text:
Dawson function

In mathematics, the Dawson function or Dawson integral (named after H. G. Dawson ) is the one-sided Fourier–Laplace sine transform of the Gaussian function.

Definition
The Dawson function is defined as either: $$D_+(x) = e^{-x^2} \int_0^x e^{t^2}\,dt,$$ also denoted as $$F(x)$$ or $$D(x),$$ or alternatively $$D_-(x) = e^{x^2} \int_0^x e^{-t^2}\,dt.\!$$

The Dawson function is the one-sided Fourier–Laplace sine transform of the Gaussian function, $$D_+(x) = \frac12 \int_0^\infty e^{-t^2/4}\,\sin(xt)\,dt.$$

It is closely related to the error function erf, as

where erfi is the imaginary error function, erfi(x) = −i erf(ix). Similarly, $$D_-(x) = \frac{\sqrt{\pi}}{2} e^{x^2} \operatorname{erf}(x)$$ in terms of the real error function, erf.

In terms of either erfi or the Faddeeva function $$w(z),$$ the Dawson function can be extended to the entire complex plane: $$F(z) = {\sqrt{\pi} \over 2} e^{-z^2} \operatorname{erfi} (z) = \frac{i\sqrt{\pi}}{2} \left[ e^{-z^2} - w(z) \right],$$ which simplifies to $$D_+(x) = F(x) = \frac{\sqrt{\pi}}{2} \operatorname{Im}[w(x)]$$ $$D_-(x) = i F(-ix) = -\frac{\sqrt{\pi}}{2} \left[ e^{x^2} - w(-ix) \right]$$ for real $$x.$$

For $$|x|$$ near zero, F(x) ≈ x. For $$|x|$$ large, F(x) ≈ 1/(2x). More specifically, near the origin it has the series expansion $$F(x) = \sum_{k=0}^\infty \frac{(-1)^k \, 2^k}{(2k+1)!!} \, x^{2k+1} = x - \frac{2}{3} x^3 + \frac{4}{15} x^5 - \cdots,$$ while for large $$x$$ it has the asymptotic expansion $$F(x) = \frac{1}{2 x} + \frac{1}{4 x^3} + \frac{3}{8 x^5} + \cdots.$$

More precisely $$\left|F(x) - \sum_{k=0}^{N} \frac{(2k-1)!!}{2^{k+1} x^{2k+1}}\right| \leq \frac{C_N}{x^{2N+3}}.$$ where $$n!!$$ is the double factorial.

$$F(x)$$ satisfies the differential equation $$\frac{dF}{dx} + 2xF = 1\,\!$$ with the initial condition $$F(0) = 0.$$ Consequently, it has extrema for $$F(x) = \frac{1}{2 x},$$ resulting in x = ±0.92413887..., F(x) = ±0.54104422....

Inflection points follow for $$F(x) = \frac{x}{2 x^2 - 1},$$ resulting in x = ±1.50197526..., F(x) = ±0.42768661.... (Apart from the trivial inflection point at $$x = 0,$$ $$F(x) = 0.$$)

Relation to Hilbert transform of Gaussian
The Hilbert transform of the Gaussian is defined as $$H(y) = \pi^{-1} \operatorname{P.V.} \int_{-\infty}^\infty \frac{e^{-x^2}}{y-x} \, dx$$

P.V. denotes the Cauchy principal value, and we restrict ourselves to real $$y.$$ $$H(y)$$ can be related to the Dawson function as follows. Inside a principal value integral, we can treat $$1/u$$ as a generalized function or distribution, and use the Fourier representation $${1 \over u} = \int_0^\infty dk \, \sin ku = \int_0^\infty dk \, \operatorname{Im} e^{iku}.$$

With $$1/u = 1/(y-x),$$ we use the exponential representation of $$\sin(ku)$$ and complete the square with respect to $$x$$ to find $$\pi H(y) = \operatorname{Im} \int_0^\infty dk \,\exp[-k^2/4+iky] \int_{-\infty}^\infty dx \, \exp[-(x+ik/2)^2].$$

We can shift the integral over $$x$$ to the real axis, and it gives $$\pi^{1/2}.$$ Thus $$\pi^{1/2} H(y) = \operatorname{Im} \int_0^\infty dk \, \exp[-k^2/4+iky].$$

We complete the square with respect to $$k$$ and obtain $$\pi^{1/2}H(y) = e^{-y^2} \operatorname{Im} \int_0^\infty dk \, \exp[-(k/2-iy)^2].$$

We change variables to $$u = ik/2+y:$$ $$\pi^{1/2}H(y) = -2e^{-y^2} \operatorname{Im} i \int_y^{i\infty+y} du\ e^{u^2}.$$

The integral can be performed as a contour integral around a rectangle in the complex plane. Taking the imaginary part of the result gives $$H(y) = 2\pi^{-1/2} F(y)$$ where $$F(y)$$ is the Dawson function as defined above.

The Hilbert transform of $$x^{2n}e^{-x^2}$$ is also related to the Dawson function. We see this with the technique of differentiating inside the integral sign. Let $$H_n = \pi^{-1} \operatorname{P.V.} \int_{-\infty}^\infty \frac{x^{2n}e^{-x^2}}{y-x} \, dx.$$

Introduce $$H_a = \pi^{-1} \operatorname{P.V.} \int_{-\infty}^\infty {e^{-ax^2} \over y-x} \, dx.$$

The $$n$$th derivative is $${\partial^nH_a \over \partial a^n} = (-1)^n\pi^{-1} \operatorname{P.V.} \int_{-\infty}^\infty \frac{x^{2n}e^{-ax^2}}{y-x} \, dx.$$

We thus find $$\left. H_n = (-1)^n \frac{\partial^nH_a}{\partial a^n} \right|_{a=1}.$$

The derivatives are performed first, then the result evaluated at $$a = 1.$$ A change of variable also gives $$H_a = 2\pi^{-1/2}F(y\sqrt a).$$ Since $$F'(y) = 1-2yF(y),$$ we can write $$H_n = P_1(y)+P_2(y)F(y)$$ where $$P_1$$ and $$P_2$$ are polynomials. For example, $$H_1 = -\pi^{-1/2}y + 2\pi^{-1/2}y^2F(y).$$ Alternatively, $$H_n$$ can be calculated using the recurrence relation (for $$n \geq 0$$) $$H_{n+1}(y) = y^2 H_n(y) - \frac{(2n-1)!!}{\sqrt{\pi} 2^n} y.$$

title: Dawson function


text:
User talk:Lupo

I'm extremely busy off-Wikipedia these days. I may or may not answer promptly, or I may not answer at all. Lupo

 2024 Monday 29 July Archives of older talk are listed on the archives page.

FAR
I have nominated Paul Kane for a featured article review here. Please join the discussion on whether this article meets featured article criteria. Articles are typically reviewed for two weeks. If substantial concerns are not addressed during the review period, the article will be moved to the Featured Article Removal Candidates list for a further period, where editors may declare "Keep" or "Delist" the article's featured status. The instructions for the review process are here.--Jarodalien (talk) 07:23, 20 March 2014 (UTC)

ImageNote
Hi Lupo, how much do you know about ImageNote? I'm asking because you edited the template on Wikipedia before, but it doesn't seem to be working here. Comparing the code of the template with that on Commons, the and  tags are used differently. I'm wondering if that's why it's not working. But there's also an extra line of code

I'm not sure what it does, but maybe it needs to be added. The page is protected at the moment because it's used by a gadget (I don't know which one), so I was hoping someone else might know how to make it work. Nev1 (talk) 23:02, 23 April 2014 (UTC)

 * 1) The added stuff you quoted above, which is present at the Commons, is just about template documentation.
 * 2) The template is not intended to be used manually.
 * 3) It's used by the ImageAnnotator gadget that was never enabled here. It was installed once for evaluation purposes.

Lupo 07:27, 24 April 2014 (UTC)

 * I manually added an annotation to a file on Wikipedia by copying the wikicode from Commons. If I changed the noinclude/includeonly tags, would it start working? Nev1 (talk) 17:13, 24 April 2014 (UTC)

 * I don't think so, even if the gadget were enabled. For files that actually reside at the Commons, the gadget uses the descriptions defined at the Commons. You don't have to copy them over. If you had chosen a file that existed locally and enabled the gadget, then it should work. However, you could then also use the gadget to add the note in the first place. (The gadget would basically make an identical edit.) If you want to play with that gadget, please read the help pages here and at the Commons. (I hope it still works here; I haven't updated the installation here in a long time.) Lupo 22:17, 24 April 2014 (UTC)

 * In fact, I just tried, and of course it didn't work anymore here. I've updated it now, and it works again. If you enable the gadget in your user.js, you should see image notes at File:Spelterini Blüemlisalp.jpg and also at File:MacLellan's Castle 20080422 - entrance.jpg. And apparently I misremembered about local notes. The gadget does take the note from the local file description page. Frankly said, I'm not sure if this is a bug... I thought it would take those from the Commons. Lupo 22:37, 24 April 2014 (UTC)

 * It is a bug. Since the gadget does not allow adding notes for remote files, it should also ignore any notes on a local description page of a remote file and should only take those from the remote description. Lupo 08:50, 25 April 2014 (UTC)

