Talk:Fresnel integral

message: Current text (definitions) says:
Some authors, including Abramowitz and Stegun, (eqs 7.3.1 – 7.3.2) use \frac{\pi}{2}t^2 for the exponent of the integrals defining S(x) and C(x)
The usage of the word "exponent" probably comes from the complex representation
:C(x)=\Re \int_0^x \exp(\mathrm{i}t^2)\,dt
(actually one would have to say \mathrm{i}\frac{\pi}{2}t^2 above in this case...). However, it is confusing when you
are using trigonometric functions.
Possible rephrasing:
Some authors, including Abramowitz and Stegun, (eqs 7.3.1 – 7.3.2) use \frac{\pi}{2}t^2 for the argument of the trig functions in the integrals defining S(x) and C(x)  —Preceding unsigned comment added by AmitAronovitch (talk • contribs) 09:07, 2 July 2008 (UTC)
thread_title:  Exponent? 

message: At the Cornu spiral: a vehicle with constant speed will not have a constant rotational acceleration, but a constant _change_ of r.a., won't it?
:: In this image of the , the radius of curvature is zero at the origin, and tends to a constant towards the two attractors ("holes"). So a vehicle travelling along this yellow-brick road would experience zero acceleration as it passes the origin, then increasing acceleration as it spirals in towards either attractor, then in the limit constant acceleration, that is, it would experience a circular orbit (within diminishing error). That isn't exactly what you said, but does it answer the question ok? Pete St.John (talk) 16:03, 10 March 2008 (UTC)
replies: text:  short answer, someone corrected that in the text already; now it says "constant change" :-) So good catch Pete St.John (talk) 16:10, 10 March 2008 (UTC)
thread_title:  Question regarding acceleration 

message: The article says that clothoid is another name for a Cornu spiral.  But the clothoid loop image is quite different to that of the spiral, and I think it must confuse readers of the article.  Also, why does the caption say "f(r)=r-\pi"?  Is the image actually a clothoid loop or is it an Archimedean spiral? JonH (talk) 12:36, 10 March 2008 (UTC)
replies: text: One image is the graph in polar coordinates, the other in cartesian coordinates. Pete St.John (talk) 16:12, 10 March 2008 (UTC)

text: The diagram caption at that section labels f as a function of r, shows a background of polar coordinates, and calls it a parameterization, so it would seem that representing the function from Cornu Spiral in polar coordinates is supposed to give the shape "Clothoid". From Archimedean spiral it would seem the Clothoid is a special case of Archimedean (the constant equals pi). I admit the section is not very clear, we might check a reference. Pete St.John (talk) 19:29, 10 March 2008 (UTC)
replies: text:  I think I've been blithe, I have to think about this some. What happens if we express r and theta in the archimedean spiral parameterized by arclength? Pete St.John (talk) 20:39, 10 March 2008 (UTC)

text: "Cornu spiral" and "clothoid" are synonyms, and refer to the curve with an S-like shape with two spiraling ends. Apparently there is such a thing as a "clothoid loop", but the article does not give a definition, nor does it explain how it is related to the clothoid curve. From some web pages I get the impression that the loop consists of two segments of the clothoid glued together (e.g. here), but I don't know of a reliable source for this definition. The "Parametric graph" sure looks like an Archimedean spiral if you let the angle run across 0. I don't get the meaning of the equation f(r)=r-\pi, which looks very wrong. Possibly r = \theta-\pi is meant.  --Lambiam 17:35, 11 March 2008 (UTC)
replies: text:  In the Archemedean spiral case where the constant is -pi, the equation would be theta = r - pi (so f(r) is theta). I think that's what was meant. I haven't had time yet to answer my own question, expressing r and theta as functions of arclength, I want to. And sorry for the primitivist typography, I never typed much math myself, it would have been nroff in my day :-( Pete St.John (talk) 02:58, 12 March 2008 (UTC)
replies: text: You mean Eqn? What will you do with your expressions of r and theta as functions of arclength, once you've got them? I think arclength can easily be expressed as a function of r or theta, but you can't invert the transcendental equation into a closed form for r and theta as functions of arclength. It is easy, though, to study the asymptotics for small or large arclengths.  --Lambiam 08:31, 12 March 2008 (UTC)
replies: text:  I meant from the Archimedean spiral /vartheta = r - /pi, write r and theta as functions of arclength. I'm wondering how to interpret C and S for this "parameterized" graph. But I'm rusty (at best) and would need some uninterupted time with pencil and paper. (And yeah, Eqn would have been available by then but I wasn't aware of it. I paid a professional who had the greek and math balls for her Selectrix :-) Pete St.John (talk) 12:46, 12 March 2008 (UTC)

