposition of a square wave using Phasor addition : It retains it’s form over several complex …. 3 Case 2: Some periodic functions (e. The Fourier series represents periodic, continuous-time signals as a weighted sum of continuous-time sinusoids. Not surprisingly, the even extension of the function into the left half plane produces a Fourier series that consists of only cos (even) terms. By the double angle formula, cos(2t) = 1 2sin2 t, so 1 + sin2 t= 3 2 1 2 cos(2t): The right hand side is a Fourier series; it happens to have only nitely many terms. A Fourier series ( / ˈfʊrieɪ, - iər /) is a sum that represents a periodic function as a sum of sine and cosine waves. The integral splits into two parts, one for each piece of , and we find that the Fourier coefficients are. Fourier Series in Mathematica Craig Beasley Department of Electrical and Systems Engineering Washington University in St. Jean Baptiste Joseph Fourier, a French mathematician and a physicist; was born in Auxerre, France. 005 (b) The Fourier series on a larger interval Figure 2. The Fourier series forthe discrete‐time periodic wave shown below: 1 Sequence x (in time domain) 0. On-Line Fourier Series Calculator is an interactive app to calculate Fourier Series coefficients (Up to 10000 elements) for user-defined piecewise …. The initial phase of the n-th oscillation θ θ n. (b) Consider the signal x(t) = sin(2πf0t), Find the complex Fourier series of x(t) and plot its frequency spectra. First, let x(t) be the zero-mean square wave. Wave Series Fourier Series Grapher; Square Wave: sin(x) + sin(3x)/3 + sin(5x)/5 + sin((2n−1)*x)/(2n−1) Sawtooth: sin(x) + sin(2x)/2 + sin(3x)/3 …. 1 Inﬁnite Sequences, Inﬁnite Series and Improper In-tegrals 1. This can be accomplished by extending the deﬁnition of the function in question to the interval [−L, 0] so that the extended function is either even (if one wants a cosine series) or odd (if one wants a sine series). As far as I know, Sage does not have a built-in method to find a “least-squares solution” to a system of linear equations. -L ≤ x ≤ L is given by: The above Fourier series formulas help in solving different types of problems easily. The periodic function shown in Fig. 23) all coefficients an vanish, the series only contains sines. HOWELL Department of Mathematical Science University of Alabama in Huntsville Principles of Fourier …. Inverse Fourier Transform maps the series of frequencies (their amplitudes and phases) back into the corresponding time series. Convert the real Fourier se-ries of the square wave f(t) to a complex series. Rad225/Bioe225 Ultrasound Fourier Series (review) Fall 2019. This section explains three Fourier series: sines, cosines, and exponentials e ikx. 5 The complex form of the Fourier series. 3 Square Wave–High Frequencies One application of Fourier series, the analysis of a “square” wave (Fig. Fourier series of the elementary waveforms. PDF The Exponential Form Fourier Series. It builds upon the Fourier Series. For the periodic bipolar, 50% duty-cycle square wave, the θ -averaging of this waveform over one θ -cycle is: QED. This subtle property is due to the symmetry of waveforms (except for the sawtooth, which is not symmetric). 01:6; >> fexact=4*(t<=3)-2*(t>=3); >> plot(t,fexact) and results in the plot: Plotting the Truncated Fourier. Joseph Fourier showed that any periodic wave can be represented by a sum of simple sine waves. 5, an =0, n ≥1, , 1 [1 cos( )] ≥ − = n n n bn π π. com analysis are described in the chapter on musical tones. Fourier series are used to approximate complex functions in many different parts of science and math. Demonstrates Taylor series expansion of complex exponentials. Recall the Fourier series, in which a function f[t] is written as a sum of sine and cosine terms: One interpretation of the above Fourier transform is that F[Z] is the frequency spectrum of a sine wave signal f[t] which is varying in time; thus Z is the angular frequency. The free space loss for electromagnetic waves spreading from a point source is The Friis’ loss formula for antenna-to-antenna loss is given by Radio wave propagation in the atmosphere: (1) space-wave propagation (e. This series of sine waves always contains a wave called the "FUNDAMENTAL", that has the same FREQUENCY (repetition rate) as the COMPLEX WAVE being created. Fourier transform spectroscopy (cont. Complex Fourier Series • Complex Fourier Analysis Example • Time Shifting • Even/Odd Symmetry • Antiperiodic ⇒ Odd Harmonics Only • Symmetry Examples • Summary E1. The fourier series of a sine wave is 100% fundamental, 0% any harmonics. If X is a multidimensional array, then fft …. Even Square Wave (Exploiting Symmetry). (here: symmetric, zero at both ends) Series –Taylor and Fourier Seismological applications The Delta function Delta function – generating series The delta function Seismological applications Fourier Integrals The basis for