Talk:Fourier transform

163: 153: 132: 31: 313:"Functional relationships, one-dimensional", property 102, time shifting of fourier transform. There should be a minus in the power of e: e^(-2*pi*i*...). That minus is missing in the entire row. I think I verified it on paper, but also with other sources, including the wikipedia fourier transform article itself (section 5.1.2 Translation / time shifting). I have no idea how to fix this. This is my first post on wikipedia ever. I hope I'm correct though and not wasting anyones time. 82: 64: 242: 22: 1229: 5346: 3263:

is negative. Engineers prefer that because the phase of cos(ωt) + sin(ωt) relative to cos(ωt) is negative. Mathematicians consider cos(ωt) and sin(ωt) as basis vectors and they plot cos(ωt) + sin(ωt) as . That is all there is to it. Engineers prefer that the Fourier component of sin(ωt) should be

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Since the source doesn’t give a reason, I will remove the reason. But the reason is this: it is arbitrary. It is just a choice of where you want your negative signs to appear. As an engineer, I have my preference mainly because that is the way I was taught and that is the way it appears in most of

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When switching to PNG I can see the sign as should. The problem is visible when using "MathML with SVG or PNG fallback", which of course is default. Just checked another thing: while zooming in and out, I saw the equations in correct form for all zooms above 125%, and also specificly for 90% (but not

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I'm not aware of our tables being incorrect nor of the inconsistencies. I looked and did not find them. So there must have been very few, and so it can more easily be addressed by changing a small number of things instead of everything else. Sorry to be disagreeable, but I also don't think 2πi is

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Aside from the last one, these are large professional orgainzations trying to make something for the masses. About the last one, I include the last one because I would like to return to the status quo for the article. I also feel it would benefit readers because they are (in my opinion) more likely

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Furthermore, the statement "every real sinusoid consists of an equal contribution of positive and negative frequency components, which is true of all real signals" is misleading. It is a cancellation, not a contribution, analogous to something like "10 apples consists of 5 apples + 5 bananas and 5

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Anyhow, we're getting a little off track. The point is that the convention chosen for the article (which we both agree with) was not an arbitrary coin toss. It might not have any consequences for EM theory, but it does have consequences for signal theory. So I added a footnote that does not need

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Hello. There is a problem with Chrome and browsers based off of Chrome such as Edge. The wizards are aware of it. As a temporary fix you can increase the zoom factor. Also, you can get the Math Anywhere extension for both Chrome and Edge that seems to take care of the problem. Or you can wait

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I have to agree with the IP editor that this is poorly written and after checking the source, I see that the source does not say what is written here. In fact, the source doesn’t give a reason, it just says that engineers prefer a certain sign convention. The source isn't even about the Fourier

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I think this article should be improved in some manner for the general public that is scientifically minded but not taking a full calculus class in college. I think the possibility of adding it to the Simple English Knowledge with easier to understand language is a good idea, in addition to the

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I think that's on the right track. I was definitely struggling with that section... I kept coming back to the question "Does it even need to be here?" IMO, the ccrma.stanford.edu viewpoint is the easy explanation, more of an engineering convenience than a true insight. I'm all in favor of

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Correct! In latex there is a minus sign. Using chrome, just checked with another pc and it's working fine. And if I look here closely I think I can see the missing sign very faintly. So it must be something with the local browser and not display settings of wikipedia. Sorry for the trouble.

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process of adding explanations and writing that is not mathematically centered. Maybe including something of this sort, "The Fourier Transform helps to transform functions such as cosine and sine into different output functions that behave differently than normal trigonometric functions."

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more common than i2π. Regardless of that, they are both common, so I think we should look at the rational for each. Besides the (very good) reason I cited in the undo, leading with the i makes it more obvious that it's a complex exponential, which is its most important attribute.

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and did not see anything there that would favor one convention for FT over the other. There is no physical requirement for one or the other. However, I will speculate. When I look at the engineers describing a simple wave traveling in the x direction, they tend to use

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It is a dense 554-page book, with not a single Fourier transform formula or even Euler's formula. My take-away is that the EM applications of Fourier transform theory don't go deep enough to matter which convention they use. To quote myself (above) "For real-valued

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I get your point. But I disagree with your statement "But the reason is this: it is arbitrary. It is just a choice of where you want your negative signs to appear." No. It comes down to your definition of the instantaneous phase and frequency of function

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which is definitely not square integrable. I don't think the citation says what the author thought it said. There are a couple of other places like this too. If I can at some point I will try to run through the citations and verify and fix what I see.

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I use the same convention that you are advocating. I have no problem with using that convention. If you want to state a reason for that convention in the article you need a reliable secondary source that states that reason. No amount of

3067: 1542:. Anyone who disagrees with that convention is in a lonely minority, and will have endless difficulty with the preponderance of relevant literature and Knowledge articles. For all the others, the conclusion is obvious. The quantity 4744:

IMO this can be handled by just inserting a footnote to the effect that a synonym (in this case at least) for transform is mapping (and vice versa). Here we prefer transform, because it is the most commonly found and widely accepted

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I would like to discuss the recent revert of my edit. The edit, in part, addresssed the comment from the IP editor about that our tables were incorrect. While looking at the article I noticed it was inconsistent in the placement of

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discussion between just 2 editors. And as noted then, ν was the original preference, replaced without a consensus. Now I am asking if there is any consensus for reverting back to ν, because ξ is unnecessarily intimidating-looking

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I conclude that those who claim otherwise have a different definition of instantaneous phase or instantaneous frequency, which puts them at odds with Knowledge's sourced articles. The burden is on them to provide contradictory

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You're right, I did reply to the edit summary stating your preferred format is more logical. Clearly, I disagree. There are plenty of these arguments around, and they all boil down to a matter of taste. This is a bit like

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in front, so I can tell immediately if it is there without searching. The fact that the figures and text disagree is not a problem. Anyone who does not realize that i2π is the same as 2πi won't understand anything

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I have no idea what your 3rd sentence means. I understand the 2nd sentence, but I disagree with "far better". Knowledge's standard is to reflect a consensus of common usage, not to create a better math

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The word "transform" (whether verb or noun) as a technical term in math is distinctly more advanced than "mapping", which all peopl exposed to even just a smidgen of math know means a function.

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By any reliable judgment, it is significant — and so worth inclusion in the article — that applying the Fourier transform four times in a row will result in the same function you started with.

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IMO, the most important attribute is to be a service to the readers, particularly the non-experts who may be thrown off by the change. They will most likely be looking at a reference where

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I find the introduction difficult to understand. It would be far better to avoid use of the word "transform" at first in the explanation, and instead say that the Fourier transform is a

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I understand you don't want a complicated discussion on complex sinusoids. I'm wondering maybe what if I move that discussion out from this article and put it in the article for either

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I removed a reference from here because the cited reference wasn't discussing the Fourier transform in the section indicated. I will try to find something more specific to this topic.

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Thank you. I can expand the upper list, if needed, but it seems to be coming down to signals and communication vs electromagnetics. Amazingly, I still have my undergrad copy of

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This has no physical effect because physical effects are caused by energy or power. The power of a Fourier transform is computed by multiplying the transform by its conjugate.

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On the other hand, there was a mistake in table item 207 as the function was not square integrable. Please go ahead and fix that while the discussion about i2π continues.

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The issue with the table being incorrect started on this talk page! A kindly IP editor spotted it, and left a comment here. I said I would research and fix it and I did.

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The reason given is nonsense. In fact, both negative and positive signs are used by different communities and there is no problem with convergence of the integral.

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In English the name is often pronounced foo-ree-ei or foor-ee-ei (the last syllable rhyming with "day", and all three syllables given approximately equal stress).

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my textbooks. You may find justifications for one choice or the other, but you will not find a definitive reason the sign must be negative or must be positive.

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you are also arbitrarily rejecting the customary definitions of instantaneous phase and frequency. Therefore you need to provide sourced reasons for that whim.

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Now let me go way off into OR la-la land to speculate why engineers prefer the usual convention. Consider the Fourier transform of cos(ωt) + sin(ωt). It is

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Yes, I think you're right. I will do a bit of research, once I find a source for this particular formula, I will cite it, move it, and include conditions.

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Since the code apparently produces a graph, perhaps Wikimedia is an appropriate host for this creation. And it welcomes the inclusion of source code.

5574:. However, this condition is not mentioned, and if this condition is included, (Line 207) becomes just a restatement of (Line 206) with the replacement 718:{\displaystyle {\begin{aligned}&{\hat {f}}(\omega )\\&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx\end{aligned}}} 6915: 6225:

My experience has an educator leads me to disagree. IMO students are more thrown off by disagreements in notation that some philosophical point about

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I was hoping to see an IPA pronunciation in the first sentence of the article (but there is not one). Wiktionary has an English pronunciation for

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Which browser are you using? Note that when you create an account and log in, you have some options to affect the appearance of text and math. -

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so I fixed it. It seems the objections are mostly about the later. For which I would like to point out, the prevailing culture is that if a

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I only use a laptop, so I'll take a look at that. Also, I'll take a look for those remaining "inconsistencies". I don't like them either.

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Indeed, with the current versions of Chrome and Edge, 4 out of 6 minus signs are missing in the table above. Firefox is doing just fine. -

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Greetings. I had not responded because it looked like we were at an impasse, but I think the conversation should continue. I looked at

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But no. The burden is one those who wish to add a fact to a Knowledge article. The burden is on them to provide a reliable source.

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Sure, but I have no better way than to copy and paste from the diff, which I cannot do from my phone, but I'll try when I am able.

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being important, so it should come first. But we may be at an impasse, hopefully someone else while chime in and help us out.

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I certainly agree: The last thing a Knowledge article on a math topic should resemble in tone or wording is a math textbook.

3095: 2266:{\displaystyle {\begin{aligned}&{\hat {f}}_{+}(\omega )=\int _{-\infty }^{\infty }f(t)e^{+i\omega x}\,dt\end{aligned}}} 2149:{\displaystyle {\begin{aligned}&{\hat {f}}_{-}(\omega )=\int _{-\infty }^{\infty }f(t)e^{-i\omega x}\,dt\end{aligned}}} 2637: 2532: 1887:{\displaystyle \int _{-\infty }^{\infty }f(x)\ e^{+i2\pi \xi x}dx=\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi (-\xi )x}\,dx} 5098:{\displaystyle e^{i2\pi \xi _{0}x}\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ \delta \left(\xi -\xi _{0}\right).} 2474:{\displaystyle {\begin{aligned}&{\hat {f}}_{-}{\hat {f}}_{-}^{*}=&{\hat {f}}_{+}{\hat {f}}_{+}^{*}\end{aligned}}} 6360: 4403: 958: 835:{\displaystyle {\begin{aligned}&{\hat {f}}(\nu )\\&=\int _{-\infty }^{\infty }f(x)e^{-i\nu x}\,dx\end{aligned}}} 44: 892: 336: 5881: 4678:

The reason in my opinion is that the audiences for the two things have very different characteristics, statistically.