Lonzo content
This is my own work, which I uploaded it myself from the photo album.......... And thats my account on that page you linked, like where it comes from.... I have every proof... thank you... — Preceding unsigned comment added by OGmuthafkkinDoc (talk • contribs) 12:27, 5 May 2014 (UTC)
 * I presume this is about c:File:The Word Class Wreckin' Cru on tour in 1988..jpg. First, complain at the Commons about image deletions that happen at the Commons. Second, the link I gave is just some fanpix site; that just shows that it's copied from the Internet, it doesn't say that site had any rights to that image. Third, if you can prove that you are the photographer of that picture, send your proof via OTRS. Lupo 12:53, 5 May 2014 (UTC)

Winter service vehicle-tehran.jpg
Hello dear Lupo. Please to see you. I suppose something wrong's been happened. the picture Winter service vehicle-tehran.jpg is already taken by me so I'm its owner. The speedy deletion tag must be removed from my work. Thanks. Milad Mosapoor 09:57, 08 May 2014 (UTC)
 * Nonsense. That was an image already published, and it showed a snow blower at Whistler Mountain, not Teheran. The image came from, photo: Mitch Winton / coastphoto.com. Lupo 04:34, 9 May 2014 (UTC)

Capitan Swan Puzzle (presented within KLM Airline for children).jpg
Hi again, I also added a new trademark tag for the image about the KLM puzzle. Milad Mosapoor 10:08, 08 May 2014 (UTC)
 * It's got nothing to do with trademarks. The image shown by that puzzle itself is copyrighted and even bears a copyright mark. So a photo of that puzzle can be published under a free license only if the copyright holder formally agrees. Lupo 04:31, 9 May 2014 (UTC)

Mawejjederrick's uploads
Mr Lupo i wanted to discuss with you a bout the issues concerning copy right, because all the images i upload on artcles like one for Radababa article are under CC BY-SA 3.0, so i wish they are not deleted because they provide a clear description of the an article  — Preceding unsigned comment added by Mawejjederrick (talk • contribs) 10:16, 12 May 2014 (UTC)
 * This is about c:Special:Log/Mawejjederrick. You upload images taken from any number of websites. None of these web sites indicate any free license for these images. In general, images you can find on the Internet are all copyrighted, all rights reserved, unless there is an explicit mention of a free license. See c:COM:NETCOPYRIGHT. Lupo 10:29, 12 May 2014 (UTC)

Why are my uploads flagged and others on the same page aren't?
Mr Lupo, why does my picture keep getting flagged but the others of games get to stay? Holland9 (talk) 14:54, 23 June 2014 (UTC)
 * Which others? Point me to them, and I'll take them down, too, if they are copyright violations. Lupo 15:33, 23 June 2014 (UTC)

User:Eagman1/Jan Estrada
Sorry to bring up something after so long, but did you conclude that this Villa player is a hoax or fantasy? Rankersbo (talk) 07:02, 18 July 2014 (UTC)
 * Yes. See Articles for deletion/Jan Estrada. Lupo 15:13, 18 July 2014 (UTC)

photographie Walter Wachsmuth
Thank you very much for your helpless words. Now, I think, the aim could be reached! Alles Gute für Sie von "Prädikat" aus Deutschland. — Preceding unsigned comment added by Prädikat (talk • contribs) 13:57, 7 August 2014 (UTC)
 * (for bystanders: this is about File:Musiker und Komponist Walter Wachsmuth, Braunschweig 1910.jpg; see commons:User talk:Prädikat) I do hope you meant "helpful". Lupo 14:52, 7 August 2014 (UTC)

FAR of Marine shrimp farming
I have nominated Marine shrimp farming for a featured article review here. Please join the discussion on whether this article meets featured article criteria. Articles are typically reviewed for two weeks. If substantial concerns are not addressed during the review period, the article will be moved to the Featured Article Removal Candidates list for a further period, where editors may declare "Keep" or "Delist" the article's featured status. The instructions for the review process are here. Sandy Georgia (Talk) 19:29, 20 December 2014 (UTC)

Photograph of Fusilier Lee Rigby
With respect to the page Murder of Lee Rigby, The of Fusilier Lee Rigby has the incorrect copyright attribution and is not a UK Crown Copyright photograph and therefore is not available under the Open Government Licence. — Preceding unsigned comment added by UKMinistryofDefence (talk • contribs) 09:55, 17 July 2015 (UTC)
 * Maybe so. As one can see at the archived original source, the MOD stated "All content is available under the Open Government Licence, except where otherwise stated", and there was no statement that the image was copyrighted by somebody else. Looks like the MOD has now taken down the image pending a resolution of a copyright claim by a photographer. Further discussions at commons:Commons:Deletion requests/File:Drummer Lee Rigby.jpg. Lupo 16:48, 29 July 2015 (UTC)

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Karte Alte Eidgenossenschaft Englisch-Übersetzung
DIe uebersetzung leventina/livinental auf livigno in englisch ist meiner Meinung falsch, da livigno ja ein Tal in Italien ist! oder sehe ich das falsch?

mein Mail ist n9d@me.com! — Preceding unsigned comment added by 178.38.81.249 (talk) 13:57, 27 December 2015 (UTC)

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Disputed non-free use rationale for File:Montreux Jazz Festival 1986 (poster).png
Thank you for uploading File:Montreux Jazz Festival 1986 (poster).png. However, there is a concern that the rationale provided for using this file on Wikipedia may not meet the criteria required by Non-free content. This can be corrected by going to the file description page and adding or clarifying the reason why the file qualifies under this policy. Adding and completing one of the templates available from Non-free use rationale guideline is an easy way to ensure that your file is in compliance with Wikipedia policy. Please be aware that a non-free use rationale is not the same as an image copyright tag; descriptions for files used under the non-free content policy require both a copyright tag and a non-free use rationale.

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Error in your LAPI.js script
Hi Lupo. I have recently turned on the gadget ImageAnnotator in my preferences, and I now get an annoying javascript warning in my console about "unreachable code after return statement" on every single page load. I have tracked down the error to the  function in the   section. In that function you include a  statement followed by a newline and then what you expected to return: I assume you know this, but in case you didn't, you cannot add a line break after a  call, even if there is no semicolon ending the line. The parser is set up to basically make semicolons optional, so a line break after a return just ends the line automatically, and the next line is not executed; the script just returns silently without a value. I believe this is what is causing the warning to show up in my console, and the way to fix it would be to simply remove the line break there. Would you be willing to do that? Thanks!--Dudemanfellabra (talk) 23:20, 6 June 2016 (UTC)
 * Good catch! Fixed this and a second occurrence of the same problem. Lupo 18:48, 10 June 2016 (UTC)
 * Thanks! I'm no longer getting the warning on every page load!--Dudemanfellabra (talk) 19:15, 10 June 2016 (UTC)

Zamba´s name from
hi my name is Jose Miguel Zamba Gomes from Angolan, and i would like to about where your name zamba from.. my skype is divanicy or j.zamba@outlook.com contact me..

Thanks; Zamba — Preceding unsigned comment added by 196.32.198.242 (talk) 11:28, 19 July 2016 (UTC)

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Hi.

A new user group, New Page Reviewer, has been created in a move to greatly improve the standard of new page patrolling. The user right can be granted by any admin at PERM. It is highly recommended that admins look beyond the simple numerical threshold and satisfy themselves that the candidates have the required skills of communication and an advanced knowledge of notability and deletion. Admins are automatically included in this user right.

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These were related to Help:Gadget-ImageAnnotator. Since the gadget was never officially installed here, I guess removing them doesn't bother anyone. Lupo 11:15, 19 April 2017 (UTC)

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Ping
Good to see you pop up on my watchlist :) Haukur (talk) 10:10, 15 July 2019 (UTC)
 * What a surprise! I thought you'd all but gone, too, but now I see you're back in force since this year. I guess my activity here will remain at current extremely low levels, though. Lupo 21:24, 15 July 2019 (UTC)
 * Yeah, I got pulled back in somehow. Mostly absent for a long time because my academic work scratches the same itch. But there's also something to be said for working on articles people might actually read.

 * I miss you at Commons which is as much a mess as ever with poor sourcing and confused copyright claims. Haukur (talk) 22:23, 15 July 2019 (UTC)

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license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:Lupo


text:
Debye function

In mathematics, the family of Debye functions is defined by $$D_n(x) = \frac{n}{x^n} \int_0^x \frac{t^n}{e^t - 1}\,dt.$$

The functions are named in honor of Peter Debye, who came across this function (with n = 3) in 1912 when he analytically computed the heat capacity of what is now called the Debye model.

Relation to other functions
The Debye functions are closely related to the polylogarithm.

Series expansion
They have the series expansion $$D_n(x) = 1 - \frac{n}{2(n+1)} x + n \sum_{k=1}^\infty \frac{B_{2k}}{(2k+n)(2k)!} x^{2k}, \quad |x| < 2\pi,\ n \ge 1,$$ where $$B_n$$ is the $n$-th Bernoulli number.