the spectral analysis (described in the continuous world) is the transform pair: Complex fourier spectrum The complex …. The conversion of complex Fourier series into standard trigonometric Fourier series is based on Euler's formulas: sinθ = 1 2j ejθ − 1 2je − jθ = ℑejθ = Imejθ, cosθ = 1 2ejθ − 1 2e − jθ = ℜejθ = Reejθ. Try another waveform, including one of the complex ones (Triangle, Sawtooth, or Square). A complex exponential einx= cosnx+isinnxhas a small-est period of 2π/n. This lesson shows you how to compute the Fourier series …. PDF General Inner Product & Fourier Series. m % Description: m-file to plot complex (exponential) Fourier Series % representation of a square wave…. Recall that the Taylor series expansion is given by f(x) = ¥ å n=0 cn(x a)n, where the expansion coefﬁcients are. 1 This pages contains exercises to practice computing the Fourier series of a CT signal. Since the coefficients of the Exponential Fourier Series of complex numbers we. Complex Fourier series Complex representation of Fourier series of a function f(t) with period T and corresponding angular frequency != 2ˇ=T: f(t) = X1 n=1 c ne in!t;where c n = 8 <: (a n ib n)=2 ;n>0 a 0=2 n= 0; (a jnj+ib jnj)=2 n<0 Note that the summation goes from 1 to 1. DC Value of a Square Wave The Fourier series coefficient for k = 0 has a special interpretation as the average value of the signal x(t). In short, the square wave’s coefficients decay more slowly with increasing frequency. Fourier series coefficients for a symmetric periodic square wave. Find the exponential Fourier series for the square wave of Figure 11. 10 Fourier Series and Transforms (2014-5543) Complex Fourier Series…. 2 Approximating the Square Wave Function using Fourier Sine Series. For the real series, we know that d = an = 0 and. Complex Fourier Series In an earlier module , we showed that a square wave could be expressed as a superposition of pulses. 1) translates into that the inverse of a complex (DFT on a square wave…. Example: Compute the Fourier series of f(t), where f(t) is the square wave with period 2π. The frequency of each wave in the sum, or harmonic, is an integer multiple of the periodic function's fundamental frequency. 11 Fourier Series of a Square Wave Co is a DC average. Fourier Series Square Wave Example The Fourier series of a square wave with period is University of California, San Diego J. Fourier series of a square wave of period T = 1. Lec1: Fourier Series Associated Prof Dr. It will definitely squander the time. Contribute to Abdul-Rahman-Ibrahim/Fourier-Series-Expansion-Complex-Coefficients development by creating an account on GitHub. Fourier analysis and predict with precision their behavior by just looking at the final expression, also called the design equation. Determine the Fourier series expansion for full wave rectified sine wave i. where the Fourier coefficients and are given by. Example: Find the complete Fourier series of the square wave function sqr(x). Keywords: Fourier analysis, Fourier coefficients, complex exponentials, discrete sine series, discrete cosine series, full and half wave rectifier, diode valve, square wave, triangular wave, and the saw-tooth wave. The complex algebra provides an elegant and compact re. The Fourier Transform is a method to single out smaller waves in a complex wave. If we let the numbers in the Fourier series get very large, we get a phenomenon, called This is called the complex Fourier series…. Fourier Series coefficients for a square wave. The main advantage of an FFT is speed, which it …. The series does not seem very useful, but we are saved by the fact that it converges rather rapidly. the Fourier representation of the square wave is given in Fig. Then mathematically, a T-periodic waveform v satisfies — a periodic waveform with period T (2) for all t. Discrete Fourier Transform (DFT)¶ From the previous section, we learned how we can easily characterize a wave with period/frequency, amplitude, phase. Show that the Fourier Series expansion of y (x) is given by y(t)= 4! sin"t+ sin3"t 3 + sin5"t 5 + # % & '(where ω = 2π/T. In this particular SPICE simulation, I’ve summed the 1st, 3rd, 5th, 7th, and 9th harmonic voltage sources in series …. Fourier Series and Coefficients Fourier series may be used to represent periodic functions as a linear combination of sine and cosine functions. 1: The cubic polynomial f(x)=−1 3 x 3 + 1 2 x 2 − 3 16 x+1on the interval [0,1], together with its Fourier series …. Example 2 Given a signal y(t) = cos(2t), find its Fourier Series coefficients. Let’s assume we have a square wave with following characteristics: P eriod = 2ms P eak−to −P eak V alue = 2 V Average V alue = 0 V P e r i o d = 2 m s P e a k − t o − P e a k V a l u e = 2 V A v e r a g e V a l u e = 0 V. The key here is that the Fourier basis is an orthogonal basis on a given interval. That is why we have programmed our free online Fourier series calculator to determine the results instantly and precisely. 