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here. I assume this is a standard ise of the term in signal processing, but agree that a reference seems desirable.

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the integrals are over the variable σ but the variable ξ appears in the integrands. Where does the ξ come from?

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in the most prominent location, and to keep 2πξ (= ω) together, especially when side-by-side in the tables with

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So what you are showing is that when its domain is bounded, it can also be recovered from discrete samples of

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So, lets look at a couple of examples. I will suppress multiplicative constants that clutter up the results.

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The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion:

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I agree. Still, I think α is assumed to be purely real, if so, it should not be in the table at that point.

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The thing is: A consensus already exists in a strong and wide area of mathematics called "Fourier analysis".

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library. So this code is very python-specific, and is not a good illustration of the Fourier transform.

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And I removed that reference to that ccrma.stanford.edu page cause what I wrote is now slightly different.

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But at 15:45 on 16 February 2023, an editor removed what appears to be a directly relevant reference: (

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I suggest initially avoiding the verb "transform" to explain the Fourier transform, and instead use the

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the convention hardly matters". The concept of negative frequency is not useful. So your statement:

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for 100%, 110% or below 80%). It seems to be some sort of a rendering issue (??) with my own browser.

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I think there may be an error in the symmetry section. For example, it says "even-symmetric function

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Hi, I think there is a small mistake in section 15 "Tables of important Fourier transforms" -: -->

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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As far as I can see, all the minus signs are there. Don't you see a minus sign in the following?

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At the very least it needs to clarified that the first equality only applies in the interval T.

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This is the other convention. It is mathematically equal to the conventional reverse transform.

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Well done, but it repeats a lot of information available thru WikiLinks, and now also added to

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might be a misleading generalization based on certain limited applications of transform theory.

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The IP editor is correct (if you read this, I hope you make an account and stick around). If

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that rotates in the positive direction (increasing vector angle = CCW) for positive frequency

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are less frequent than those that avoid complex number all together by using sine and cosine.

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an external citation. All it relies on is a Wikilink to our instantaneous frequency article.

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function whose Fourier series was referred to ... I realized that the article refers to the

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That sounds about right, based on my distant memories (circa 1968) of one EM theory course.

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the convention hardly matters, because every frequency has a positive equivalent (e.g. see

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conveniences, but I'm also in favor of distinguishing them from the underlying realities.

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redirect page could become its own page that discusses both real and complex sinusoids.

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A quick impression is that this section could be simplified, perhaps making use of the

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And that according to a version of the Fourier transform F favored by mathematicians,

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A Commons file used on this page or its Wikidata item has been nominated for deletion

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is listed as a square integrable function (Line 207). I think this is only true if

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will change that. However, I do not mind dabbling in OR here on the talk page.

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I should have added: please do not try to fix it by modifying the LaTeX markup.

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I hope someone knowledgeable about this subject can fix this apparent problem.

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is considered a positive or a negative frequency. The customary definition of

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Thus the results of these two conventions are simply conjugates of each other.

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Transform, as a verb, is more meaningful to the typical reader than mapping.

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Section: Fourier transform for functions that are zero outside an interval

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And as a proportion of total references those that put an i before the

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isn't even symmetric, right? Shouldn't the even symmetric function be

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Excluding the few that I couldn't get access to, in the ones that use

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I'm sorry. Thank you for pointing that out. For now I've rephrased as:

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Press, William H.; Teukolsky, Saul A.; Vetterling, William T. (2007),

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by 45°. When an engineer plots this in Cartesian space, it is . The

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Reitz, John R.; Milford, Frederick J.; Christy, Robert W. (1993),

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Fourier transform for functions that are zero outside an interval

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I went through several of my text books. Here is what I found.

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waveforms. However, the Fourier transform is able to represent

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Section: Use of complex sinusoids to represent real sinusoids

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taking an integrable function on R to an integrable function.

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One is periodic and the other is zero outside the interval .

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As I tried to write a few clarifying sentences in the section

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The last sentence is complete mathematical nonsense, and is

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Oppenheim, Alan V.; Willsky, Alan S.; Young, Ian T. (1983),

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That is what's in the 102 item, and here's the entire line:

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is the appropriate nomenclature. Similarly, the quantity

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all your talk page messages with four tildes (~~~~) — See

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This usage is supported by Oppenheim and Schafer, fwiw.

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Complex-valued function modulating a complex sinusoid.svg

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Cutting to the chase, the convention determines whether

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Consequently, you've made a mountain out of a molehill.

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I am not in favor of this. This is a bit like avoiding

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And I think it serves the non-experts more to put the

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I believe α is assumed to be purely real. Im(α) = 0.

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The result of this section seems to be the statement:

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Principles and Applications of Electromagnetic fields

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Under appropriate conditions, the Fourier series of

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Use of complex sinusoids to represent real sinusoids

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The Fourier transform under the other convention is

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The Fourier transform under the usual convention is

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The Fourier transform under the other convention is

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The Fourier transform under the usual convention is

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Definition section: the reason for the negative sign