Limiting values
$$\lim_{x \to 0} D_n(x) = 1.$$ If $$\Gamma$$ is the gamma function and $$\zeta$$ is the Riemann zeta function, then, for $$x \gg 0$$, $$D_n(x) = \frac{n}{x^n} \int_0^x \frac{t^n\,dt}{e^t-1} \sim \frac{n}{x^n}\Gamma(n + 1) \zeta(n + 1), \qquad \operatorname{Re} n > 0,$$

Derivative
The derivative obeys the relation $$x D^{\prime}_n(x) = n \left(B(x) - D_n(x)\right),$$ where $$B(x) = x/(e^x-1)$$ is the Bernoulli function.

The Debye model
The Debye model has a density of vibrational states $$g_\text{D}(\omega) = \frac{9\omega^2}{\omega_\text{D}^3} \,, \qquad 0\le\omega\le\omega_\text{D}$$ with the $ω_{D}$.

Internal energy and heat capacity
Inserting $g$ into the internal energy $$U = \int_0^\infty d\omega\,g(\omega)\,\hbar\omega\,n(\omega)$$ with the Bose–Einstein distribution $$n(\omega) = \frac{1}{\exp(\hbar\omega / k_\text{B} T)-1}.$$ one obtains $$U = 3 k_\text{B}T \, D_3(\hbar\omega_\text{D} / k_\text{B}T).$$ The heat capacity is the derivative thereof.

Mean squared displacement
The intensity of X-ray diffraction or neutron diffraction at wavenumber q is given by the Debye-Waller factor or the Lamb-Mössbauer factor. For isotropic systems it takes the form $$\exp(-2W(q)) = \exp\left(-q^2\langle u_x^2\rangle\right).$$ In this expression, the mean squared displacement refers to just once Cartesian component $u_{x}$ of the vector $u$ that describes the displacement of atoms from their equilibrium positions. Assuming harmonicity and developing into normal modes, one obtains $$2W(q) = \frac{\hbar^2 q^2}{6M k_\text{B}T} \int_0^\infty d\omega \frac{k_\text{B}T}{\hbar\omega}g(\omega) \coth\frac{\hbar\omega}{2k_\text{B}T}=\frac{\hbar^2 q^2}{6M k_\text{B}T} \int_0^\infty d\omega \frac{k_\text{B}T}{\hbar\omega} g(\omega) \left[\frac{2}{\exp(\hbar\omega/k_\text{B}T)-1}+1\right].$$ Inserting the density of states from the Debye model, one obtains $$2W(q) = \frac{3}{2} \frac{\hbar^2 q^2}{M\hbar\omega_\text{D}} \left[2\left(\frac{k_\text{B}T}{\hbar\omega_\text{D}}\right) D_1{\left(\frac{\hbar\omega_\text{D}}{k_\text{B}T}\right)} + \frac{1}{2}\right].$$ From the above power series expansion of $$D_1$$ follows that the mean square displacement at high temperatures is linear in temperature $$2W(q) = \frac{3 k_\text{B}T q^2}{M\omega_\text{D}^2}.$$ The absence of $$\hbar$$ indicates that this is a classical result. Because $$D_1(x)$$ goes to zero for $$x \to \infty$$ it follows that for $$T = 0$$ $$2W(q)=\frac{3}{4}\frac{\hbar^2 q^2}{M\hbar\omega_\text{D}}$$ (zero-point motion).

Implementations

 * Fortran 77 code
 * Fortran 90 version
 * C version of the GNU Scientific Library
 * Fortran 90 version
 * C version of the GNU Scientific Library
 * C version of the GNU Scientific Library
 * C version of the GNU Scientific Library

title: Debye function


text:
Yagan

Yagan (c. 1795 – 11 July 1833) was an Aboriginal Australian warrior from the Noongar people. Yagan was pursued by the local authorities after he killed Erin Entwhistle, a servant of farmer Archibald Butler. It was an act of retaliation after Thomas Smedley, another of Butler's servants, shot at a group of Noongar people stealing potatoes and fowls, killing one of them. The government offered a bounty for Yagan's capture, dead or alive, and a young settler, William Keats, shot and killed him. He is considered a legendary figure by the Noongar.

After his shooting, settlers removed Yagan's head to claim the bounty. Later, an official sent it to London, where it was exhibited as an "anthropological curiosity" and eventually given to a museum in Liverpool. It held the head in storage for more than a century before burying it with other remains in an unmarked grave in Liverpool in 1964. Over the years, the Noongar asked for repatriation of the head, both for religious reasons and because of Yagan's traditional stature. The burial site was identified in 1993; officials exhumed the head four years later and repatriated it to Australia. After years of debate within the Noongar community on the appropriate final resting place, Yagan's head was buried in a traditional ceremony in the Swan Valley in July 2010, 177 years after his death.

Early life
A member of the Whadjuk Noongar people, Yagan belonged to a tribe of around 60 people whose name, according to Robert Lyon, was Beeliar. Scholars now believe that the Beeliar people may have been a family subgroup (or clan) of a larger tribe whom Daisy Bates called Beelgar. According to Lyon, the Beeliar people occupied the land south of the Swan and Canning rivers, as far south as Mangles Bay. The group had customary land usage rights over a much larger area than this, extending north as far as Lake Monger and northeast to the Helena River. The group also had an unusual degree of freedom to move over their neighbours' land, possibly due to kinship and marriage ties with neighbouring groups.

Yagan is thought to have been born around 1795. His father was Midgegooroo, an elder of the Beeliar people; his mother was one of Midgegooroo's three wives. Yagan was probably a Ballaroke in the Noongar classification.

Marriage and family
According to the historian Neville Green, Yagan had a wife and two children. A report in the Perth Gazette in 1833 gives the names of his children as "Naral", age 9, and "Willim", age 11, but most other sources state that the warrior was unmarried and childless. When his entry in the Australian Dictionary of Biography was rewritten in 2019, Reece suggested those said to have been his sons may have been his younger brothers. Described as taller than average with an impressive burly physique, Yagan had a distinctive tribal tattoo on his right shoulder, which identified him as "a man of high degree in tribal law". He was generally acknowledged to be the most physically powerful of his tribe, and was said to have been able to spear another stick from a distance of 25 yards or penetrate a tree from a distance of 60 yards.

Relations with settlers
Yagan would have been about 35 years old in 1829 when British settlers landed in the area and established the Swan River Colony. For the first two years of the colony, relations between settlers and Noongar were generally amicable, as there was little competition for resources. The Noongar welcomed the white settlers as, the returned spirits of their dead. Historical reports noted the two groups shared fish. As time passed, conflicts between the two cultures gradually became more frequent. The settlers thought that the Noongar were nomads who had no claim to the land over which they roamed. Colonists fenced off land for grazing and farming according to their traditional practices of land use.

As the colonists fenced off more land, the Noongar were increasingly denied access to their traditional hunting grounds and sacred sites. In search of food, the Noongar raided the settlers' crops and killed their cattle. They also developed a taste for the settlers' supplies, and began to take flour and other food, which became a serious problem for the colony. In addition, the Noongar practice of firestick farming, or lighting the bush to flush out game threatened the settlers' crops and houses.

In December 1831 Yagan and his father led the first significant Aboriginal resistance to white settlement in Western Australia. Thomas Smedley, a servant of farmer Archibald Butler, apprehended some natives who were raiding a potato patch, and killed one of Yagan's family group. A few days later, Yagan, Midgegooroo and others stormed the farmhouse and, finding the door locked, began to break through the mud-brick walls. Inside were Butler's servant Erin Entwhistle and his two sons Enion and Ralph. After hiding his sons under the bed, Entwhistle opened the door to parley and was killed by Yagan and Midgegooroo. Noongar tribal law required that murders be avenged by the killing of a member of the murderer's tribal group, not necessarily the murderer. The Noongar considered servants and employees to be part of the settlers' groups. Historians believe the Noongar attack on Entwhistle was retribution under their tribal law. Not understanding tribal law (and unlikely to agree with its concepts), the white settlers took the killing to be an unprovoked murder and dispatched a force to arrest Yagan's group, without success.

In June 1832 Yagan led a party of Noongar in attacking two labourers sowing a field of wheat alongside the Canning River near Kelmscott. One of the men, John Thomas, escaped, but the other, William Gaze, was wounded and later died as a result. The settlement declared Yagan an outlaw and offered a reward of £20 for his capture. He avoided capture until early October 1832. A group of fishermen enticed Yagan and two companions into their boat, then pushed off into deep water. The fishermen took the three Noongar men to the Perth guardhouse, from which they were transferred to the Round House at Fremantle. Yagan was sentenced to death, but he was saved by the intercession of settler Robert Lyon. Arguing that Yagan was defending his land against invasion, Lyon said Yagan should not be considered a criminal but a prisoner of war and suggested he should be treated as such. At the recommendation of John Septimus Roe, the Surveyor-General of Western Australia, Yagan and his men were exiled on Carnac Island under the supervision of Lyon and two soldiers.

Lyon thought he could teach Yagan British ways and convert him to Christianity. He hoped to gain his cooperation and use his tribal stature to persuade the Noongar to accept colonial authority. Lyon spent many hours with Yagan learning his language and customs. After a month, Yagan and his companions escaped by stealing an unattended dinghy and rowing to Woodman Point on the mainland. The Government did not pursue them; apparently its officials considered they had been sufficiently punished.