6 Example: Fourier Series for Square Wave. Find the Fourier Complex Fourier Series . In the processing of audio signals (although it can be used for radio waves, light waves, seismic waves, and even images), Fourier analysis can isolate individual components of a continuous complex waveform, and concentrate. The function is reconstructed by the following summations over the fourier coefficients. In general square integrability will not guarantee convergence of the Fourier series to the original function. Fourier series are useful in a wide range of fields including acoustics, with representation of musical sounds as sums of waves of various frequencies (Nearing, 2020) or quantum mechanics, for the quantum wave function of a particle in a box. Step 3: Finally, substituting all the coefficients in Fourier …. Check back soon! Problem 5 Show that the complex Fourier series …. Any function can be written as the sum of an even and an odd function [ ( )]/2 A complex Lorentzian! Example: the Fourier …. Square WaveSeries Coefficients 11. The study of Fourier series is a branch of Fourier analysis. 6 Fourier Coefficients for the Complex Conjugate of a Signal. 1: Fourier series approximation to s q ( t). Once the complex wave is broken up, we can analyze the results using the simpler sine waves Consider a square wave as shown in Fig. But these are easy for simple periodic signal, such as sine or cosine waves. org odic if it repeats itself identically after a period of time. Every circle rotating translates to a simple sin or cosine wave. If a function is periodic and follows below 2 conditions, then the Fourier series for such a function exists. Thus a function or signal f(t) with period T 0 can be expressed as [0 < t < T 0] where is called the fundamental frequency or base frequency (first resonant frequency = 1/T) and all other nw 0 frequencies are called harmonics (every other component of. Fourier series in 1-D, 2-D and 3-D. A Fourier series is that series of sine waves; and we use Fourier analysis or spectrum analysis to deconstruct a signal into its individual sine wave components. Fourier series and fourier solutions square wave inverters are some mathematically and capacitors work, the average signal is our scope of a closer approximation graph the. (This is analogous to the fact that the Maclaurin series of any polynomial function is just the polynomial itself, which is a sum of finitely many powers of x. Start by forming a time vector running from 0 to 10 in steps of 0. The Fourier transform is an extension of the Fourier series that results when the period of the represented function is lengthened and allowed to approach infinity. When we break a signal down into its composite sine waves, we call it a Fourier series. A steady musical tone from an instrument or a voice has, in most cases, quite a complicated wave shape. As can clearly be seen it looks like a wave with different frequencies. The corresponding analysis equations for the Fourier series are usually written in terms of the period of the waveform, denoted by T, rather than the fundamental frequency, f (where f = 1/T). 1: Square Wave Then the Fourier Series …. Fourier Series Least Squares Curve Fit; Fourier Series Time Shift; Fourier Series Frequency Shift; Fourier Series, 4 segment; Fourier Series, var. The following method makes use of logical operators. 10 Fourier Series and Transforms (2014-5543) Complex Fourier Series: 3 - 2 / 12 Euler's Equation: eiθ =cosθ +isinθ [see RHB 3. The original signal x (t) is an square …. It can usually be solved with some Fourier series: Square wave example. So is periodic with period and its graph is shown in . The two round markers in the imaginary. which is the Fourier expansion of a square wave …. The Fourier Series (an infinite sum of trigonometric terms) gave us that formula. %Examples of Fourier Series Square Wave …. We choose xˆ[k] = 1 N X8 n=−2 x[n]e−2πiknN = 1 11 X2 n=−2 e−2πikn 11 The sum. This representation is done by summing sine and cosine functions or with complex exponentials of different amplitudes and phases. Step 2: Estimate for n=0, n=1, etc. For example, to ﬁnd the Fourier series for a triangular wave …. Okay, in the previous two sections we’ve looked at Fourier sine and Fourier cosine series. PDF Interpretation of the Physical Significance of the Fourier. The Fourier series represents the sum of smooth sinusoids, but a square wave …. As a tool to better understand the Fourier series coefﬁcients for the square …. Every 2πperiodic function that is analytic in a neighborhood of the real axis has a Fourier series …. In the interval (c, c+2ℓ), the complex form of Fourier series is given by. More formally, it decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex …. ∴ Given waveform is Non-periodic so, the Fourier series will NOT exist. Left click and drag the [ball, green] circles to change the magnitude of each Fourier functions [Sin nf, Cos nf]. B Square Wave The square or rectangular waveform is similar to the sawtooth in that the amplitudes of the har-monics follow the A N = A 1=N dependence. The Fourier transform is represented as spikes in the frequency domain, the height of the spike showing the amplitude of the wave of that …. More information about the Gibbs phenomenon. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Imagine a thin piece of wire, which only gains or loses heat through its ends. If you ally infatuation such a referred fourier series examples and solutions square wave ebook that will provide you worth, get the extremely best seller from us currently from several preferred authors. The coefﬁcients fb ng1 n=1 in a Fourier sine series F(x) are determined by. 23 ), are a measure of the extent to which has components along each of the basis vectors. The diagram above shows a "saw-tooth" wave …. entities represented by symbols such as ∞ n=−∞ a n, ∞ n=−∞ f n(x), and ∞ −∞ f(x) dx are central to Fourier Analysis. Fourier series Fourier series in higher dimensions (vector notation) Complex Exponentials In 2-D, the building blocks for periodic function f(x 1;x 2) are the product of complex exponentials in one variable. INTRODUCTION During Napoleon's reign, Joseph Fourier (1768-1830) was one of the French scientists who took French science to new heights. First we see that fcan be expressed in terms of the standard square wave as f(t) = 1 + sq t+ ˇ 2 : Now (see overleaf) the Fourier series for sq(t. width; Fourier Series, Continuous; Fourier Series, double Pulse; Fourier Series Calculation Square; Fourier Series Calculation Triangle; Fourier Series Square to Triangle Wave; Fourier Series …. Film or TV show with spooky dead trees and singing Explicit triples of isomorphic Riemann surfaces Typeset sudoku grid using tabular syntax. Aug 15, 2013 - The first one is the exponential form of the Fourier series and the. It is typical ( but as far as I know not required) that complex …. We look at a spike, a step function, and a ramp—and smoother functions too. The computer algorithm for Fourier transforms is called an FFT (Fast Fourier …. Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products. Determine the Fourier series coefficients of ( ) 𝑎 ( ). If you work through the math, you can find the optimal values for cn using equation [3]: [Equation 4] Note that the Fourier coefficients are complex numbers, even though the series in Equation [1], evaluated with the coefficients in Equation [4], result in a real function. the same way that it did for the complex Fourier series we talked about earlier, only we have to replace an integral with a sum. In other words, Fourier series can be used to express a function in terms of the frequencies () it is composed of. The Fourier transform tells us what frequency components are present in a given signal. How do I plot the Fourier series for a square wave? [closed] Ask Question Asked 4 years, 6 months ago. A Fourier series (/ ˈ f ʊr i eɪ,-i ər /) is a sum using only basic waves chosen to mathematically represent the waveform for almost any periodic function. That sawtooth ramp RR is the integral of the square wave. 6 Complex Fourier Series The exponential form of sine and cosine can be use to give: f(x) = X r c re 2ˇirx L The coe cients can be calculated from: c r= 1 L Z x o+L x o. Wave Equation and Fourier Series…. A complex waveform can be constructed from, or decomposed into, sine (and cosine) waves of various amplitude and phase relationships. As useful as this decomposition was in this example, it does not generalize well to other periodic signals: How can a superposition of pulses equal a smooth signal like a sinusoid?. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory. The Fourier series can be applied to periodic signals only but the Fourier transform can also be applied to non-periodic functions like rectangular pulse, step functions, ramp function etc. In this section we define the Fourier Cosine Series, i. You can use the following commands to calculate the nth partial sum of the Fourier series of the expression f on the interval [-L,L] syms x k L n. Similar expressions hold for more general intervals [ a, b] by shifting and scaling appropriately. a demo in Maple of complex fourier series. The formula for the fourier series of the function f(x) in the interval [-L, L], i. As promised in the first part of the Fourier series we will now demonstrate a simple example of constructing a periodic signal using the, none other then, Fourier series…. The amplitude of the n-th harmonic oscillation A n. f (t) has a finite number of maxima. An in nite sum as in formula (1) is called a Fourier series (after the French engineer Fourier who rst considered properties of these series. 