180:, a collaborative effort to improve the coverage of 93:, a collaborative effort to improve the coverage of 3356:is the derivative of instantaneous phase, which is 1007:{\displaystyle e^{-ia\omega }{\hat {f}}(\omega )\,} 6766: 6692: 6623: 6551: 6197: 6161: 6123: 6091: 6065: 6039: 5999: 5922: 5850: 5827: 5801: 5728: 5672: 5595: 5564: 5523: 5466:which, by the way, is the dual of the time domain 5458: 5417: 5376: 5340: 5097: 4988: 4849:Section: Fourier transform for periodic functions 4825: 4469:waveforms as well. It achieves this by applying a 4196: 4152: 4068: 3855: 3752: 3675: 3634: 3561: 3459: 3397: 3374: 3344: 3304: 3251: 3213: 3143: 3084: 3061: 2953: 2930: 2907: 2839: 2816: 2793: 2724: 2682: 2623: 2600: 2577: 2517: 2473: 2345: 2265: 2148: 2004: 1963: 1934: 1886: 1729: 1687: 1658: 1616: 1534: 1514: 1440: 1066: 1006: 947:{\displaystyle e^{-2\pi ia\xi }{\hat {f}}(\xi )\,} 946: 880: 834: 717: 585: 462: 391:{\displaystyle e^{-2\pi ia\xi }{\hat {f}}(\xi )\,} 390: 4719:Mapping is discussed in the body of the article. 3469:is a positive frequency. And its measurement is 1894:is a measure of the relative amount of component 1624:is a measure of the relative amount of component 5887:The first few articles I looked at from the IEEE 6931:Knowledge level-4 vital articles in Mathematics 6879:Even_and_odd_functions#Complex-valued_functions 6794:Even_and_odd_functions#Complex-valued_functions 4999:is no better than relying on the transform pair 4271:Fourier_analysis#(Continuous)_Fourier_transform 6351:Participate in the deletion discussion at the 5882:The makers of FFTW, the open source FFT engine 4477:and then treating that as a periodic waveform. 4254:) 01:38, 14 October 2021 (UTC)ScientistBuilder 3460:{\displaystyle f(t)=e^{i\omega t},\omega : --> 3069:. The component at the positive frequency of 6838:I don't think I have access to the reference. 6829:harvnb error: no target: CITEREFProakis1996 ( 1067:{\displaystyle e^{-ia\nu }{\hat {f}}(\nu )\,} 8: 4596:Surely this is worthy of prominent mention. 4517:harvnb error: no target: CITEREFTaneja2008 ( 4027:Plonsey, Robert; Collin, Robert E. (1961). 3214:{\displaystyle cos(\omega t)+sin(\omega t)} 2156:This is the conventional forward transform. 6379: 5565:{\displaystyle \mathrm {Im} (\alpha ): --> 4660:from a function space to a function space. 4473:to lengthen the period of any waveform to 126: 58: 6743: 6723: 6676: 6672: 6649: 6645: 6636: 6607: 6603: 6580: 6576: 6567: 6535: 6531: 6508: 6504: 6495: 6180: 6174: 6154: 6113: 6081: 6055: 6012: 5972: 5909: 5840: 5814: 5788: 5715: 5707: 5696: 5687: 5685: 5647: 5645: 5579: 5539: 5537: 5513: 5502: 5496: 5433: 5432: 5430: 5392: 5391: 5389: 5360: 5318: 5304: 5289: 5271: 5270: 5264: 5250: 5224: 5210: 5198: 5188: 5174: 5153: 5081: 5052: 5051: 5046: 5044: 5043: 5030: 5016: 5010: 4975: 4948: 4934: 4915: 4894: 4884: 4870: 4864: 4818: 4165: 4121: 4052: 3925:Hayt, William; Kemmerly, Jack E. (1971), 3827: 3803: 3795: 3768: 3767: 3765: 3735: 3729: 3650: 3649: 3647: 3612: 3588: 3580: 3574: 3536: 3512: 3504: 3477: 3476: 3474: 3433: 3412: 3387: 3361: 3330: 3324: 3288: 3226: 3164: 3097: 3074: 2970: 2943: 2920: 2855: 2829: 2806: 2741: 2699: 2690:. The result is exactly the same result. 2639: 2613: 2590: 2534: 2492: 2487:First, consider the Fourier transform of 2461: 2456: 2445: 2444: 2437: 2426: 2425: 2413: 2408: 2397: 2396: 2389: 2378: 2377: 2370: 2368: 2333: 2328: 2317: 2316: 2306: 2295: 2294: 2287: 2285: 2236: 2214: 2206: 2184: 2173: 2172: 2165: 2163: 2119: 2097: 2089: 2067: 2056: 2055: 2048: 2046: 1979: 1978: 1976: 1947: 1905: 1899: 1846: 1822: 1814: 1780: 1756: 1748: 1742: 1707: 1706: 1704: 1671: 1635: 1629: 1585: 1561: 1553: 1547: 1527: 1453: 1360: 1354: 1042: 1041: 1026: 1020: 982: 981: 966: 960: 922: 921: 900: 894: 856: 805: 783: 775: 740: 739: 733: 731: 688: 666: 658: 639: 608: 607: 601: 599: 550: 528: 520: 485: 484: 478: 476: 444: 366: 365: 344: 338: 6767:{\displaystyle f(-x)={\overline {f(x)}}} 6693:{\displaystyle (f_{_{RE}}+i\ f_{_{IE}})} 6624:{\displaystyle (f_{_{RE}}+i\ f_{_{IO}})} 6552:{\displaystyle (f_{_{RE}}+i\ f_{_{IO}})} 5673:{\displaystyle \mathrm {Im} (\alpha )=0} 3918:Stein, Seymour; Jones, J. Jones (1967), 2694:Next, consider the Fourier transform of 409: 6921:Knowledge vital articles in Mathematics 6824: 6703:Is there a reference for this section? 6029: 5977: 5899:when it was changed without discussion. 5204: 4505: 4304:. I'll try to give that some thought. 3956:(3rd ed.), Cambridge University Press, 3842: 3624: 3551: 2251: 2134: 1876: 1606: 1062: 1002: 942: 876: 820: 703: 571: 458: 386: 308:Maybe a mistake in time shift property? 128: 107:Knowledge:WikiProject Signal Processing 60: 19: 5862:to see this I would like to point to: 4770:Fourier transform#The analysis formula 4512: 3314:Aliasing#Sampling_sinusoidal_functions 1152:No problem. What happens if you go to 110:Template:WikiProject Signal Processing 6936:B-Class vital articles in Mathematics 5729:{\displaystyle |e^{i\alpha x^{2}}|=1} 4707:2601:200:C000:1A0:FDA6:8A26:FCCA:2C1B 4598:2601:200:C000:1A0:FDA6:8A26:FCCA:2C1B 4561:2601:200:C000:1A0:BC00:5039:DB55:E9EC 4488:2601:200:C000:1A0:BC00:5039:DB55:E9EC 4419:2601:200:C000:1A0:BC00:5039:DB55:E9EC 4394:(as a surname), which might apply to 3989:Foundations of Electromagnetic Theory 2915:. It has Fourier components at both 7: 6877:Thank you. I added the citation to 5487:Erroneous square integrable function 4656:"mapping" to say that the f.t. is a 3676:{\displaystyle {\hat {f}}(-\omega )} 2801:. It has Fourier components at both 2585:. It has Fourier components at both 1226: 174:This article is within the scope of 87:This article is within the scope of 5596:{\displaystyle \alpha \to i\alpha } 3984:, McGraw-Hill page 294, equation 47 3968:Classical Electrodynamics (3rd ed.) 3922:, McGraw-Hill, page 4, equation 1-5 49:It is of interest to the following 6718:"Even symmetric" apparently means 5651: 5648: 5543: 5540: 5329: 5265: 5260: 5189: 5184: 4949: 4944: 4885: 4880: 4160:whereas the EM guys tend to write 3907:Analog & Digital Communication 3804: 3799: 3589: 3584: 3513: 3508: 2360:Again, I hope it is obvious that 2215: 2210: 2098: 2093: 1935:{\displaystyle e^{i2\pi (-\xi )x}} 1823: 1818: 1757: 1752: 1562: 1557: 784: 779: 667: 662: 529: 524: 14: 6946:Top-priority mathematics articles 5891:This article, for a minimum of a 5524:{\displaystyle e^{i\alpha x^{2}}} 5459:{\displaystyle {\hat {f}}(\xi ),} 5418:{\displaystyle {\hat {f}}(\xi ).} 4461:, Fourier series can't represent 4197:{\displaystyle \cos(kx-\omega t)} 4153:{\displaystyle \cos(\omega t-kx)} 3159:. Engineers prefer this because 3151:. Notice in particular that the 2005:{\displaystyle {\hat {f}}(-\xi )} 194:Knowledge:WikiProject Mathematics 6916:Knowledge level-4 vital articles 6073:just over 80% place the i after. 4532:Possible problem may need fixing 4486:taken from the cited reference. 3920:Modern Communnication Principles 3884:Using the engineering convention 3264:negative at positive frequency. 2012:is the appropriate nomenclature. 1730:{\displaystyle {\hat {f}}(\xi )} 1659:{\displaystyle e^{+i2\pi \xi x}} 1257:until Chrome fixes the problem. 