In January 1833 two Noongar, Gyallipert and Manyat, visited Perth from King George Sound, where relations between settlers and natives were amicable. Two settlers, Richard Dale and George Smythe, arranged for the men to meet a party of local Noongar to encourage friendly relations in the Swan River Colony. On 26 January Yagan led a group of ten formally armed Noongars in greeting the two men near Lake Monger. The men exchanged weapons and held a corroboree, though the groups did not appear to share a language. Yagan and Gyallipert competed at spear throwing. As an example of his prowess, Yagan struck a walking stick from a distance of 25 m.

Gyallipert and Manyat remained in Perth for some time. On 3 March, Yagan obtained permission to hold another corroboree, this time in the Post Office garden in Perth. The Perth and King George Sound men met at dusk, chalked their bodies, and performed a number of dances including a kangaroo hunt dance. The Perth Gazette wrote that Yagan "was master of ceremonies and acquitted himself with infinite grace and dignity".

During February and March, Yagan was involved in a series of minor conflicts with settlers. In February William Watson complained that Yagan had pushed open his door, demanded a gun, and taken handkerchiefs. Watson had to give him and his companions flour and bread. The following month, Yagan was among a group who received biscuits from a military contingent under Lieutenant Norcott; when Norcott tried to restrict his supply, Yagan threatened him with his spear. Later that month, Yagan was with a group of Noongar who entered Watson's house while he was away. The group left after Watson's wife called on neighbours for help. The next day Captain Ellis lectured the Noongar about their behaviour. The frequent incidents prompted The Perth Gazette to remark on "the reckless daring of this desperado who sets his life at a pin's fee ... For the most trivial offence ... he would take the life of any man who provoked him. He is at the head and front of any mischief."

Wanted dead or alive
On the night of 29 April, a party of Noongar broke into a Fremantle store to steal flour and they were shot at by the caretaker Peter Chidlow. Domjum, a brother of Yagan, was badly injured and died in jail a few days later. The rest of the party moved from Fremantle to Preston Point, where Yagan reportedly vowed vengeance for the death. Between 50 and 60 Noongar gathered at Bull Creek, where they met a party of settlers who were loading carts with provisions. Later that day, the group ambushed the lead cart, killing two settlers, Tom and John Velvick. Tribal law required only a single death for vengeance. Some historians have speculated that the Velvicks were targeted because they had previously been convicted for assaulting Aboriginal people and coloured seamen. Alexandra Hasluck has also argued that stealing provisions was an important motive in the attack, but this has been refuted elsewhere.

For the killing of the Velvicks, the Lieutenant-Governor Frederick Irwin declared Yagan, Midgegooroo and Munday to be outlaws, offering rewards of £20 each for the capture of Midgegooroo and Munday, and a reward of £30 for Yagan's capture, dead or alive. Munday successfully appealed against his proscription. Midgegooroo, Yagan and their group immediately moved from their territory north towards the Helena Valley. On 17 May, Midgegooroo was captured on the Helena River. After a brief, informal trial, he was executed by firing squad. Yagan remained at large for over two months.

Late in May, George Fletcher Moore reported seeing Yagan on his property and talking with him in pidgin English. Moore wrote in the Perth Gazette: "Yagan stepped forward and leaning with his left hand on my shoulder while he gesticulated with the right, delivered a sort of recitation, looking earnestly in my face. I regret I could not understand it, I thought from the tone and manner that the purport was this: "'You came to our country — you have driven us from our haunts, and disturbed us in our occupations. As we walk in our own country we are fired upon by the white men; why should the white men treat us so?'"Since Moore had little knowledge of Yagan's native language, the historian Hasluck suggests that this account is probably more indicative of "a feeling of conscience on the part of the white men" than an accurate rendering of Yagan's state of mind."

Yagan asked Moore whether Midgegooroo was dead or alive. Moore gave no reply, but a servant answered that Midgegooroo was a prisoner on Carnac Island. Yagan warned, "White man shoot Midgegooroo, Yagan kill three." Moore reported the encounter but made no attempt to restrain Yagan. He later wrote, "The truth is, every one wishes him taken, but no one likes to be the captor ... there is something in his daring which one is forced to admire."

Death
On 11 July 1833, two teenage brothers named William and James Keates were herding cattle along the Swan River north of Guildford when a group of Noongar approached while en route to collect flour rations from Henry Bull's house. The Keates brothers suggested Yagan remain with them to avoid arrest. While he was staying with them during the morning, the brothers decided to kill the warrior and claim the reward. When the natives were ready to depart, the Keateses took their last opportunity. William Keates shot Yagan, and James shot Heegan, another native, in the act of throwing his spear. The brothers ran away, but other Noongar overtook William and speared him to death. James escaped by swimming the river. Shortly afterward he returned with a party of armed settlers from Bull's estate.

When the party of settlers arrived, they found Yagan dead and Heegan dying. Heegan "was groaning and his brains were partly out when the party came, and whether humanity or brutality, a man put a gun to his head and blew it to pieces." The settlers cut Yagan's head from his body, and skinned his back to obtain his tribal markings as a trophy. They buried the bodies a short distance away.

James Keates claimed the reward, but his conduct was widely criticised. The Perth Gazette referred to Yagan's killing as "a wild and treacherous act ... it is revolting to hear this lauded as a meritorious deed." However, Daisy Bates understood that "he was killed in self-defence by the young lad." Keates left the colony the following month; it is possible that he left from fear of being murdered in tribal retaliation.

Exhibition and burial
Yagan's head was initially taken to Henry Bull's house. Moore saw it there and sketched the head a number of times in his unpublished, handwritten diary, commenting that "possibly it may yet figure in some museum at home." The head was preserved by smoking.

In September 1833, Governor Irwin sailed for London, partly to give his own account of the events leading up to the killing. This was an unusual measure, especially given his regiment was about to leave for a tour of duty in India. The Colonial Office indicated satisfaction with Irwin's administration of the colony.

Travelling with Irwin was Ensign Robert Dale, who had somehow acquired Yagan's head. According to the historian Paul Turnbull, Dale appears to have persuaded Irwin to let him have the head as an "anthropological curiosity". After arriving in London, Dale tried to sell the head to scientists, approaching a number of anatomists and phrenologists. His price of £20 failed to find a buyer, so he made an agreement with Thomas Pettigrew for the exclusive use of the head for 18 months. Pettigrew, a surgeon and antiquarian, was well known in the London social scene for holding private parties at which he unrolled and autopsied ancient Egyptian mummies. He displayed the head on a table in front of a panoramic view of King George Sound reproduced from Dale's sketches. For effect, the head was adorned with a fresh corded headband and feathers of the red-tailed black cockatoo.

Pettigrew had the head examined by a phrenologist. Examination was considered difficult because of the large fracture across the back of the head caused by the gunshot. His conclusions were consistent with contemporary European opinion of Indigenous Australians. Dale published these in a pamphlet entitled Descriptive Account of the Panoramic View &c. of King George's Sound and the Adjacent Country, which Pettigrew encouraged his guests to buy as a souvenir of their evening. The frontispiece of the pamphlet was a hand-coloured aquatint print of Yagan's head by the artist George Cruikshank.

Early in October 1835, Yagan's head and the panoramic view were returned to Dale, then living in Liverpool. On 12 October he presented them to the Liverpool Royal Institution, where the head may have been displayed in a case along with some other preserved heads and wax models illustrating cranial anatomy. In 1894 the Institution's collections were dispersed, and Yagan's head was lent to the Liverpool Museum; it is thought not to have been put on display there. By the 1960s Yagan's head was badly deteriorated. In April 1964 the museum decided to dispose of it. It arranged burial of the head on 10 April 1964, together with a Peruvian mummy and a Māori head. They were buried in Everton Cemetery's General Section 16, grave number 296. In later years a number of burials were made around the grave. For example, in 1968 a local hospital buried directly over the box, 20 stillborn babies and two infants who died soon after birth.

Lobbying for repatriation
For many years beginning in the early 1980s, a number of Noongar groups sought the return of Yagan's head to Australia. "It is Aboriginal belief that because Yagan's skeletal remains are incomplete, his spirit is earthbound. The uniting of his head and torso will immediately set his spirit free to continue its eternal journey."

At the time, there was no historical trail for the head after Pettigrew passed it on. Tribal elders entrusted the Aboriginal leader Ken Colbung with the search. In the early 1990s, Colbung enlisted the aid of University of London archaeologist Peter Ucko. One of Ucko's researchers, Cressida Fforde, conducted a literature search for information on the head. Fforde successfully traced the head in December 1993. The following April, Colbung applied to exhume the remains under Section 25 of the Burial Act 1857. Home Office regulations required next of kin consent before disturbing the remains of the 22 infants. Colbung's solicitors requested waiver of this condition on grounds that the exhumation would be of great personal significance to Yagan's living relatives, and great national importance to Australia.