14 The Complex Fourier Series 138 4. It establishes a relation between a function in the domain of time and a function in the domain of frequency. representing a function with a series in the form Sum( A_n cos(n pi x / L) ) from n=0 to n=infinity. More instructional engineering videos can be found . (a) The function and its Fourier series 0 0. This last line is the complex Fourier series. A square wave (represented as the blue dot) is approximated by its sixth partial sum (represented as the purple dot), formed by summing the first six terms (represented as arrows) of the square wave's Fourier series…. Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 1978 The use of Fourier analyzed square waves in the determination of the modulation tra. Given the mathematical theorem of a Fourier series, the period function f(t) can be written as follows: f(t) = A 0 + A 1 cosωt + A 2 cos2ωt + … + B 1 sinωt + B 2 sin2ωt + … where: A 0 = The DC component of the original wave. So I’ve just started learning about the Fourier series and wish to calculate one for a square wave function, my working is as follows. In some simple cases, Fourier series can be found using purely algebraic methods. approximations to the continuous Fourier series in Chapter 2. Also recall that the real part u and the imaginary part v of an analytic function f = u+iv are harmonic. If history were more logical they might have been found this way. Duration: 11:46 Complex Fourier Series - Square Wave …. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation. In this method, if N harmonics are included in the truncated Fourier series, then the amplitude of the kth harmonic is multiplied by (N - k)/N. Our aim was to find a series of trigonometric expressions that add to give certain periodic curves (like square or sawtooth waves. The result of the FFT is an array of complex numbers. The fine oscillations at the edges do not disappear even if the Fourier series takes many more terms. 1 The Fourier Series Components of an Even Square Wave Components of cos(nt) found in the approximation to an even square wave may be calculated generally as a function of n for all n > 0. Y = fft (X) computes the discrete Fourier transform (DFT) of X using a fast Fourier transform (FFT) algorithm. The Fourier Transform can be used for this purpose, which it decompose any signal into a sum of simple sine and cosine waves that we can easily measure the frequency, amplitude and phase. 5, and the one term expansion along with the function is shown in Figure 2: Figure 2. 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Here is why a square wave is a good test of high frequencies: The Fourier series corresponding to the square wave includes an infinite number of odd-harmonic sine wave components. The input to a discrete function must be a whole number. Examples of the Fourier Series · Rectangular Pulse Train · Impulse Train · Decaying Exponential Pulse Train · Even Square Wave · Odd Square Wave. Here, I’ll use square brackets, [], instead of parentheses, (), to show discrete vs continuous time functions. In this Tutorial, we consider working out Fourier series for func-tions f(x) with period L = 2π. Experts are tested by Chegg as specialists in their subject area. The vowel signal and the square-wave are both examples that suggest the idea ofapproximating a periodic On the other hand, when the positive and negative frequency terms in the Fourier Series are combined we add a complex …. Let the integer m become a real number and let the coefficients, F m, become a function F(m). 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A square wave is a type of waveform where the signal has only two levels. A Fourier series represents a function as an infinite sum of trigonometric functions: You can often use the complex exponential (DeMoivre's formula) to simplify computations: Specifically, let This graph is called a square wave…. The resulting series is known as Fourier series. Algorithm: Fourier Transform. This gif demonstrates how combinations of Fourier series mathematical functions can be used to create complex animated surfaces like these: Fourier Series, & Rectify. What we are extracting in these cases are the coefficients for the Fourier …. Consider the square wave: f(x) = 1 0 ≤ x < π = 0 −π ≤ x < 0 f(x) = f(x+2π) This appears to be a di cult case - the rather angular square wave does not look as …. In the previous lab, we implemented the trigonometric form of the Fourier Series, in this lab we will implement the complex form of the Fourier Series while learning some additional features of Simulink. Fourier Series Representation • Fourier Series Representation of CT Periodic Signals A periodic signal 𝑥𝑥(𝑡𝑡) can be represented as a linear combination of harmonically related complex exponentials (or sinusoids) 𝑎𝑎. The inverse Fourier transform is just to reconstruct the original function. Some operational formulas : 70. Graph of the function and its Fourier ….