1227: 240: 197:Template:WikiProject Mathematics 161: 151: 130: 80: 62: 29: 20: 4791:It has been a decade since the 4437:The section idiotically titled 3980:Stratton, Julius Adams (1941), 3760:When you "arbitrarily" choose 1166:MathML with SVG or PNG fallback 214:This article has been rated as 6926:B-Class level-4 vital articles 6796:. That's where the reference 6755: 6749: 6737: 6728: 6687: 6638: 6618: 6569: 6546: 6497: 6198:{\displaystyle e^{i\omega t}.} 6026: 6020: 5994: 5988: 5716: 5688: 5661: 5655: 5584: 5553: 5547: 5450: 5444: 5438: 5409: 5403: 5397: 5371: 5365: 5355:But also we already know that 5295: 5282: 5276: 5164: 5158: 5047: 4983: 4957: 4758:14:38, 23 September 2022 (UTC) 4734:17:10, 23 September 2022 (UTC) 4715:00:33, 23 September 2022 (UTC) 4644:19:03, 22 September 2022 (UTC) 4625:13:32, 22 September 2022 (UTC) 4606:04:39, 22 September 2022 (UTC) 4191: 4173: 4147: 4129: 4063: 4057: 3818: 3812: 3785: 3779: 3773: 3753:{\displaystyle e^{i\omega t}.} 3670: 3661: 3655: 3603: 3597: 3527: 3521: 3494: 3488: 3482: 3423: 3417: 3299: 3293: 3246: 3237: 3208: 3199: 3184: 3175: 3138: 3126: 3114: 3102: 3056: 3044: 3032: 3020: 3008: 2996: 2987: 2975: 2902: 2890: 2878: 2866: 2788: 2776: 2764: 2752: 2719: 2710: 2677: 2665: 2656: 2644: 2572: 2560: 2551: 2539: 2512: 2503: 2450: 2431: 2402: 2383: 2322: 2300: 2229: 2223: 2196: 2190: 2178: 2112: 2106: 2079: 2073: 2061: 1999: 1990: 1984: 1958: 1952: 1924: 1915: 1868: 1859: 1837: 1831: 1771: 1765: 1724: 1718: 1712: 1682: 1676: 1576: 1570: 1509: 1506: 1491: 1479: 1464: 1455: 1435: 1420: 1405: 1390: 1059: 1053: 1047: 999: 993: 987: 939: 933: 927: 873: 861: 798: 792: 757: 751: 745: 681: 675: 625: 619: 613: 543: 537: 502: 496: 490: 455: 449: 430:non-unitary, angular frequency 383: 377: 371: 1: 6365:14:53, 17 February 2023 (UTC) 6332:20:47, 16 February 2023 (UTC) 6312:13:22, 16 February 2023 (UTC) 6291:02:57, 16 February 2023 (UTC) 6269:01:38, 16 February 2023 (UTC) 6239:00:25, 16 February 2023 (UTC) 6217:23:19, 15 February 2023 (UTC) 6141:21:27, 15 February 2023 (UTC) 6040:{\displaystyle \int f(x)\,dx} 6000:{\displaystyle \int \,dxf(x)} 5959:18:47, 15 February 2023 (UTC) 5940:15:47, 15 February 2023 (UTC) 5768:20:17, 13 February 2023 (UTC) 5754:17:25, 13 February 2023 (UTC) 5632:17:57, 10 February 2023 (UTC) 5613:16:40, 10 February 2023 (UTC) 5482:09:00, 27 December 2022 (UTC) 5120:01:59, 25 December 2022 (UTC) 4843:21:30, 15 February 2023 (UTC) 4808:00:59, 19 December 2022 (UTC) 4792: 4782:16:47, 12 December 2022 (UTC) 4099:12:30, 18 November 2021 (UTC) 4018:17:28, 17 November 2021 (UTC) 3875:05:10, 15 November 2021 (UTC) 3718:04:21, 15 November 2021 (UTC) 3698:04:38, 15 November 2021 (UTC) 3345:{\displaystyle e^{i\omega t}} 3279:22:03, 13 November 2021 (UTC) 3252:{\displaystyle cos(\omega t)} 2725:{\displaystyle sin(\omega t)} 2518:{\displaystyle cos(\omega t)} 2025:16:41, 13 November 2021 (UTC) 188:and see a list of open tasks. 101:and see a list of open tasks. 90:WikiProject Signal Processing 6941:B-Class mathematics articles 6759: 6437:Here is a link to an example 5930:in books/papers/references. 4548:when discussing the Fourier 4310:Negative_frequency#Sinusoids 4286:12:29, 16 October 2021 (UTC) 4264:01:38, 14 October 2021 (UTC) 3927:Engineering Circuit Analysis 1310:16:49, 13 January 2021 (UTC) 1287:16:22, 13 January 2021 (UTC) 1272:16:12, 13 January 2021 (UTC) 1243:13:45, 13 January 2021 (UTC) 1205:13:34, 13 January 2021 (UTC) 1178:13:22, 13 January 2021 (UTC) 1126:13:14, 13 January 2021 (UTC) 1097:12:55, 13 January 2021 (UTC) 326:12:23, 13 January 2021 (UTC) 6394:21:25, 21 August 2023 (UTC) 6370:Error in inversion section? 4554:It's not the same function. 4308:apples - 5 bananas". (See 3966:Jackson, John Davd (1999), 2277:I hope it is obvious that 1345:12:26, 8 October 2021 (UTC) 420:unitary, ordinary frequency 6962: 6486:Error in Symmetry Section? 5858:is used, it is before the 4574:Disappointing introduction 4540:, in order to specify the 4379:11:45, 17 March 2022 (UTC) 4361:02:15, 17 March 2022 (UTC) 4325:12:34, 16 March 2022 (UTC) 4236:11:47, 16 March 2022 (UTC) 4219:05:04, 16 March 2022 (UTC) 3944:Using the other convention 3896:(1st ed.), Prentice-Hall, 425:unitary, angular frequency 113:Signal Processing articles 6894:12:58, 20 June 2024 (UTC) 6865:11:03, 20 June 2024 (UTC) 6851:10:32, 20 June 2024 (UTC) 6813:11:42, 19 June 2024 (UTC) 6784:22:56, 17 June 2024 (UTC) 6713:22:01, 17 June 2024 (UTC) 5872:Mathematica Documentation 4569:20:08, 29 July 2022 (UTC) 4496:19:13, 29 July 2022 (UTC) 4427:22:23, 29 July 2022 (UTC) 4412:21:30, 11 July 2022 (UTC) 4031:. New York: McGraw-Hill. 3998:, page 607, equation VI-2 3977:, page 372, equation 8.89 3936:, page 535, equation 8b. 3375:{\displaystyle \omega t,} 3155:of the imaginary part is 1697:Fourier_series#Definition 1225:OK, happy experimenting! 213: 146: 75: 57: 6478:13:09, 23 May 2024 (UTC) 6452:12:13, 23 May 2024 (UTC) 6430:10:42, 23 May 2024 (UTC) 5137:will equal the function 4454:waveforms as the sum of 3929:(2nd ed.), McGraw-Hill, 3905:Gregg, W. David (1977), 3398:{\displaystyle \omega .} 1448:corresponds to a vector 881:{\displaystyle f(x-a)\,} 220:project's priority scale 6247:. I prefer to see the 4826:{\displaystyle \theta } 4441:contains this passage: 4113:instantaneous frequency 3354:instantaneous frequency 177:WikiProject Mathematics 6911:B-Class vital articles 6768: 6694: 6625: 6553: 6414:fast Fourier transform 6199: 6163: 6125: 6093: 6067: 6041: 6001: 5924: 5923:{\displaystyle 2\pi i} 5852: 5829: 5828:{\displaystyle i2\pi } 5803: 5802:{\displaystyle 2\pi i} 5730: 5674: 5597: 5567: 5525: 5460: 5419: 5384:can be recovered from 5378: 5350: 5342: 5269: 5193: 5099: 4990: 4953: 4889: 4827: 4241:Language for Beginners 4198: 4154: 4070: 3982:Electromagnetic Theory 3857: 3754: 3677: 3636: 3563: 3462: 3399: 3376: 3346: 3306: 3253: 3215: 3145: 3086: 3063: 2955: 2932: 2909: 2841: 2818: 2795: 2726: 2684: 2625: 2602: 2579: 2519: 2475: 2347: 2267: 2150: 2006: 1965: 1936: 1888: 1731: 1689: 1660: 1618: 1536: 1516: 1442: 1068: 1008: 948: 882: 836: 719: 587: 464: 463:{\displaystyle f(x)\,} 392: 6792:The article links to 6769: 6695: 6626: 6554: 6200: 6164: 6126: 6124:{\displaystyle 2\pi } 6094: 6092:{\displaystyle 2\pi } 6068: 6066:{\displaystyle 2\pi } 6042: 6002: 5925: 5853: 5851:{\displaystyle 2\pi } 5830: 5804: 5731: 5675: 5598: 5568: 5526: 5461: 5420: 5379: 5343: 5246: 5170: 5131: 5100: 4991: 4930: 4866: 4828: 4199: 4155: 4071: 3858: 3755: 3678: 3637: 3564: 3463: 3400: 3377: 3347: 3307: 3254: 3216: 3146: 3087: 3064: 2956: 2933: 2910: 2842: 2819: 2796: 2727: 2685: 2626: 2603: 2580: 2520: 2476: 2348: 2268: 2151: 2007: 1966: 1937: 1889: 1732: 1690: 1661: 1619: 1537: 1517: 1443: 1146:Help:Using talk pages 1076:Shift in time domain 1069: 1009: 949: 883: 837: 720: 588: 465: 393: 36:level-4 vital article 6722: 6635: 6566: 6494: 6173: 6153: 6112: 6080: 6054: 6011: 5971: 5908: 5867:MATLAB Documentation 5839: 5813: 5787: 5684: 5644: 5578: 5536: 5495: 5429: 5388: 5377:{\displaystyle f(x)} 5359: 5152: 5009: 4863: 4817: 4456:harmonically-related 4164: 4120: 4069:{\displaystyle f(t)} 4051: 3764: 3728: 3646: 3573: 3473: 3411: 3386: 3382:whose derivative is 3360: 3323: 3305:{\displaystyle f(t)} 3287: 3225: 3163: 3096: 3073: 2969: 2942: 2919: 2854: 2828: 2805: 2740: 2698: 2638: 2612: 2589: 2533: 2491: 2367: 2284: 2162: 2045: 1975: 1964:{\displaystyle f(x)} 1946: 1898: 1741: 1703: 1688:{\displaystyle f(x)} 1670: 1628: 1546: 