Meanwhile, divisions in the Noongar community in Perth began to develop. Some elders questioned Colbung's role and one Noongar registered a complaint with the Liverpool City Council over his involvement. Media reports indicated acrimonious debate within the Noongar community about who had the best cultural qualifications to take possession of the head. The academic Hannah McGlade claims that these divisions were largely manufactured by the media, particularly The West Australian, which "aimed to and successfully represented the Nyungar community in terms of disharmony and dissent". She alleges that one West reporter contacted Noongar who were known to be in disagreement, and quoted one to the other, so as to elicit provocative responses. The disputes were "trumpeted" by The West, allowing it to "preach" against the infighting.

On 25 July a public meeting was held in Perth. All parties agreed to put aside their differences and co-operate to ensure that the repatriation was a "national success". A Yagan Steering Committee was established to co-ordinate the repatriation, and Colbung's application was allowed to proceed. In January 1995 the Home Office advised Colbung that it was unable to waive the requirement to obtain next of kin consent for the exhumation. It contacted the five relatives whose addresses were known, and received unconditional consent from only one. Accordingly, on 30 June 1995, Colbung and the other interested parties were advised that the application for exhumation had been rejected.

Meeting on 21 September, the Yagan Steering Committee decided to lobby Australian and British politicians for support. In 1997 Colbung was invited to visit the United Kingdom at the British government's expense and he arrived on 20 May. His visit attracted substantial media coverage, and increased the political pressure on the British Government. He secured the support of the Prime Minister of Australia, John Howard, after gate crashing the Prime Minister's June visit to the United Kingdom.

Exhumation
While Colbung was in the United Kingdom, Martin and Richard Bates were engaged to undertake a geophysical survey of the grave site. Using electromagnetic and ground penetrating radar techniques, they identified an approximate position of the box that suggested it could be accessed from the side via the adjacent plot. A report of the survey was passed to the Home Office, prompting further discussions between the British and Australian Governments.

Of concern to the Home Office were an undisclosed number of letters that it had received objecting to Colbung's involvement in the repatriation process; it therefore sought assurances from the Australian Government that Colbung was a correct applicant. In response Colbung asked his elders to ask the Aboriginal and Torres Strait Islander Commission (ATSIC) to tell the British Home Office that he was the correct applicant. ATSIC then convened a meeting in Perth at which it was again resolved that Colbung's application could proceed.

Colbung continued to press for the exhumation, asking that it be performed before the 164th anniversary of Yagan's death on 11 July, so that the anniversary could be the occasion of a celebration. His request was not met, and on the anniversary of Yagan's death, Colbung conducted a short memorial service at the burial plot in Everton. He returned to Australia empty-handed on 15 July.

The exhumation of Yagan's head eventually proceeded, without Colbung's knowledge, by excavating 6 ft down the side of the grave, then tunnelling horizontally to the location of the box. Thus the exhumation was performed without disturbing any other remains. The following day, a forensic palaeontologist from the University of Bradford positively identified the skull as Yagan's by correlating the fractures with those described in Pettigrew's report. The skull was then kept at the museum until 29 August, when it was handed over to the Liverpool City Council.

Repatriation
On 27 August 1997, a delegation of Noongars consisting of Ken Colbung, Robert Bropho, Richard Wilkes and Mingli Wanjurri-Nungala arrived in the UK to collect Yagan's head. The delegation was to have been larger, but Commonwealth funding was withdrawn at the last minute. The handover of Yagan's skull was further delayed when a Noongar named Corrie Bodney applied to the Supreme Court of Western Australia for an injunction against the handover. Claiming that his family group has sole responsibility for Yagan's remains, Bodney declared the exhumation illegal and denied the existence of any tradition or belief necessitating the head's exhumation and removal to Australia. On 29 August, Justice Henry Wallwork rejected the injunction application, on the grounds that Bodney had previously agreed to the current arrangements, and on the evidence of another Noongar elder (Albert Corunna, who claimed to be a closer relation of Yagan) and anthropologist Pat Baines, both of whom refuted Bodney's claim to sole responsibility.

Yagan's skull was handed over to the Noongar delegation at a ceremony at Liverpool Town Hall on 31 August 1997. In accepting the skull, Colbung made comments that were interpreted as linking Yagan's death with the death of Diana, Princess of Wales, earlier that day: "That is how nature goes ... Nature is a carrier of all good things and all bad things. And because the Poms did the wrong thing, they now have to suffer." Colbung's comments prompted a media furore throughout Australia, with newspapers receiving many letters from the public expressing shock and anger at the comments. Colbung later claimed that his comments had been misinterpreted.

Throughout the repatriation process, many sections of the international media treated the story as a joke. For example, U.S. News & World Report ran a story under the headline Raiders of the Lost Conk, in which Yagan's head was referred to as a "pickled curio", and Colbung's actions were treated as a publicity stunt.

Preparations for reburial
On its return to Perth, Yagan's head continued to be a source of controversy and conflict. Responsibility for reburial of the head was given to a "Committee for the Reburial of Yagan's Kaat", headed by Richard Wilkes. The reburial was delayed by disputes between elders over the burial location, mainly due to uncertainty of the whereabouts of the rest of his body, and disagreement about the importance of burying the head with the body.

A number of attempts were made to locate the remains of Yagan's body, which were believed to be on Lot 39 West Swan Road in the outer Perth suburb of Belhus. A remote sensing survey of the site was carried out in 1998, but no remains were found. An archaeological survey of the area was undertaken two years later, but this also was unsuccessful. Disputes then arose over whether the head could be buried separately from the body. Wilkes has claimed that it can, so long as it is placed where Yagan was killed, so that Dreamtime spirits can reunite the remains.

In 1998 the Western Australian Planning Commission and the Department of Aboriginal Affairs jointly published a document entitled Yagan's Gravesite Master Plan, which discussed "matters of ownership, management, development and future use" of the property on which Yagan's remains are believed to be buried. Under consideration was the possibility of turning the site into an Indigenous burial site, to be managed by the Metropolitan Cemeteries Board.

Yagan's head spent some time in storage in a bank vault before being handed over to forensics experts who reconstructed a model from it. After that it was held in storage at Western Australia's state mortuary. Plans to re-bury the head were repeatedly deferred, causing ongoing conflict between Noongar groups. In September 2008 it was reported that Yagan's head would be reburied in November, and a Yagan Memorial Park created as a projected cost of A$996,000; but in November it was announced that the reburial had been rescheduled for July 2009 because of logistical problems. In March 2009, it was announced that the Department of Indigenous Affairs had given the City of Swan more than A$500,000 to develop the park.

Reburial
The head was finally buried in a private ceremony attended only by invited Noongar elders, on 10 July 2010, the anniversary of the last full day he lived and one day before the end of NAIDOC Week 2010. The site in Belhus was chosen as it is believed to be near to where the rest of Yagan's body was buried. The burial coincided with a ceremony to mark the opening of the Yagan Memorial Park, which was attended by around 300 people, including Noongar elders and state government representatives. Premier Colin Barnett described the occasion as "a wonderful day for all West Australians".

The art works for the Yagan Memorial Park were designed by Peter Farmer, Sandra Hill, Jenny Dawson and Kylie Ricks. Dawson and Hill created an entry wall of Yagan's story; Farmer designed the park entry statements and Ricks the female coolamon.

Legacy
In the Australian Dictionary of Biography, Reece wrote that Yagan "was not the brutal, indiscriminate killer most settlers thought him to be", but sought to enforce the Noongar system of retributive justice "as the only basis for a resolution of conflict between Noongars and colonists". Yagan was a lone actor in this regard:"Yagan cannot properly be described as a ‘resistance leader’ when the Noongars offered no organised and sustained opposition to the settlers. He was more of a maverick, a bold and courageous warrior whose actions on behalf of his people and their rights made him notorious."The repatriation of Yagan's head increased the Aboriginal leader's notability. He is considered a famous historical figure throughout Australia, with material about him appearing in such publications as the Australian Dictionary of Biography, and Western Australia's school curriculum. He is of greatest significance to the Noongar people, for whom he is "a revered, cherished and heroic individual ... patriot and visionary hero of WA's South-West". The return of his head was likened by some Indigenous Australians to the November 1993 ceremonial repatriation from Gallipoli of Australia's unknown soldier.

The former Upper Swan Bridge, which carries the Great Northern Highway over the Swan River at Belhus, was renamed the Yagan Bridge in 2010.

An open plaza in the Perth central business district, constructed as part of the Perth City Link urban renewal project, was named Yagan Square. Featuring the 9 m statue "Wirin", the plaza, located adjacent to the Horseshoe Bridge, was opened on 3 March 2018.

Alas Poor Yagan
On 6 September 1997 The West Australian published a Dean Alston cartoon entitled Alas Poor Yagan, which was critical of the fact that the return of Yagan's head had become a source of conflict between Noongars instead of fostering unity. The cartoon was interpreted by some as insulting aspects of Noongar culture, and casting aspersions on the motives and legitimacy of Indigenous Australians with mixed racial heritage. The content of the cartoon offended many Indigenous Australians, and a group of Noongar elders complained about the cartoon to the Human Rights and Equal Opportunity Commission. The commission ruled that the cartoon made inappropriate references to Noongar beliefs but was not in breach of the Racial Discrimination Act 1975 because it was "an artistic work" that was published "reasonably and in good faith", and was therefore exempt. This ruling was upheld on appeal by the Federal Court of Australia. Some academic commentators have since expressed concern that the protections offered under the act have been undermined by the ruling's broad interpretation of the exemptions.