text: Update: I've now found a cite for the use of the term "clothoid loop" as used in rollercoaster design: I've put it in the vertical loop article and made the term redirect there, and adjusted this article accordingly. The image stays out: I'm not sure what it illustrates, but since it seems neither to be a clothoid curve or a rollercoaster "clothoid loop", and does not have any idenfiable mathematical significance, it does not seem relevant to either article. -- The Anome (talk) 12:02, 13 March 2008 (UTC)
thread_title:  Cornu spiral images 

message: In the picture the parameterization of gamma 3 should be z=t*exp(i*pi/4)  —Preceding unsigned comment added by 74.69.117.252 (talk) 17:44, 6 August 2008 (UTC)
thread_title:  Evaluation 

message: See for example the French Clothoïde versus
French Intégrale_de_Fresnel
The German page Klothoide would be particularly
worth mining for its civil engineering applications of clothoidal spirals, which are useful
but would be out of place in an article on Fresnel integrals and functions.
Note also that a number of the language links to this page are inappropriately clothoidal
(eg ttp://fr.wikipedia.org/wiki/Clothoïde  )
rather than Fresnel-Integral (eg http://fr.wikipedia.org/wiki/Intégrale_de_Fresnel ); this ought to be cleaned up once an English-language
Clothoid page is created.  —Preceding unsigned comment added by 76.191.212.109 (talk) 02:36, 20 September 2008 (UTC)
thread_title:  Clothoid/Cornu Spiral section should be a separate page 

message: I glanced through this article, and when I came to the graph, I immediately thought "black hole", however, I would like this to be run by a physicist or a mathematition before it is added. --Wyatt915✍ 15:10, 13 October 2008 (UTC)
thread_title:  Notice a similarity? 

message: There seems to be an inconsistency in the variable that is used. For example
S(x) = integral(sin(t^2)dt.
The variable of interest should either be t or x. I think that x would be best as it is more general (t implies 'time' to a certain extent).  —Preceding unsigned comment added by Jezwells (talk • contribs) 10:48, 15 March 2010 (UTC)
replies: text: No, it is right: S(x) = integral from 0 to x sin(t^2)dt, so that's how S is a function of x.  t is commonly used for the parameter in a parametric plot (probably because it often represents time, admittedly). ChrisHodgesUK (talk) 15:18, 15 March 2010 (UTC)
thread_title:  t or x? 

message: The convention with a π/2 factor in the integrand argument seems to be far more common.  e.g. A&S, Mathematica, Mathworld, and Maple all use it.   Arguably the Wikipedia article should use this convention as well.
— Steven G. Johnson (talk) 20:29, 24 December 2012 (UTC)
replies: text: Small correction in the code. janboc (talk) 10:51, 04 September 2018 (UTC)
thread_title:  switch to &pi;/2 convention 

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thread_title:  External links modified 

message: Corresponding to the formula,
S(z) + i C(z) = \sqrt{\frac{\pi}{2}}\frac{1+i}{2} \operatorname{erf}\left(\frac{1+i}{\sqrt{2}}z\right)
I believe it is also true that,
C(z) + i S(z) = \sqrt{\frac{\pi}{2}}\frac{i+1}{2} \operatorname{erf}\left(\frac{1-i}{\sqrt{2}}z\right)
This is probably of more interest than the original formula as it parallels Euler's formula.  Perhaps it should be included?
Skinnerd (talk) 21:42, 30 April 2016 (UTC)
replies: text: I've added it to the article.  Ozob (talk) 01:42, 1 May 2016 (UTC)
thread_title:  C(z) + i S(z) 

message: I think that there should be a source for the claim that the integrals "cannot be evaluated in the closed form in terms of elementary functions" found early in the "Limits as x approaches infinity" section. I spent a short while Googling and wasn't able to find a good one, though. The best I could come up with is the following answer on Math Stack Exchange: https://math.stackexchange.com/a/265884. I'm not sure if such a source would be appropriate for this site, and the answer only deals with S(x) besides, so I figured it would be best to post here. WallAdhesion (talk) 23:54, 16 April 2019 (UTC)
thread_title:  No elementary antiderivative 

message: If we had a shorter name for \frac{1+i}{\sqrt2} = \exp \tfrac14 \pi i = \sqrt[4]{-1}, we could make a bunch of the formulas here much more legible. I proposed \sqrt{i} to indicate the principle square root of i, because this is compact while still being clear and not needing too much separate definition, but User:Jasper Deng didn’t like it. Maybe we can just give it some arbitrary symbol, maybe a Greek letter along the lines of: let \eta = \exp \tfrac14 \pi i,\ \bar\eta = \exp\bigl({-\tfrac14 \pi i}\bigr)? –jacobolus (t) 01:57, 16 July 2022 (UTC)
thread_title:  Can we come up with a more concise way of writing (1 + i)/√2, etc.?