1535:{\displaystyle \xi } 1526: 1452: 1353: 1019: 959: 893: 855: 730: 598: 475: 443: 337: 200:mathematics articles 5876:Details and Options 4300:concept instead of 3894:Signals and Systems 3808: 3593: 3517: 2466: 2418: 2338: 2219: 2102: 1827: 1761: 1566: 788: 671: 533: 6764: 6690: 6621: 6549: 6357:Community Tech bot 6195: 6159: 6121: 6089: 6063: 6037: 6030: 5997: 5978: 5920: 5848: 5825: 5799: 5726: 5670: 5593: 5562: 5521: 5456: 5415: 5374: 5338: 5205: 5141:. In other words, 5095: 4986: 4823: 4787:ξ vs ν ? revisited 4433:Nonsense statement 4400:excarnateSojourner 4341:negative frequency 4194: 4150: 4066: 3991:, Addison-Wesley, 3853: 3843: 3791: 3750: 3673: 3632: 3625: 3576: 3559: 3552: 3500: 3457: 3395: 3372: 3342: 3302: 3261:principal argument 3249: 3211: 3141: 3085:{\displaystyle +a} 3082: 3059: 2954:{\displaystyle -a} 2951: 2931:{\displaystyle +a} 2928: 2905: 2840:{\displaystyle -a} 2837: 2817:{\displaystyle +a} 2814: 2791: 2722: 2680: 2624:{\displaystyle -a} 2621: 2601:{\displaystyle +a} 2598: 2575: 2515: 2471: 2469: 2443: 2395: 2343: 2341: 2315: 2263: 2261: 2252: 2202: 2146: 2144: 2135: 2085: 2002: 1961: 1932: 1884: 1877: 1810: 1744: 1727: 1685: 1656: 1614: 1607: 1549: 1532: 1512: 1438: 1064: 1063: 1004: 1003: 944: 943: 878: 877: 832: 830: 821: 771: 715: 713: 704: 654: 583: 581: 572: 516: 460: 459: 428:Fourier transform 423:Fourier transform 418:Fourier transform 388: 387: 169:Mathematics portal 45:content assessment 6762: 6667: 6598: 6526: 6396: 6384:comment added by 6376:inversion section 6162:{\displaystyle i} 5441: 5400: 5299: 5279: 5232: 5062: 5058: 5042: 4928: 4396:Fourier transform 4269:Is this better: 3954:Numerical Recipes 3822: 3776: 3658: 3607: 3531: 3485: 2453: 2434: 2405: 2386: 2325: 2303: 2181: 2064: 1987: 1971:. And therefore 1841: 1775: 1715: 1580: 1149: 1112:comment added by 1080: 1079: 1050: 990: 930: 748: 652: 651: 616: 493: 374: 305: 304: 286: 285: 234: 233: 230: 229: 226: 225: 125: 124: 121: 120: 104:Signal Processing 95:signal processing 70:Signal Processing 6953: 6834: 6773: 6771: 6770: 6765: 6763: 6758: 6744: 6699: 6697: 6696: 6691: 6686: 6685: 6684: 6683: 6666: 6659: 6658: 6657: 6656: 6630: 6628: 6627: 6622: 6617: 6616: 6615: 6614: 6597: 6590: 6589: 6588: 6587: 6558: 6556: 6555: 6550: 6545: 6544: 6543: 6542: 6525: 6518: 6517: 6516: 6515: 6463: 6318:Symmetry section 6204: 6202: 6201: 6196: 6191: 6190: 6168: 6166: 6165: 6160: 6130: 6128: 6127: 6122: 6098: 6096: 6095: 6090: 6072: 6070: 6069: 6064: 6046: 6044: 6043: 6038: 6006: 6004: 6003: 5998: 5929: 5927: 5926: 5921: 5857: 5855: 5854: 5849: 5834: 5832: 5831: 5826: 5808: 5806: 5805: 5800: 5735: 5733: 5732: 5727: 5719: 5714: 5713: 5712: 5711: 5691: 5679: 5677: 5676: 5671: 5654: 5602: 5600: 5599: 5594: 5573: 5570: 5569: 5563: 5546: 5530: 5528: 5527: 5522: 5520: 5519: 5518: 5517: 5468:sampling theorem 5465: 5463: 5462: 5457: 5443: 5442: 5434: 5424: 5422: 5421: 5416: 5402: 5401: 5393: 5383: 5381: 5380: 5375: 5347: 5345: 5344: 5339: 5328: 5327: 5323: 5322: 5298: 5294: 5293: 5281: 5280: 5272: 5268: 5263: 5242: 5241: 5237: 5233: 5225: 5203: 5202: 5192: 5187: 5145:can be written: 5144: 5140: 5136: 5104: 5102: 5101: 5096: 5091: 5087: 5086: 5085: 5061: 5060: 5059: 5057: 5056: 5050: 5045: 5041: 5040: 5039: 5035: 5034: 4995: 4993: 4992: 4987: 4979: 4952: 4947: 4929: 4924: 4916: 4911: 4910: 4888: 4883: 4832: 4830: 4829: 4824: 4524: 4522: 4510: 4471:limiting process 4347:. Or maybe the 4256:ScientistBuilder 4248:ScientistBuilder 4203: 4201: 4200: 4195: 4159: 4157: 4156: 4151: 4110: 4075: 4073: 4072: 4067: 4041: 3862: 3860: 3859: 3854: 3841: 3840: 3821: 3807: 3802: 3778: 3777: 3769: 3759: 3757: 3756: 3751: 3746: 3745: 3682: 3680: 3679: 3674: 3660: 3659: 3651: 3641: 3639: 3638: 3633: 3623: 3622: 3606: 3592: 3587: 3568: 3566: 3565: 3560: 3550: 3549: 3530: 3516: 3511: 3487: 3486: 3478: 3468: 3465: 3464: 3458: 3444: 3443: 3404: 3402: 3401: 3396: 3381: 3379: 3378: 3373: 3351: 3349: 3348: 3343: 3341: 3340: 3311: 3309: 3308: 3303: 3283:For real-valued 3258: 3256: 3255: 3250: 3220: 3218: 3217: 3212: 3150: 3148: 3147: 3142: 3091: 3089: 3088: 3083: 3068: 3066: 3065: 3060: 2960: 2958: 2957: 2952: 2937: 2935: 2934: 2929: 2914: 2912: 2911: 2906: 2846: 2844: 2843: 2838: 2823: 2821: 2820: 2815: 2800: 2798: 2797: 2792: 2731: 2729: 2728: 2723: 2689: 2687: 2686: 2681: 2630: 2628: 2627: 2622: 2607: 2605: 2604: 2599: 2584: 2582: 2581: 2576: 2524: 2522: 2521: 2516: 2480: 2478: 2477: 2472: 2470: 2465: 2460: 2455: 2454: 2446: 2442: 2441: 2436: 2435: 2427: 2417: 2412: 2407: 2406: 2398: 2394: 2393: 2388: 2387: 2379: 2352: 2350: 2349: 2344: 2342: 2337: 2332: 2327: 2326: 2318: 2311: 2310: 2305: 2304: 2296: 2272: 2270: 2269: 2264: 2262: 2250: 2249: 2218: 2213: 2189: 2188: 2183: 2182: 2174: 2155: 2153: 2152: 2147: 2145: 2133: 2132: 2101: 2096: 2072: 2071: 2066: 2065: 2057: 2011: 2009: 2008: 2003: 1989: 1988: 1980: 1970: 1968: 1967: 1962: 1941: 1939: 1938: 1933: 1931: 1930: 1893: 1891: 1890: 1885: 1875: 1874: 1840: 1826: 1821: 1800: 1799: 1774: 1760: 1755: 1736: 1734: 1733: 1728: 1717: 1716: 1708: 1694: 1692: 1691: 1686: 1665: 1663: 1662: 1657: 1655: 1654: 1623: 1621: 1620: 1615: 1605: 1604: 1579: 1565: 1560: 1541: 1539: 1538: 1533: 1521: 1519: 1518: 1515:{\displaystyle } 1513: 1447: 1445: 1444: 1439: 1380: 1379: 1232: 1231: 1230: 1139: 1128: 1073: 1071: 1070: 1065: 1052: 1051: 1043: 1040: 1039: 1013: 1011: 1010: 1005: 992: 991: 983: 980: 979: 953: 951: 950: 945: 932: 931: 923: 920: 919: 887: 885: 884: 879: 841: 839: 838: 833: 831: 819: 818: 787: 782: 750: 749: 741: 724: 722: 721: 716: 714: 702: 701: 670: 665: 653: 644: 640: 618: 617: 609: 592: 590: 589: 584: 582: 570: 569: 532: 527: 495: 494: 486: 469: 467: 466: 461: 410: 397: 395: 394: 389: 376: 375: 367: 364: 363: 300: 255: 254: 244: 236: 202: 201: 198: 195: 192: 171: 166: 165: 155: 148: 147: 142: 134: 127: 115: 114: 111: 108: 105: 84: 77: 76: 66: 59: 42: 33: 32: 25: 24: 16: 6961: 6960: 6956: 6955: 6954: 6952: 6951: 6950: 6901: 6900: 6828: 6745: 6720: 6719: 6673: 6668: 6646: 6641: 6633: 6632: 6604: 6599: 6577: 6572: 6564: 6563: 6532: 6527: 6505: 6500: 6492: 6491: 6488: 6457: 6402: 6372: 6353:nomination page 6339: 6320: 6176: 6171: 6170: 6151: 6150: 6110: 6109: 6078: 6077: 6052: 6051: 6009: 6008: 5969: 5968: 5906: 5905: 5837: 5836: 5811: 5810: 5785: 5784: 5780: 5703: 5692: 5682: 5681: 5642: 5641: 5576: 5575: 5533: 5532: 5509: 5498: 5493: 5492: 5489: 5427: 5426: 5386: 5385: 5357: 5356: 5314: 5300: 5285: 5220: 5206: 5194: 5150: 5149: 5142: 5138: 5134: 5127: 5077: 5070: 5066: 5026: 5012: 5007: 5006: 4917: 4890: 4861: 4860: 4853:The reliance on 4851: 4815: 4814: 4789: 4766: 4576: 4534: 4529: 4528: 4527: 4516: 4511: 4507: 4435: 4386: 4298:Analytic signal 4294: 4243: 4162: 4161: 4118: 4117: 4104: 4049: 4048: 4038: 4026: 3823: 3762: 3761: 3731: 3726: 3725: 3644: 3643: 3608: 3571: 3570: 3532: 3471: 3470: 3429: 3408: 3407: 3384: 3383: 3358: 3357: 3326: 3321: 3320: 3285: 3284: 3223: 3222: 3161: 3160: 3094: 3093: 3071: 3070: 2967: 2966: 2940: 2939: 2917: 2916: 2852: 2851: 2826: 2825: 2803: 2802: 2738: 2737: 2696: 2695: 2636: 2635: 2610: 2609: 2587: 2586: 2531: 2530: 2489: 2488: 2468: 2467: 2424: 2422: 2376: 2374: 2365: 2364: 2340: 2339: 2293: 2291: 2282: 2281: 2260: 2259: 2232: 2171: 2169: 2160: 2159: 2143: 2142: 2115: 2054: 2052: 2043: 2042: 1973: 1972: 1944: 1943: 1901: 1896: 