# Optimization problem (rectangle inscribed below parabola) - possible textbook mistake

I have been working on a textbook's Optimization problem but the answer that I got does not match the textbook's answer. I would like to make sure I got it right (I can't find any mistakes on my solution), so I would like to ask for someone's help. I would appreciate that.

The problem:

A rectangle is located below a parabola, which is given by $$y = 3x- \frac{x^2}{2}$$ in such a way that its two superior vertexes are placed on the parabola and its two inferior vertexes are located on the $$x$$ axis. The left, inferior vertex, is placed on the point $$(c,0)$$. That said: a) Show that the area of the rectangle can be represented by the equation $$A(c) = c^3 -9c^2 + 18c$$ b) Find the rectangle's height and width given that is has maximum possible area.

c) What is that area?

Instead of typing the whole solution I will post an image with it (sorry for that but latex-ing it would take a lot of time!). You will find my solution below.

If that's helpful, the textbook's answer for b) and c) are $$3-\sqrt{3} \times 3$$ and $$9 + 9\sqrt{3}$$ respectively.

Thank you.

• $base=2\sqrt{3}$ – Aleksas Domarkas Jan 6 at 8:19

You made a slight error when trying to find the base:

$$c = 3-\sqrt{3} \implies b = 2(3-c) = \color{blue}{2\left[3-\left(3-\sqrt{3}\right)\right]} = 2\sqrt{3}$$

You forgot the $$3$$ in $$(\color{blue}{3}-c)$$ and found $$b = 2c$$ instead.

Addition: From here, using $$h = 3$$ as you found, you get

$$A = bh \iff A = 3\left(2\sqrt{3}\right) = 6\sqrt{3}$$

Try plugging in $$c = 3-\sqrt{3}$$ in $$f(c)$$:

$$f\left(3-\sqrt{3}\right) = \left(3-\sqrt{3}\right)^3-9\left(3-\sqrt{3}\right)^2+18\left(3-\sqrt{3}\right) = 6\sqrt{3}$$

Through confirmation, you can see this point coincides with the local maximum.

• thank you I've fixed that mistake. Now I have $2 \sqrt{3}$ as a base and $3$ as the rectangle's height, which gives me an area of $A = 6 \sqrt{3} \ m^2$. Am I good? Thanks for your help! – bru1987 Jan 6 at 9:16
• Yes, that’s the correct answer! – KM101 Jan 6 at 9:24
• thank you and have a great day =) – bru1987 Jan 6 at 9:26
• Glad to have helped! :-) You may want to mark the answer as accepted if it has helped resolved the issue. – KM101 Jan 6 at 9:31


# Parity of the multiplicative order of 2 modulo p

Let $$\operatorname{ord}_p(2)$$ be the order of 2 in the multiplicative group modulo $$p$$. Let $$A$$ be the subset of primes $$p$$ where $$\operatorname{ord}_p(2)$$ is odd, and let $$B$$ be the subset of primes $$p$$ where $$\operatorname{ord}_p(2)$$ is even. Then how large is $$A$$ compared to $$B$$?