Statue
From the mid-1970s, members of the Noongar community lobbied for the erection of a statue of Yagan as part of the WAY 1979 sesquicentennial celebrations. Their requests were refused, however, after the Premier, Charles Court, was advised by one prominent historian that Yagan was not important enough to warrant a statue. Colbung claims "Court was more interested in spending tax payers' money on refurbishing the badly neglected burial place of Captain James Stirling, WA's first governor." Despite this setback, the Noongar community persisted, establishing a Yagan Committee and running a number of fund-raising drives. Eventually, sufficient funds were collected to allow the commissioning of Australian sculptor Robert Hitchcock to create a statue. The result was a life-size statue in bronze, depicting Yagan standing naked with a spear held across his shoulders. Hitchcock's statue of Yagan was officially opened by Yagan Committee chairperson Elizabeth Hanson on 11 September 1984. It stands on Heirisson Island in the Swan River near Perth.

In 1997, within a week of the return of Yagan's head to Perth, vandals beheaded the statue using an angle grinder. Soon after a replacement head was installed and it too was detached and stolen. Credit for the act was anonymously claimed by a "British loyalist" as an act of retaliation for Colbung's comments about Diana, Princess of Wales. The Western Australia Police did not succeed in identifying the vandals, nor in recovering the heads, and deemed it infeasible to have the statue fenced off or placed under guard.

Commentary on the beheadings varied widely. One column in The West Australian found humour in them, referring to the head as a "bonce" and a "noggin", and finished with a pun on "skullduggery". Stephen Muecke calls this the "satirical trivialising of Aboriginal concerns", and Adam Shoemaker writes "This is the stuff of light humour and comic relief. There is no sense of the decapitation as being an act of vandalism, even less that it could have been motivated by malevolence ... [T]he piece has a definite authorising function." On the other hand, academic analysis has treated the act with much more gravity. In 2007, for example, David Martin described the decapitation as "an act which speaks not only to the continuance of white settler racism, but also to the power of mimesis to invigorate our modern memorials and monuments with a life of their own".

In 2002, Janet Woollard, the member for Alfred Cove, called for the statue's genitalia to be covered up. In November 2005, Richard Wilkes also called for the statue's groin to be covered on the grounds that such a depiction would be more historically accurate, as Yagan would have worn a covering for most of the year. Also under consideration is the creation of a new statue with a head shape that accords better with the forensic reconstruction of Yagan's head.

Literature and film
Mary Durack published a fictionalised account of Yagan's life in her 1964 children's novel The Courteous Savage: Yagan of the Swan River, which was renamed Yagan of the Bibbulmun on reissue in 1976.

The repeated beheading of Yagan's statue in 1997 prompted Aboriginal writer Archie Weller to write a short story entitled Confessions of a Headhunter. Weller later worked with film director Sally Riley to adapt the story into a script, and in 2000 a 35-minute movie, also named Confessions of a Headhunter, was released. Directed by Sally Riley, the movie won Best Short Fiction Film at the 2000 AFI Awards. The following year the script won the Script Award in the 2001 Western Australian Premier's Book Awards.

In 2002, the South African-born Australian poet John Mateer published his fourth collection of poems, entitled Loanwords. The collection is divided into four sections, of which the third, In the Presence of a Severed Head, has Yagan as its subject.

Other cultural references
A section of Kullark, a play by Jack Davis, explores the deteriorating relationship between Yagan and a settler couple.

In September 1989 an early maturing cultivar of barley, bred by the Western Australian Department of Agriculture for performance on sandy soils, was released under the name "Hordeum vulgare (Barley) c.v. Yagan". Commonly referred to simply as "Yagan", the cultivar is named for Yagan, continuing a tradition of labelling Western Australian grain cultivars after historic people of Western Australia.

title: Yagan


text:
Legendre form

In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because the second kind gives the arc length of an ellipse of unit semi-major axis and eccentricity $$\scriptstyle{k}$$ (the ellipse being defined parametrically by $$\scriptstyle{x = \sqrt{1 - k^{2}} \cos(t)}$$, $$\scriptstyle{y = \sin(t)}$$).

In modern times the Legendre forms have largely been supplanted by an alternative canonical set, the Carlson symmetric forms. A more detailed treatment of the Legendre forms is given in the main article on elliptic integrals.

Definition
The incomplete elliptic integral of the first kind is defined as,

 * $$F(\phi,k) = \int_0^\phi \frac{1}{\sqrt{1 - k^2 \sin^2(t)}} dt,$$

the second kind as

 * $$E(\phi,k) = \int_0^\phi \sqrt{1 - k^2 \sin^2(t)}\,dt,$$

and the third kind as

 * $$\Pi(\phi,n,k) = \int_0^\phi \frac{1}{(1 - n \sin^2(t))\sqrt{1 - k^2 \sin^2(t)}}\,dt.$$

The argument n of the third kind of integral is known as the characteristic, which in different notational conventions can appear as either the first, second or third argument of Π and furthermore is sometimes defined with the opposite sign. The argument order shown above is that of Gradshteyn and Ryzhik as well as Numerical Recipes. The choice of sign is that of Abramowitz and Stegun as well as Gradshteyn and Ryzhik, but corresponds to the $$\scriptstyle{\Pi(\phi,-n,k)}$$ of Numerical Recipes.

The respective complete elliptic integrals are obtained by setting the amplitude, $$\scriptstyle{\phi}$$, the upper limit of the integrals, to $$\scriptstyle{\pi / 2}$$.

The Legendre form of an elliptic curve is given by

 * $$y^2 = x(x - 1)(x - \lambda)$$

Numerical evaluation
The classic method of evaluation is by means of Landen's transformations. Descending Landen transformation decreases the modulus $$\scriptstyle{k}$$ towards zero, while increasing the amplitude $$\scriptstyle{\phi}$$. Conversely, ascending transformation increases the modulus towards unity, while decreasing the amplitude. In either limit of $$\scriptstyle{k}$$ approaching zero or one, the integral is readily evaluated.

Most modern authors recommend evaluation in terms of the Carlson symmetric forms, for which there exist efficient, robust and relatively simple algorithms. This approach has been adopted by Boost C++ Libraries, GNU Scientific Library and Numerical Recipes.

title: Legendre form


text:
The Tall Guy

The Tall Guy is a 1989 British romantic comedy and the feature film debut of screenwriter Richard Curtis and director Mel Smith. It was produced by London Weekend Television for theatrical release and stars Jeff Goldblum, Emma Thompson and Rowan Atkinson. Curtis's script draws from his experiences as straight man to long-time collaborator Rowan Atkinson.

Plot
The protagonist and narrator is Dexter King (Goldblum), an American actor working in London and living platonically in Camden Town with his "educated, charming... nymphomaniac" landlady (played by Geraldine James). He has just finished his sixth year playing "The Tall Guy", a straight man in a two-man, long-running comedy revue starring (and dominated by) Ron Anderson (Rowan Atkinson, playing a role based on himself).

Chronic hay fever prompts him to see a doctor, where he meets and falls quickly in love with Kate (played by Emma Thompson), who works there as a nurse.

Soon after meeting Kate, Dexter is fired by Ron. After being rejected for a role in a new Steven Berkoff play for "lacking anger", Dexter wins the title role in a new Royal Shakespeare Company musical based on The Elephant Man. It's "a sparkily nasty send-up of Andrew Lloyd Webber" called Elephant! which features a song called "He’s Packing His Trunk" and a finale which ends with the lyric "Somewhere up in heaven there's an angel with big ears!"

During rehearsal, Dexter succumbs to the advances of a married co-star (played by Kim Thomson). On the new musical's opening night, Kate puts together evidence of the affair from a few subtle clues, and leaves Dexter without further ado.

After seeing a scene in a televised award show that suggests Ron is now dating Kate, Dexter impulsively gives up his role in Elephant! just before the curtain rises, with plans to make an impassioned plea to Kate to take him back. With Ron's involuntary help (Dexter ties him up in his dressing room and steals his car), Dexter presents his case to Kate in a busy hospital ward. Kate agrees to give him another chance.

Cast
The cast includes:

 * Jeff Goldblum as Dexter King
 * Rowan Atkinson as Ron Anderson
 * Emma Thompson as Kate Lemmon
 * Geraldine James as Carmen
 * Anna Massey as Mary
 * Kim Thomson as Cheryl
 * Hugh Thomas as Dr. Karabekian
 * Emil Wolk as Cyprus Charlie
 * Tim Barlow as Mr. Morrow
 * Harold Innocent as Timothy
 * Joanna Kanska as Tamara
 * Angus Deayton as Actor in Agent's Office
 * Jason Isaacs as 2nd Doctor

The film also includes cameo appearances from John Inman, Melvyn Bragg, Suggs, Jonathan Ross and director Mel Smith.

Production
Principal photography occurred in 1988. Nearly twenty years later, Mel Smith, calling his directorial debut the high point of his career, commented on the directing experience: "I didn't know enough about the film business, and so it seemed wonderfully easy."