1895: 1842: 1776: 1739: 1738: 1701: 1700: 1668: 1667: 1631: 1626: 1625: 1581: 1544: 1543: 1524: 1523: 1450: 1449: 1356: 1351: 1350: 1320: 1228: 1107: 1022: 1017: 1016: 962: 957: 956: 896: 891: 890: 853: 852: 829: 828: 801: 764: 761: 760: 737: 728: 727: 712: 711: 684: 632: 629: 628: 605: 596: 595: 580: 579: 546: 509: 506: 505: 482: 473: 472: 441: 440: 340: 335: 334: 310: 301: 295: 249: 199: 196: 193: 190: 189: 167: 160: 140: 112: 109: 106: 103: 102: 43:on Knowledge's 40: 30: 12: 11: 5: 6959: 6957: 6949: 6948: 6943: 6938: 6933: 6928: 6923: 6918: 6913: 6903: 6902: 6899: 6898: 6897: 6896: 6882: 6872: 6871: 6870: 6869: 6868: 6867: 6839: 6836: 6818: 6817: 6816: 6815: 6801: 6787: 6786: 6761: 6757: 6754: 6751: 6748: 6742: 6739: 6736: 6733: 6730: 6727: 6689: 6682: 6679: 6675: 6671: 6665: 6662: 6655: 6652: 6648: 6644: 6640: 6620: 6613: 6610: 6606: 6602: 6596: 6593: 6586: 6583: 6579: 6575: 6571: 6548: 6541: 6538: 6534: 6530: 6524: 6521: 6514: 6511: 6507: 6503: 6499: 6487: 6484: 6483: 6482: 6481: 6480: 6440: 6401: 6398: 6371: 6368: 6349: 6348: 6338: 6335: 6319: 6316: 6315: 6314: 6300: 6296: 6295: 6294: 6293: 6276: 6275: 6274: 6273: 6272: 6271: 6253: 6220: 6219: 6205: 6194: 6189: 6186: 6183: 6179: 6158: 6146: 6145: 6144: 6143: 6120: 6117: 6108:following the 6102: 6101: 6100: 6088: 6085: 6074: 6062: 6059: 6036: 6033: 6028: 6025: 6022: 6019: 6016: 5996: 5993: 5990: 5987: 5984: 5981: 5976: 5964: 5947: 5919: 5916: 5913: 5901: 5900: 5889: 5884: 5879: 5869: 5847: 5844: 5824: 5821: 5818: 5798: 5795: 5792: 5779: 5776: 5775: 5774: 5773: 5772: 5771: 5770: 5756: 5725: 5722: 5718: 5710: 5706: 5702: 5699: 5695: 5690: 5669: 5666: 5663: 5660: 5657: 5653: 5650: 5635: 5634: 5592: 5589: 5586: 5583: 5561: 5558: 5555: 5552: 5549: 5545: 5542: 5516: 5512: 5508: 5505: 5501: 5488: 5485: 5471: 5455: 5452: 5449: 5446: 5440: 5437: 5414: 5411: 5408: 5405: 5399: 5396: 5373: 5370: 5367: 5364: 5349: 5348: 5337: 5334: 5331: 5326: 5321: 5317: 5313: 5310: 5307: 5303: 5297: 5292: 5288: 5284: 5278: 5275: 5267: 5262: 5259: 5256: 5253: 5249: 5245: 5240: 5236: 5231: 5228: 5223: 5219: 5216: 5213: 5209: 5201: 5197: 5191: 5186: 5183: 5180: 5177: 5173: 5169: 5166: 5163: 5160: 5157: 5126: 5123: 5109: 5106: 5105: 5094: 5090: 5084: 5080: 5076: 5073: 5069: 5065: 5055: 5049: 5038: 5033: 5029: 5025: 5022: 5019: 5015: 4997: 4996: 4985: 4982: 4978: 4974: 4971: 4968: 4965: 4962: 4959: 4956: 4951: 4946: 4943: 4940: 4937: 4933: 4927: 4923: 4920: 4914: 4909: 4906: 4903: 4900: 4897: 4893: 4887: 4882: 4879: 4876: 4873: 4869: 4850: 4847: 4846: 4845: 4833:for an angle. 4822: 4797: 4788: 4785: 4765: 4762: 4761: 4760: 4746: 4741: 4740: 4739: 4738: 4737: 4736: 4700: 4699: 4698: 4697: 4691: 4690: 4689: 4688: 4682: 4681: 4680: 4679: 4673: 4672: 4671: 4670: 4664: 4663: 4662: 4661: 4647: 4646: 4627: 4613: 4575: 4572: 4533: 4530: 4526: 4525: 4504: 4503: 4499: 4450:can represent 4448:Fourier series 4434: 4431: 4430: 4429: 4385: 4382: 4368: 4364: 4363: 4337: 4334: 4331: 4302:Fourier series 4293: 4290: 4289: 4288: 4274: 4242: 4239: 4225: 4222: 4221: 4193: 4190: 4187: 4184: 4181: 4178: 4175: 4172: 4169: 4149: 4146: 4143: 4140: 4137: 4134: 4131: 4128: 4125: 4088: 4085: 4082: 4081: 4065: 4062: 4059: 4056: 4044: 4043: 4042: 4036: 4002: 4001: 4000: 3999: 3985: 3978: 3970:, John-Wiley, 3964: 3947: 3946: 3940: 3939: 3938: 3937: 3923: 3916: 3909:, John Wiley, 3903: 3887: 3886: 3864: 3852: 3849: 3846: 3839: 3836: 3833: 3830: 3826: 3820: 3817: 3814: 3811: 3806: 3801: 3798: 3794: 3790: 3787: 3784: 3781: 3775: 3772: 3749: 3744: 3741: 3738: 3734: 3721: 3720: 3687: 3684: 3672: 3669: 3666: 3663: 3657: 3654: 3631: 3628: 3621: 3618: 3615: 3611: 3605: 3602: 3599: 3596: 3591: 3586: 3583: 3579: 3558: 3555: 3548: 3545: 3542: 3539: 3535: 3529: 3526: 3523: 3520: 3515: 3510: 3507: 3503: 3499: 3496: 3493: 3490: 3484: 3481: 3456: 3453: 3450: 3447: 3442: 3439: 3436: 3432: 3428: 3425: 3422: 3419: 3416: 3405: 3394: 3391: 3371: 3368: 3365: 3339: 3336: 3333: 3329: 3301: 3298: 3295: 3292: 3248: 3245: 3242: 3239: 3236: 3233: 3230: 3210: 3207: 3204: 3201: 3198: 3195: 3192: 3189: 3186: 3183: 3180: 3177: 3174: 3171: 3168: 3140: 3137: 3134: 3131: 3128: 3125: 3122: 3119: 3116: 3113: 3110: 3107: 3104: 3101: 3081: 3078: 3058: 3055: 3052: 3049: 3046: 3043: 3040: 3037: 3034: 3031: 3028: 3025: 3022: 3019: 3016: 3013: 3010: 3007: 3004: 3001: 2998: 2995: 2992: 2989: 2986: 2983: 2980: 2977: 2974: 2963: 2962: 2950: 2947: 2927: 2924: 2904: 2901: 2898: 2895: 2892: 2889: 2886: 2883: 2880: 2877: 2874: 2871: 2868: 2865: 2862: 2859: 2848: 2836: 2833: 2813: 2810: 2790: 2787: 2784: 2781: 2778: 2775: 2772: 2769: 2766: 2763: 2760: 2757: 2754: 2751: 2748: 2745: 2721: 2718: 2715: 2712: 2709: 2706: 2703: 2692: 2691: 2679: 2676: 2673: 2670: 2667: 2664: 2661: 2658: 2655: 2652: 2649: 2646: 2643: 2632: 2620: 2617: 2597: 2594: 2574: 2571: 2568: 2565: 2562: 2559: 2556: 2553: 2550: 2547: 2544: 2541: 2538: 2514: 2511: 2508: 2505: 2502: 2499: 2496: 2482: 2481: 2464: 2459: 2452: 2449: 2440: 2433: 2430: 2423: 2421: 2416: 2411: 2404: 2401: 2392: 2385: 2382: 2375: 2373: 2372: 2355: 2354: 2336: 2331: 2324: 2321: 2314: 2309: 2302: 2299: 2292: 2290: 2289: 2275: 2274: 2258: 2255: 2248: 2245: 2242: 2239: 2235: 2231: 2228: 2225: 2222: 2217: 2212: 2209: 2205: 2201: 2198: 2195: 2192: 2187: 2180: 2177: 2170: 2168: 2167: 2157: 2141: 2138: 2131: 2128: 2125: 2122: 2118: 2114: 2111: 2108: 2105: 2100: 2095: 2092: 2088: 2084: 2081: 2078: 2075: 2070: 2063: 2060: 2053: 2051: 2050: 2028: 2027: 2013: 2001: 1998: 1995: 1992: 1986: 1983: 1960: 1957: 1954: 1951: 1929: 1926: 1923: 1920: 1917: 1914: 1911: 1908: 1904: 1883: 1880: 1873: 1870: 1867: 1864: 1861: 1858: 1855: 1852: 1849: 1845: 1839: 1836: 1833: 1830: 1825: 1820: 1817: 1813: 1809: 1806: 1803: 1798: 1795: 1792: 1789: 1786: 1783: 1779: 1773: 1770: 1767: 1764: 1759: 1754: 1751: 1747: 1726: 1723: 1720: 1714: 1711: 1699:). Therefore 1684: 1681: 1678: 1675: 1653: 1650: 1647: 1644: 1641: 1638: 1634: 1613: 1610: 1603: 1600: 1597: 1594: 1591: 1588: 1584: 1578: 1575: 1572: 1569: 1564: 1559: 1556: 1552: 1531: 1511: 1508: 1505: 1502: 1499: 1496: 1493: 1490: 1487: 1484: 1481: 1478: 1475: 1472: 1469: 1466: 1463: 1460: 1457: 1437: 1434: 1431: 1428: 1425: 1422: 1419: 1416: 1413: 1410: 1407: 1404: 1401: 1398: 1395: 1392: 1389: 1386: 1383: 1378: 1375: 1372: 1369: 1366: 1363: 1359: 1319: 1316: 1315: 1314: 1313: 1312: 1290: 1289: 1254: 1253: 1252: 1251: 1250: 1249: 1248: 1247: 1246: 1245: 1214: 1213: 1212: 1211: 1210: 1209: 1208: 1207: 1185: 1184: 1183: 1182: 1181: 1180: 1150: 1132: 1131: 1130: 1129: 1100: 1099: 1084: 1083: 1082: 1081: 1078: 1077: 1074: 1061: 1058: 1055: 1049: 1046: 1038: 1035: 1032: 1029: 1025: 1014: 1001: 998: 995: 989: 986: 978: 975: 972: 969: 965: 954: 941: 938: 935: 929: 926: 918: 915: 912: 909: 906: 903: 899: 888: 875: 872: 869: 866: 863: 860: 850: 846: 845: 842: 827: 824: 817: 814: 811: 808: 804: 800: 797: 794: 791: 786: 781: 778: 774: 770: 767: 765: 763: 762: 759: 756: 753: 747: 744: 738: 736: 735: 725: 710: 707: 700: 697: 694: 691: 687: 683: 680: 677: 674: 669: 664: 661: 657: 650: 647: 643: 638: 635: 633: 631: 630: 627: 624: 621: 615: 612: 606: 604: 603: 593: 578: 575: 568: 565: 562: 559: 556: 553: 549: 545: 542: 539: 536: 531: 526: 523: 519: 515: 512: 510: 508: 507: 504: 501: 498: 492: 489: 483: 481: 480: 470: 457: 454: 451: 448: 438: 435: 434: 431: 429: 