• $A/(A+B)$ tends to $7/24$ ? (not proved yet). – Henri Cohen Sep 23 at 21:39
• Seems like an interesting question, and clearly generalizable quite a lot. However, if you're going to ask many questions on this site, it would be a good idea to learn a little bit of TeX formatting. I've fixed the formatting of your question, so if you click on "edit", you'll be able to see what I did to make it more readable. I also changed the title of your question to make it even clearer what you're asking. – Joe Silverman Sep 23 at 21:54
• @HenriCohen how did you determine $A/(A+B)$ to be $7/24$ while also writing "not proved yet"? The proportion of $p \leq 100000$ for which $2 \bmod p$ has odd order is $2797/9591$, which as a continued fraction is $[0,3,2,3,44,9]$, and the truncated continued fraction $[0,3,2,3]$ is $7/24$. I'd be interested to know if you did that or something else. – KConrad Sep 24 at 3:26
• Note: the set $A$ is at OEIS: oeis.org/A014663, while its complement $B$ is oeis.org/A091317. Among the 46 primes below 200, $A$ consists of the 14 primes 7, 23, 31, 47, 71, 73, 79, 89, 103, 127, 151, 167, 191, 199. – YCor Sep 24 at 9:44
• see this answer – René Gy Sep 24 at 15:39

This problem was asked by Sierpinski in 1958 and answered by Hasse in the 1960s.

For each nonzero rational number $$a$$ (take $$a \in \mathbf Z$$ if you wish) and each prime $$\ell$$, let $$S_{a,\ell}$$ be the set of primes $$p$$ not dividing the numerator or denominator of $$a$$ such that $$a \bmod p$$ has multiplicative order divisible by $$\ell$$. When $$a = \pm 1$$, $$S_{a,\ell}$$ is empty except that $$S_{-1,2}$$ is all odd primes. From now on, suppose $$a \not= \pm 1$$.

In Math. Ann. 162 (1965/66), 74–76 (the paper is at https://eudml.org/doc/161322 and on MathSciNet see MR0186653) Hasse treated the case $$\ell \not= 2$$. Let $$e$$ be the largest nonnegative integer such that $$a$$ in $$\mathbf Q$$ is an $$\ell^e$$-th power. (For example, if $$a$$ is squarefree then $$e = 0$$ for every $$\ell$$ not dividing $$a$$.) The density of $$S_{a,\ell}$$ is $$\ell/(\ell^e(\ell^2-1))$$. This is $$\ell/(\ell^2-1)$$ when $$e = 0$$ and $$1/(\ell^2-1)$$ when $$e = 1$$.

In Math. Ann. 166 (1966), 19–23 (the paper is at https://eudml.org/doc/161442 and on MathSciNet see MR0205975) Hasse treated the case $$\ell = 2$$. The general answer in this case is more complicated, as issues involving $$\ell$$-th roots of unity in the ground field (like $$\pm 1$$ in $$\mathbf Q$$ when $$\ell = 2$$) often are. The density of $$S_{a,2}$$ for "typical" $$a$$ is $$1/3$$, such as when $$a \geq 3$$ is squarefree. But $$S_{2,2}$$ has density 17/24, so the set of $$p$$ for which $$2 \bmod p$$ has even order has density $$17/24$$ and the set of $$p$$ for which $$2 \bmod p$$ has odd order has density $$1 - 17/24 = 7/24$$.

For example, there are $$167$$ odd primes up to $$1000$$, $$1228$$ odd primes up to $$10000$$, and $$9591$$ odd primes up to $$100000$$. There are $$117$$ odd primes $$p \leq 1000$$ such that $$2 \bmod p$$ has even order, $$878$$ odd primes $$p \leq 10000$$ such that $$2 \bmod p$$ has even order, and $$6794$$ odd primes $$p \leq 100000$$ such that $$2 \bmod p$$ has even order. The proportion of odd primes up $$1000$$, $$10000$$, and $$100000$$ for which $$2 \bmod p$$ has even order is $$117/167 \approx .700059$$, $$878/1228 \approx .71498$$, and $$6794/9591 \approx .70837$$, while $$17/24 \approx .70833$$.

The math.stackexchange page here treats $$S_{7,2}$$ in some detail and at the end mentions the case of $$S_{2,2}$$.

• Thanks @KConrad for the expression. I was thinking about a combinatorial property of cyclic groups of prime order(called acyclic matching property) and I proved the above mentioned sequence of primes does not hold it. See Proposition 2.3 of core.ac.uk/download/pdf/33123051.pdf Anyway I will like to mention this result in my ongoing research work as a remark, and hence I ask your permission for the same, of course with acknowledgment. – Mohsen Sep 25 at 18:09
• Since the result is due to Hasse, cite his paper when you want to indicate who first showed that the density exists and what its value is. – KConrad Sep 25 at 19:35