Critical reception
Upon its September 1990 US release, Entertainment Weekly gave it a B−, describing it as "mildly charming and mostly too broad" and accusing it of overplaying "Dexter's dorkiness in the same way it overplays the big sex scene, the romantic montage, the breakup scene…" Caryn James of The New York Times wrote "even when its bright theatrical satire gives way to men dressed as nuns dancing in wimples and red sequined shorts, this modest comedy is always wickedly endearing, thanks to the off-kilter characters played by Mr. Goldblum and Emma Thompson as the unlikely woman of his dreams." Roger Ebert called the film a "sweet, whimsical and surprisingly intelligent comedy" whose "last third ...turns into a hilarious sendup of the modern musical" that "must be the funniest deliberately bad play in a movie since Mel Brooks' "Springtime for Hitler" in The Producers." The Deseret News called it a "most uneven romantic comedy," saying "If you're a Monty Python fan, lower your expectations a notch. We're more in Benny Hill territory here....The highlights here are easily the staged "Elephant!" sequences, with some very funny sendups of the gargantuan musicals of Andrew Lloyd Webber."

On review aggregator Rotten Tomatoes, the film has an approval rating of 85% based on 20 reviews and an average rating of 6.7/10. In a 2003 mid-career retrospective about Richard Curtis, The Guardian described the film as being "patronised in one sense by critics while not patronised in the other by audiences." It also identified several tropes from The Tall Guy that would be utilised in his subsequent romantic comedies, (Four Weddings and a Funeral, Notting Hill and Love Actually):
 * romantic lightning strikes that "Curtis seems to believe in as much as any figure in history apart from Cupid";
 * the "willingness to sacrifice realism to a gag" (e.g. turning cartwheels in front of a giant Moon to show that Dexter is in love);
 * the "wacky but wise" flatmate;
 * the use of eccentric obscenities (e.g. Ron's question "What in the name of Judas Iscariot's bumboy is going on?").

Box office
The film grossed $5 million worldwide, including £1,586,383 in the United Kingdom and $510,712 in the United States and Canada.

Soundtrack
The score is by Peter Brewis. The soundtrack also includes Labi Siffre's "It Must Be Love", performed by Madness. The film features a montage in which various characters sing along to this song, and which includes a cameo by Madness frontman Suggs. Other tracks include "Let the Heartaches Begin" (Long John Baldry), "Heartbreak Hotel" (Sam Williams), "Breaking Up Is Hard to Do" and "Crying in the Rain" (both performed by Phil Pope).

Home video
According to WorldCat, the film was released on VHS and Laserdisc in 1990 by RCA/Columbia Pictures Home Video. It was released on VHS in the UK by Virgin Vision, and on DVD in 2003 by Buena Vista Home Entertainment.

title: The Tall Guy


text:
Operation Iron Hammer

Operation Iron Hammer may refer to:

 * Operation Eisenhammer (Iron Hammer), a planned military operation in World War II
 * Operation Iron Hammer (Iraq 2003), a military operation of the multinational force during the Iraq War
 * Operation Iron Hammer (Iraq 2005), a joint U.S.–Iraqi military operation during the subsequent insurgency

title: Operation Iron Hammer


text:
Carlson symmetric form

In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the Legendre forms. The Legendre forms may be expressed in terms of the Carlson forms and vice versa.

The Carlson elliptic integrals are: $$R_F(x,y,z) = \tfrac{1}{2}\int_0^\infty \frac{dt}{\sqrt{(t+x)(t+y)(t+z)}}$$ $$R_J(x,y,z,p) = \tfrac{3}{2}\int_0^\infty \frac{dt}{(t+p)\sqrt{(t+x)(t+y)(t+z)}}$$ $$R_G(x,y,z) = \tfrac{1}{4}\int_0^\infty\frac{1}{\sqrt{(t+x)(t+y)(t+z)}} \biggl(\frac{x}{t+x} + \frac{y}{t+y} + \frac{z}{t+z} \biggr) t\,dt $$ $$R_C(x,y) = R_F(x,y,y) = \tfrac{1}{2} \int_0^\infty \frac{dt}{(t+y)\sqrt{(t+x)}}$$ $$R_D(x,y,z) = R_J(x,y,z,z) = \tfrac{3}{2} \int_0^\infty \frac{dt}{ (t+z) \,\sqrt{(t+x)(t+y)(t+z)}}$$

Since $$R_C$$ and $$R_D$$ are special cases of $$R_F$$ and $$R_J$$, all elliptic integrals can ultimately be evaluated in terms of just $$R_F$$, $$R_J$$, and $$R_G$$.

The term symmetric refers to the fact that in contrast to the Legendre forms, these functions are unchanged by the exchange of certain subsets of their arguments. The value of $$R_F(x,y,z)$$ is the same for any permutation of its arguments, and the value of $$R_J(x,y,z,p)$$ is the same for any permutation of its first three arguments.

The Carlson elliptic integrals are named after Bille C. Carlson (1924-2013).

Incomplete elliptic integrals
Incomplete elliptic integrals can be calculated easily using Carlson symmetric forms:

 * $$F(\phi,k)=\sin\phi R_F\left(\cos^2\phi,1-k^2\sin^2\phi,1\right) $$

 * $$E(\phi,k)=\sin\phi R_F\left(\cos^2\phi,1-k^2\sin^2\phi,1\right) -\tfrac{1}{3}k^2\sin^3\phi R_D\left(\cos^2\phi,1-k^2\sin^2\phi,1\right)$$

 * $$\Pi(\phi,n,k)=\sin\phi R_F\left(\cos^2\phi,1-k^2\sin^2\phi,1\right)+

\tfrac{1}{3}n\sin^3\phi R_J\left(\cos^2\phi,1-k^2\sin^2\phi,1,1-n\sin^2\phi\right)$$

(Note: the above are only valid for $$-\frac{\pi}2\le\phi\le\frac{\pi}2$$ and $$0\le k^2\sin^2\phi\le1$$)

Complete elliptic integrals
Complete elliptic integrals can be calculated by substituting φ = $1/2$π:

 * $$K(k)=R_F\left(0,1-k^2,1\right) $$

 * $$E(k)=R_F\left(0,1-k^2,1\right)-\tfrac{1}{3}k^2 R_D\left(0,1-k^2,1\right)$$

 * $$\Pi(n,k)=R_F\left(0,1-k^2,1\right)+\tfrac{1}{3}n R_J \left(0,1-k^2,1,1-n\right) $$

Special cases
When any two, or all three of the arguments of $$R_F$$ are the same, then a substitution of $$\sqrt{t + x} = u$$ renders the integrand rational. The integral can then be expressed in terms of elementary transcendental functions.

 * $$R_{C}(x,y)

= R_{F}(x,y,y) = \frac{1}{2} \int _{0}^{\infty} \frac{dt}{\sqrt{t + x} (t + y)} = \int _{\sqrt{x}}^{\infty} \frac{du}{u^{2} - x + y} = \begin{cases} \frac{\arccos \sqrt{{x}/{y}}}{\sqrt{y - x}}, & x < y \\ \frac{1}{\sqrt{y}}, & x = y \\ \frac{\operatorname{arcosh} \sqrt{{x}/{y}}}{\sqrt{x - y}}, & x > y \\ \end{cases}$$

Similarly, when at least two of the first three arguments of $$R_J$$ are the same,

 * $$R_{J}(x,y,y,p)

= 3 \int _{\sqrt{x}}^{\infty} \frac{du}{(u^{2} - x + y) (u^{2} - x + p)} = \begin{cases} \frac{3}{p - y} (R_{C}(x,y) - R_{C}(x,p)), & y \ne p \\ \frac{3}{2 (y - x)} \left( R_{C}(x,y) - \frac{1}{y} \sqrt{x}\right), & y = p \ne x \\ \frac{1}{y^{{3}/{2}}}, &y = p = x \\ \end{cases}$$

Homogeneity
By substituting in the integral definitions $$t = \kappa u$$ for any constant $$\kappa$$, it is found that

 * $$R_F\left(\kappa x,\kappa y,\kappa z\right)=\kappa^{-1/2}R_F(x,y,z)$$

 * $$R_J\left(\kappa x,\kappa y,\kappa z,\kappa p\right)=\kappa^{-3/2}R_J(x,y,z,p)$$

Duplication theorem

 * $$R_F(x,y,z)=2R_F(x+\lambda,y+\lambda,z+\lambda)=

R_F\left(\frac{x+\lambda}{4},\frac{y+\lambda}{4},\frac{z+\lambda}{4}\right),$$

where $$\lambda=\sqrt{x}\sqrt{y}+\sqrt{y}\sqrt{z}+\sqrt{z}\sqrt{x}$$.