426: 424: 421: 419: 416: 413: 404: 403: 400: 399: 398: 385: 382: 379: 373: 370: 362: 359: 356: 353: 350: 347: 343: 309: 306: 303: 302: 297: 293: 291: 288: 287: 284: 283: 278: 272: 271: 266: 261: 251: 250: 245: 239: 232: 231: 228: 227: 224: 223: 212: 206: 205: 203: 186:the discussion 173: 172: 156: 144: 143: 135: 123: 122: 119: 118: 116: 99:the discussion 85: 73: 72: 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 6958: 6947: 6944: 6942: 6939: 6937: 6934: 6932: 6929: 6927: 6924: 6922: 6919: 6917: 6914: 6912: 6909: 6908: 6906: 6895: 6891: 6887: 6883: 6880: 6876: 6875: 6874: 6873: 6866: 6862: 6858: 6854: 6853: 6852: 6848: 6844: 6840: 6837: 6832: 6826: 6822: 6821: 6820: 6819: 6814: 6810: 6806: 6802: 6799: 6795: 6791: 6790: 6789: 6788: 6785: 6781: 6777: 6752: 6746: 6740: 6734: 6731: 6725: 6717: 6716: 6715: 6714: 6710: 6706: 6701: 6680: 6677: 6674: 6669: 6663: 6660: 6653: 6650: 6647: 6642: 6611: 6608: 6605: 6600: 6594: 6591: 6584: 6581: 6578: 6573: 6560: 6539: 6536: 6533: 6528: 6522: 6519: 6512: 6509: 6506: 6501: 6485: 6479: 6475: 6471: 6470: 6469: 6461: 6456:I agree with 6455: 6454: 6453: 6449: 6445: 6441: 6438: 6434: 6433: 6432: 6431: 6427: 6423: 6419: 6415: 6411: 6407: 6399: 6397: 6395: 6391: 6387: 6383: 6377: 6369: 6367: 6366: 6362: 6358: 6354: 6347: 6344: 6343: 6342: 6336: 6334: 6333: 6329: 6325: 6317: 6313: 6309: 6305: 6301: 6298: 6297: 6292: 6288: 6284: 6280: 6279: 6278: 6277: 6270: 6266: 6262: 6261: 6260: 6254: 6250: 6246: 6243:I agree with 6242: 6241: 6240: 6236: 6232: 6228: 6224: 6223: 6222: 6221: 6218: 6214: 6210: 6206: 6192: 6187: 6184: 6181: 6177: 6156: 6148: 6147: 6142: 6138: 6134: 6118: 6115: 6107: 6103: 6086: 6083: 6075: 6060: 6057: 6049: 6048: 6034: 6031: 6023: 6017: 6014: 5991: 5985: 5982: 5979: 5974: 5965: 5962: 5961: 5960: 5956: 5952: 5948: 5944: 5943: 5942: 5941: 5937: 5933: 5917: 5914: 5911: 5898: 5894: 5890: 5888: 5885: 5883: 5880: 5877: 5873: 5870: 5868: 5865: 5864: 5863: 5861: 5845: 5842: 5822: 5819: 5816: 5796: 5793: 5790: 5778:Recent revert 5777: 5769: 5765: 5761: 5757: 5755: 5751: 5747: 5746: 5745: 5739: 5738: 5723: 5720: 5708: 5704: 5700: 5697: 5693: 5667: 5664: 5658: 5639: 5638: 5637: 5636: 5633: 5629: 5625: 5624: 5623: 5617: 5616: 5615: 5614: 5610: 5606: 5590: 5587: 5581: 5559: 5556: 5550: 5514: 5510: 5506: 5503: 5499: 5491:The function 5486: 5484: 5483: 5479: 5475: 5469: 5453: 5447: 5435: 5412: 5406: 5394: 5368: 5362: 5353: 5335: 5332: 5324: 5319: 5315: 5311: 5308: 5305: 5301: 5290: 5286: 5273: 5257: 5254: 5251: 5247: 5243: 5238: 5234: 5229: 5226: 5221: 5217: 5214: 5211: 5207: 5199: 5195: 5181: 5178: 5175: 5171: 5167: 5161: 5155: 5148: 5147: 5146: 5130: 5124: 5122: 5121: 5117: 5113: 5092: 5088: 5082: 5078: 5074: 5071: 5067: 5063: 5036: 5031: 5027: 5023: 5020: 5017: 5013: 5005: 5004: 5003: 5002: 4980: 4976: 4972: 4969: 4966: 4963: 4960: 4954: 4941: 4938: 4935: 4931: 4925: 4921: 4918: 4912: 4907: 4904: 4901: 4898: 4895: 4891: 4877: 4874: 4871: 4867: 4859: 4858: 4857: 4856: 4848: 4844: 4840: 4836: 4820: 4812: 4811: 4810: 4809: 4805: 4801: 4794: 4786: 4784: 4783: 4779: 4775: 4771: 4763: 4759: 4755: 4751: 4747: 4743: 4742: 4735: 4731: 4727: 4726: 4725: 4718: 4717: 4716: 4712: 4708: 4704: 4703: 4702: 4701: 4695: 4694: 4693: 4692: 4686: 4685: 4684: 4683: 4677: 4676: 4675: 4674: 4668: 4667: 4666: 4665: 4659: 4655: 4651: 4650: 4649: 4648: 4645: 4641: 4637: 4636: 4635: 4628: 4626: 4622: 4618: 4614: 4610: 4609: 4608: 4607: 4603: 4599: 4594: 4592: 4588: 4583: 4581: 4573: 4571: 4570: 4566: 4562: 4557: 4555: 4551: 4547: 4546:same function 4543: 4539: 4531: 4520: 4514: 4509: 4506: 4502: 4498: 4497: 4493: 4489: 4485: 4480: 4478: 4476: 4472: 4466: 4462: 4460: 4457: 4453: 4449: 4442: 4440: 4432: 4428: 4424: 4420: 4416: 4415: 4414: 4413: 4409: 4405: 4401: 4397: 4393: 4392: 4384:Pronunciation 4383: 4381: 4380: 4376: 4372: 4362: 4358: 4354: 4350: 4346: 4342: 4338: 4335: 4332: 4329: 4328: 4327: 4326: 4322: 4318: 4313: 4311: 4305: 4303: 4299: 4291: 4287: 4283: 4279: 4275: 4272: 4268: 4267: 4266: 4265: 4261: 4257: 4253: 4249: 4240: 4238: 4237: 4233: 4229: 4220: 4216: 4212: 4211: 4210: 4188: 4185: 4182: 4179: 4176: 4170: 4167: 4144: 4141: 4138: 4135: 4132: 4126: 4123: 4114: 4108: 4103: 4102: 4101: 4100: 4096: 4092: 4079: 4078: 4077: 4060: 4054: 4039: 4034: 4030: 4025: 4024: 4023: 4020: 4019: 4015: 4011: 4010: 4009: 3997: 3994: 3990: 3986: 3983: 3979: 3976: 3973: 3969: 3965: 3962: 3961:9780521880688 3959: 3955: 3951: 3950: 3949: 3948: 3945: 3942: 3941: 3935: 3932: 3928: 3924: 3921: 3917: 3915: 3912: 3908: 3904: 3902: 3899: 3895: 3891: 3890: 3889: 3888: 3885: 3882: 3881: 3880: 3877: 3876: 3872: 3868: 3850: 3847: 3844: 3837: 3834: 3831: 3828: 3824: 3815: 3809: 3796: 3792: 3788: 3782: 3770: 3747: 3742: 3739: 3736: 3732: 3719: 3715: 3711: 3710: 3709: 3702: 3701: 3700: 3699: 3695: 3691: 3667: 3664: 3652: 3629: 3626: 3619: 3616: 3613: 3609: 3600: 3594: 3581: 3577: 3569:(which means 3556: 3553: 3546: 3543: 3540: 3537: 3533: 3524: 3518: 3505: 3501: 3497: 3491: 3479: 3454: 3451: 3448: 3445: 3440: 3437: 3434: 3430: 3426: 3420: 3414: 3392: 3389: 3369: 3366: 3363: 3355: 3337: 3334: 3331: 3327: 3317: 3315: 3296: 3290: 3281: 3280: 3276: 3272: 3271: 3270: 3262: 3243: 3240: 3234: 3231: 3228: 3205: 3202: 3196: 3193: 3190: 3187: 3181: 3178: 3172: 3169: 3166: 3158: 3154: 3135: 3132: 3129: 3123: 3120: 3117: 3111: 3108: 3105: 3099: 3079: 3076: 3053: 3050: 3047: 3041: 3038: 3035: 3029: 3026: 3023: 3017: 3014: 3011: 3005: 3002: 2999: 2993: 2990: 2984: 2981: 2978: 2972: 2948: 2945: 2925: 2922: 2899: 2896: 2893: 2887: 2884: 2881: 2875: 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1721: 1709: 1698: 1679: 1673: 1651: 1648: 1645: 1642: 1639: 1636: 1632: 1611: 1608: 1601: 1598: 1595: 1592: 1589: 1586: 1582: 1573: 1567: 1554: 1550: 1529: 1503: 1500: 1497: 1494: 1488: 1485: 1482: 1476: 1473: 1470: 1467: 1461: 1458: 1432: 1429: 1426: 1423: 1417: 1414: 1411: 1408: 1402: 1399: 1396: 1393: 1387: 1384: 1381: 1376: 1373: 1370: 1367: 1364: 1361: 1357: 1349: 1348: 1347: 1346: 1342: 1338: 1337: 1336: 1327: 1324: 1317: 1311: 1307: 1303: 1302: 1301: 1294: 1293: 1292: 1291: 1288: 1284: 1280: 1276: 1275: 1274: 1273: 1269: 1265: 1264: 1263: 1244: 1240: 1236: 1224: 1223: 1222: 1221: 1220: 1219: 1218: 1217: 1216: 1215: 1206: 1202: 1198: 1193: 1192: 1191: 1190: 1189: 1188: 1187: 1186: 1179: 1175: 1171: 1167: 1163: 1159: 1155: 1151: 1147: 1143: 1138: 1137: 1136: 1135: 1134: 1133: 1127: 1123: 1119: 1115: 1111: 1104: 1103: 1102: 1101: 1098: 1094: 1090: 1086: 1085: 1075: 1056: 1044: 1036: 1033: 1030: 1027: 1023: 1015: 996: 984: 976: 973: 970: 967: 963: 955: 936: 924: 916: 913: 910: 907: 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reverted 6400:Code sample 6259:Constant314 5895:until last 5744:Constant314 5622:Constant314 4513:Taneja 2008 3963:, page 692. 1323:transform. 1164:and select 1154:Preferences 844:Definition 191:Mathematics 182:mathematics 138:Mathematics 6905:Categories 6705:Jackmjackm 5572:0}" /: --> 4501:References 4037:0070503400 3996:0201526247 3975:047130932X 3934:0070273820 3914:0471326615 3901:0138097313 3467:0}" /: --> 3406:Therefore 1158:Appearance 6406:this edit 6324:Thenub314 6283:Thenub314 6231:Thenub314 6133:Thenub314 5932:Thenub314 5760:Thenub314 4835:Thenub314 4612:textbook. 4550:transform 4459:sinusoids 4446:Although 4345:Sine wave 1197:OmerLauer 1148:. 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- 1162:Math 1142:sign 1118:talk 1093:talk 1089:DVdm 849:102 322:talk 6800:be. 6355:. — 6007:vs 5897:May 5809:vs 4724:314 4634:314 4484:not 4343:or 4209:314 4168:cos 4124:cos 4008:314 3708:314 3269:314 3092:is 1486:sin 1459:cos 1415:sin 1385:cos 1335:314 1300:314 1262:314 210:Top 6907:: 6892:) 6884:-- 6863:) 6849:) 6841:-- 6811:) 6803:-- 6782:) 6760:¯ 6732:− 6711:) 6700:? 6476:) 6450:) 6442:-- 6428:) 6392:) 6363:) 6330:) 6310:) 6302:-- 6289:) 6267:) 6237:) 6215:) 6207:-- 6185:ω 6139:) 6131:. 6119:π 6087:π 6061:π 6015:∫ 5975:∫ 5957:) 5949:-- 5938:) 5915:π 5846:π 5823:π 5794:π 5766:) 5752:) 5701:α 5659:α 5630:) 5611:) 5603:. 5591:α 5585:→ 5582:α 5566:0} 5551:α 5507:α 5480:) 5472:-- 5448:ξ 5439:^ 5407:ξ 5398:^ 5333:ξ 5330:Δ 5316:ξ 5309:π 5287:ξ 5277:^ 5266:∞ 5261:∞ 5258:− 5248:∑ 5215:π 5190:∞ 5185:∞ 5182:− 5172:∑ 5118:) 5110:-- 5079:ξ 5075:− 5072:ξ 5064:δ 5048:⟷ 5028:ξ 5024:π 4970:π 4964:− 4961:ω 4955:δ 4950:∞ 4945:∞ 4942:− 4932:∑ 4922:π 4902:ω 4896:− 4886:∞ 4881:∞ 4878:− 4868:∑ 4841:) 4821:θ 4806:) 4798:-- 4780:) 4756:) 4748:-- 4732:) 4713:) 4642:) 4623:) 4615:-- 4604:) 4593:. 4589:= 4567:) 4494:) 4479:" 4425:) 4410:) 4377:) 4369:-- 4359:) 4323:) 4315:-- 4312:) 4284:) 4276:-- 4262:) 4234:) 4226:-- 4217:) 4186:ω 4183:− 4171:⁡ 4139:− 4133:ω 4127:⁡ 4097:) 4089:-- 4016:) 3873:) 3865:-- 3835:ω 3805:∞ 3800:∞ 3797:− 3793:∫ 3783:ω 3774:^ 3740:ω 3716:) 3696:) 3688:-- 3683:). 3668:ω 3665:− 3656:^ 3617:ω 3590:∞ 3585:∞ 3582:− 3578:∫ 3544:ω 3538:− 3514:∞ 3509:∞ 3506:− 3502:∫ 3492:ω 3483:^ 3461:0} 3449:ω 3438:ω 3390:ω 3364:ω 3335:ω 3277:) 3241:ω 3203:ω 3179:ω 3133:− 3130:ω 3124:δ 3118:− 3109:− 3106:ω 3100:δ 3048:ω 3042:δ 3027:− 3024:ω 3018:δ 3012:− 3000:ω 2994:δ 2982:− 2979:ω 2973:δ 2946:− 2897:− 2894:ω 2888:δ 2870:ω 2864:δ 2858:− 2832:− 2780:ω 2774:δ 2759:− 2756:ω 2750:δ 2744:− 2732:. 2714:ω 2672:− 2669:ω 2663:δ 2648:ω 2642:δ 2616:− 2564:ω 2558:δ 2546:− 2543:ω 2537:δ 2525:. 2507:ω 2463:∗ 2451:^ 2432:^ 2415:∗ 2410:− 2403:^ 2391:− 2384:^ 2335:∗ 2323:^ 2308:− 2301:^ 2244:ω 2216:∞ 2211:∞ 2208:− 2204:∫ 2194:ω 2179:^ 2127:ω 2121:− 2099:∞ 2094:∞ 2091:− 2087:∫ 2077:ω 2069:− 2062:^ 2023:) 2015:-- 1997:ξ 1994:− 1985:^ 1922:ξ 1919:− 1913:π 1866:ξ 1863:− 1857:π 1848:− 1824:∞ 1819:∞ 1816:− 1812:∫ 1794:ξ 1791:π 1758:∞ 1753:∞ 1750:− 1746:∫ 1722:ξ 1713:^ 1649:ξ 1646:π 1599:ξ 1596:π 1587:− 1563:∞ 1558:∞ 1555:− 1551:∫ 1530:ξ 1501:ξ 1498:π 1489:⁡ 1474:ξ 1471:π 1462:⁡ 1430:ξ 1427:π 1418:⁡ 1400:ξ 1397:π 1388:⁡ 1374:ξ 1371:π 1343:) 1308:) 1285:) 1270:) 1241:) 1233:- 1203:) 1176:) 1160:, 1156:, 1124:) 1120:• 1095:) 1057:ν 1048:^ 1037:ν 1028:− 997:ω 988:^ 977:ω 968:− 937:ξ 928:^ 917:ξ 908:π 902:− 868:− 813:ν 807:− 785:∞ 780:∞ 777:− 773:∫ 755:ν 746:^ 696:ω 690:− 668:∞ 663:∞ 660:− 656:∫ 649:π 623:ω 614:^ 567:ξ 558:π 552:− 530:∞ 525:∞ 522:− 518:∫ 500:ξ 491:^ 381:ξ 372:^ 361:ξ 352:π 346:− 324:) 316:-- 6888:( 6859:( 6845:( 6835:) 6833:) 6807:( 6778:( 6756:) 6753:x 6750:( 6747:f 6741:= 6738:) 6735:x 6729:( 6726:f 6707:( 6688:) 6681:E 6678:I 6670:f 6664:i 6661:+ 6654:E 6651:R 6643:f 6639:( 6619:) 6612:O 6609:I 6601:f 6595:i 6592:+ 6585:E 6582:R 6574:f 6570:( 6547:) 6540:O 6537:I 6529:f 6523:i 6520:+ 6513:E 6510:R 6502:f 6498:( 6472:( 6462:: 6458:@ 6446:( 6424:( 6388:( 6359:( 6326:( 6306:( 6285:( 6263:( 6249:i 6233:( 6227:i 6211:( 6193:. 6188:t 6182:i 6178:e 6157:i 6135:( 6116:2 6106:i 6084:2 6058:2 6035:x 6032:d 6027:) 6024:x 6021:( 6018:f 5995:) 5992:x 5989:( 5986:f 5983:x 5980:d 5953:( 5934:( 5918:i 5912:2 5878:) 5860:i 5843:2 5820:2 5817:i 5797:i 5791:2 5762:( 5748:( 5724:1 5721:= 5717:| 5709:2 5705:x 5698:i 5694:e 5689:| 5668:0 5665:= 5662:) 5656:( 5652:m 5649:I 5626:( 5607:( 5588:i 5560:0 5554:) 5548:( 5544:m 5541:I 5515:2 5511:x 5504:i 5500:e 5476:( 5454:, 5451:) 5445:( 5436:f 5413:. 5410:) 5404:( 5395:f 5372:) 5369:x 5366:( 5363:f 5336:, 5325:x 5320:n 5312:i 5306:2 5302:e 5296:) 5291:n 5283:( 5274:f 5255:= 5252:n 5244:= 5239:x 5235:) 5230:T 5227:n 5222:( 5218:i 5212:2 5208:e 5200:n 5196:c 5179:= 5176:n 5168:= 5165:) 5162:x 5159:( 5156:f 5143:f 5139:f 5135:f 5114:( 5093:. 5089:) 5083:0 5068:( 5054:F 5037:x 5032:0 5021:2 5018:i 5014:e 5001:: 4984:) 4981:T 4977:/ 4973:k 4967:2 4958:( 4939:= 4936:k 4926:T 4919:2 4913:= 4908:T 4905:n 4899:i 4892:e 4875:= 4872:n 4855:: 4837:( 4802:( 4776:( 4752:( 4728:( 4709:( 4638:( 4619:( 4600:( 4591:I 4587:F 4563:( 4523:. 4521:) 4490:( 4444:" 4421:( 4406:| 4402:( 4373:( 4355:( 4319:( 4280:( 4273:? 4258:( 4250:( 4230:( 4213:( 4192:) 4189:t 4180:x 4177:k 4174:( 4148:) 4145:x 4142:k 4136:t 4130:( 4109:: 4105:@ 4093:( 4064:) 4061:t 4058:( 4055:f 4040:. 4012:( 3869:( 3851:, 3848:t 3845:d 3838:t 3832:i 3829:+ 3825:e 3819:) 3816:t 3813:( 3810:f 3789:= 3786:) 3780:( 3771:f 3748:. 3743:t 3737:i 3733:e 3712:( 3692:( 3671:) 3662:( 3653:f 3630:t 3627:d 3620:t 3614:i 3610:e 3604:) 3601:t 3598:( 3595:f 3557:t 3554:d 3547:t 3541:i 3534:e 3528:) 3525:t 3522:( 3519:f 3498:= 3495:) 3489:( 3480:f 3455:0 3446:, 3441:t 3435:i 3431:e 3427:= 3424:) 3421:t 3418:( 3415:f 3393:. 3370:, 3367:t 3338:t 3332:i 3328:e 3300:) 3297:t 3294:( 3291:f 3273:( 3247:) 3244:t 3238:( 3235:s 3232:o 3229:c 3209:) 3206:t 3200:( 3197:n 3194:i 3191:s 3188:+ 3185:) 3182:t 3176:( 3173:s 3170:o 3167:c 3139:) 3136:a 3127:( 3121:i 3115:) 3112:a 3103:( 3080:a 3077:+ 3057:) 3054:a 3051:+ 3045:( 3039:i 3036:+ 3033:) 3030:a 3021:( 3015:i 3009:) 3006:a 3003:+ 2997:( 2991:+ 2988:) 2985:a 2976:( 2949:a 2926:a 2923:+ 2903:) 2900:a 2891:( 2885:i 2882:+ 2879:) 2876:a 2873:+ 2867:( 2861:i 2847:. 2835:a 2812:a 2809:+ 2789:) 2786:a 2783:+ 2777:( 2771:i 2768:+ 2765:) 2762:a 2753:( 2747:i 2720:) 2717:t 2711:( 2708:n 2705:i 2702:s 2678:) 2675:a 2666:( 2660:+ 2657:) 2654:a 2651:+ 2645:( 2631:. 2619:a 2596:a 2593:+ 2573:) 2570:a 2567:+ 2561:( 2555:+ 2552:) 2549:a 2540:( 2513:) 2510:t 2504:( 2501:s 2498:o 2495:c 2458:+ 2448:f 2439:+ 2429:f 2420:= 2400:f 2381:f 2330:+ 2320:f 2313:= 2298:f 2257:t 2254:d 2247:x 2241:i 2238:+ 2234:e 2230:) 2227:t 2224:( 2221:f 2200:= 2197:) 2191:( 2186:+ 2176:f 2140:t 2137:d 2130:x 2124:i 2117:e 2113:) 2110:t 2107:( 2104:f 2083:= 2080:) 2074:( 2059:f 2019:( 2000:) 1991:( 1982:f 1959:) 1956:x 1953:( 1950:f 1928:x 1925:) 1916:( 1910:2 1907:i 1903:e 1882:x 1879:d 1872:x 1869:) 1860:( 1854:2 1851:i 1844:e 1838:) 1835:x 1832:( 1829:f 1808:= 1805:x 1802:d 1797:x 1788:2 1785:i 1782:+ 1778:e 1772:) 1769:x 1766:( 1763:f 1725:) 1719:( 1710:f 1683:) 1680:x 1677:( 1674:f 1652:x 1643:2 1640:i 1637:+ 1633:e 1612:x 1609:d 1602:x 1593:2 1590:i 1583:e 1577:) 1574:x 1571:( 1568:f 1510:] 1507:) 1504:x 1495:2 1492:( 1483:, 1480:) 1477:x 1468:2 1465:( 1456:[ 1436:) 1433:x 1424:2 1421:( 1412:i 1409:+ 1406:) 1403:x 1394:2 1391:( 1382:= 1377:x 1368:2 1365:i 1362:+ 1358:e 1339:( 1304:( 1281:( 1266:( 1237:( 1199:( 1172:( 1116:( 1091:( 1060:) 1054:( 1045:f 1034:a 1031:i 1024:e 1000:) 994:( 985:f 974:a 971:i 964:e 940:) 934:( 925:f 914:a 911:i 905:2 898:e 874:) 871:a 865:x 862:( 859:f 826:x 823:d 816:x 810:i 803:e 799:) 796:x 793:( 790:f 769:= 758:) 752:( 743:f 709:x 706:d 699:x 693:i 686:e 682:) 679:x 676:( 673:f 646:2 642:1 637:= 626:) 620:( 611:f 577:x 574:d 564:x 561:i 555:2 548:e 544:) 541:x 538:( 535:f 514:= 503:) 497:( 488:f 456:) 453:x 450:( 447:f 384:) 378:( 369:f 358:a 355:i 349:2 342:e 320:( 222:. 53::

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