 * $$\begin{align}R_{J}(x,y,z,p) & = 2 R_{J}(x + \lambda,y + \lambda,z + \lambda,p + \lambda) + 6 R_{C}(d^{2},d^{2} + (p - x) (p - y) (p - z)) \\

& = \frac{1}{4} R_{J}\left( \frac{x + \lambda}{4},\frac{y + \lambda}{4},\frac{z + \lambda}{4},\frac{p + \lambda}{4}\right) + 6 R_{C}(d^{2},d^{2} + (p - x) (p - y) (p - z)) \end{align}$$

where $$d = (\sqrt{p} + \sqrt{x}) (\sqrt{p} + \sqrt{y}) (\sqrt{p} + \sqrt{z})$$ and $$\lambda =\sqrt{x}\sqrt{y}+\sqrt{y}\sqrt{z}+\sqrt{z}\sqrt{x}$$

Series Expansion
In obtaining a Taylor series expansion for $$R_{F}$$ or $$R_{J}$$ it proves convenient to expand about the mean value of the several arguments. So for $$R_{F}$$, letting the mean value of the arguments be $$A = (x + y + z)/3$$, and using homogeneity, define $$\Delta x$$, $$\Delta y$$ and $$\Delta z$$ by

 * $$\begin{align}R_{F}(x,y,z) & = R_{F}(A (1 - \Delta x),A (1 - \Delta y),A (1 - \Delta z)) \\

& = \frac{1}{\sqrt{A}} R_{F}(1 - \Delta x,1 - \Delta y,1 - \Delta z) \end{align}$$

that is $$\Delta x = 1 - x/A$$ etc. The differences $$\Delta x$$, $$\Delta y$$ and $$\Delta z$$ are defined with this sign (such that they are subtracted), in order to be in agreement with Carlson's papers. Since $$R_{F}(x,y,z)$$ is symmetric under permutation of $$x$$, $$y$$ and $$z$$, it is also symmetric in the quantities $$\Delta x$$, $$\Delta y$$ and $$\Delta z$$. It follows that both the integrand of $$R_{F}$$ and its integral can be expressed as functions of the elementary symmetric polynomials in $$\Delta x$$, $$\Delta y$$ and $$\Delta z$$ which are

 * $$E_{1} = \Delta x + \Delta y + \Delta z = 0$$

 * $$E_{2} = \Delta x \Delta y + \Delta y \Delta z + \Delta z \Delta x$$

 * $$E_{3} = \Delta x \Delta y \Delta z$$

Expressing the integrand in terms of these polynomials, performing a multidimensional Taylor expansion and integrating term-by-term...

 * $$\begin{align}R_{F}(x,y,z) & = \frac{1}{2 \sqrt{A}} \int _{0}^{\infty}\frac{1}{\sqrt{(t + 1)^{3} - (t + 1)^{2} E_{1} + (t + 1) E_{2} - E_{3}}} dt \\

& = \frac{1}{2 \sqrt{A}} \int _{0}^{\infty}\left( \frac{1}{(t + 1)^{\frac{3}{2}}} - \frac{E_{2}}{2 (t + 1)^{\frac{7}{2}}} + \frac{E_{3}}{2 (t + 1)^{\frac{9}{2}}} + \frac{3 E_{2}^{2}}{8 (t + 1)^{\frac{11}{2}}} - \frac{3 E_{2} E_{3}}{4 (t + 1)^{\frac{13}{2}}} + O(E_{1}) + O(\Delta^{6})\right) dt \\ & = \frac{1}{\sqrt{A}} \left( 1 - \frac{1}{10} E_{2} + \frac{1}{14} E_{3} + \frac{1}{24} E_{2}^{2} - \frac{3}{44} E_{2} E_{3} + O(E_{1}) + O(\Delta^{6})\right) \end{align}$$

The advantage of expanding about the mean value of the arguments is now apparent; it reduces $$E_{1}$$ identically to zero, and so eliminates all terms involving $$E_{1}$$ - which otherwise would be the most numerous.

An ascending series for $$R_{J}$$ may be found in a similar way. There is a slight difficulty because $$R_{J}$$ is not fully symmetric; its dependence on its fourth argument, $$p$$, is different from its dependence on $$x$$, $$y$$ and $$z$$. This is overcome by treating $$R_{J}$$ as a fully symmetric function of five arguments, two of which happen to have the same value $$p$$. The mean value of the arguments is therefore taken to be

 * $$A = \frac{x + y + z + 2 p}{5}$$

and the differences $$\Delta x$$, $$\Delta y$$ $$\Delta z$$ and $$\Delta p$$ defined by

 * $$\begin{align}R_{J}(x,y,z,p) & = R_{J}(A (1 - \Delta x),A (1 - \Delta y),A (1 - \Delta z),A (1 - \Delta p)) \\

& = \frac{1}{A^{\frac{3}{2}}} R_{J}(1 - \Delta x,1 - \Delta y,1 - \Delta z,1 - \Delta p) \end{align}$$

The elementary symmetric polynomials in $$\Delta x$$, $$\Delta y$$, $$\Delta z$$, $$\Delta p$$ and (again) $$\Delta p$$ are in full

 * $$E_{1} = \Delta x + \Delta y + \Delta z + 2 \Delta p = 0$$

 * $$E_{2} = \Delta x \Delta y + \Delta y \Delta z + 2 \Delta z \Delta p + \Delta p^{2} + 2 \Delta p \Delta x + \Delta x \Delta z + 2 \Delta y \Delta p$$

 * $$E_{3} = \Delta z \Delta p^{2} + \Delta x \Delta p^{2} + 2 \Delta x \Delta y \Delta p + \Delta x \Delta y \Delta z + 2 \Delta y \Delta z \Delta p + \Delta y \Delta p^{2} + 2 \Delta x \Delta z \Delta p$$

 * $$E_{4} = \Delta y \Delta z \Delta p^{2} + \Delta x \Delta z \Delta p^{2} + \Delta x \Delta y \Delta p^{2} + 2 \Delta x \Delta y \Delta z \Delta p$$

 * $$E_{5} = \Delta x \Delta y \Delta z \Delta p^{2}$$

However, it is possible to simplify the formulae for $$E_{2}$$, $$E_{3}$$ and $$E_{4}$$ using the fact that $$E_{1} = 0$$. Expressing the integrand in terms of these polynomials, performing a multidimensional Taylor expansion and integrating term-by-term as before...

 * $$\begin{align}R_{J}(x,y,z,p) & = \frac{3}{2 A^{\frac{3}{2}}} \int _{0}^{\infty}\frac{1}{\sqrt{(t + 1)^{5} - (t + 1)^{4} E_{1} + (t + 1)^{3} E_{2} - (t + 1)^{2} E_{3} + (t + 1) E_{4} - E_{5}}} dt \\

& = \frac{3}{2 A^{\frac{3}{2}}} \int _{0}^{\infty}\left( \frac{1}{(t + 1)^{\frac{5}{2}}} - \frac{E_{2}}{2 (t + 1)^{\frac{9}{2}}} + \frac{E_{3}}{2 (t + 1)^{\frac{11}{2}}} + \frac{3 E_{2}^{2} - 4 E_{4}}{8 (t + 1)^{\frac{13}{2}}} + \frac{2 E_{5} - 3 E_{2} E_{3}}{4 (t + 1)^{\frac{15}{2}}} + O(E_{1}) + O(\Delta^{6})\right) dt \\ & = \frac{1}{A^{\frac{3}{2}}} \left( 1 - \frac{3}{14} E_{2} + \frac{1}{6} E_{3} + \frac{9}{88} E_{2}^{2} - \frac{3}{22} E_{4} - \frac{9}{52} E_{2} E_{3} + \frac{3}{26} E_{5} + O(E_{1}) + O(\Delta^{6})\right) \end{align}$$

As with $$R_{J}$$, by expanding about the mean value of the arguments, more than half the terms (those involving $$E_{1}$$) are eliminated.

Negative arguments
In general, the arguments x, y, z of Carlson's integrals may not be real and negative, as this would place a branch point on the path of integration, making the integral ambiguous. However, if the second argument of $$R_C$$, or the fourth argument, p, of $$R_J$$ is negative, then this results in a simple pole on the path of integration. In these cases the Cauchy principal value (finite part) of the integrals may be of interest; these are

 * $$\mathrm{p.v.}\; R_C(x, -y) = \sqrt{\frac{x}{x + y}}\,R_C(x + y, y),$$

and

 * $$\begin{align}\mathrm{p.v.}\; R_{J}(x,y,z,-p) & = \frac{(q - y) R_{J}(x,y,z,q) - 3 R_{F}(x,y,z) + 3 \sqrt{y} R_{C}(x z,- p q)}{y + p} \\

& = \frac{(q - y) R_{J}(x,y,z,q) - 3 R_{F}(x,y,z) + 3 \sqrt{\frac{x y z}{x z + p q}} R_{C}(x z + p q,p q)}{y + p} \end{align}$$ where

 * $$q = y + \frac{(z - y) (y - x)}{y + p}.$$

which must be greater than zero for $$R_{J}(x,y,z,q)$$ to be evaluated. This may be arranged by permuting x, y and z so that the value of y is between that of x and z.

Numerical evaluation
The duplication theorem can be used for a fast and robust evaluation of the Carlson symmetric form of elliptic integrals and therefore also for the evaluation of Legendre-form of elliptic integrals. Let us calculate $$R_F(x,y,z)$$: first, define $$x_0=x$$, $$y_0=y$$ and $$z_0=z$$. Then iterate the series

 * $$\lambda_n=\sqrt{x_n}\sqrt{y_n}+\sqrt{y_n}\sqrt{z_n}+\sqrt{z_n}\sqrt{x_n},$$
 * $$x_{n+1}=\frac{x_n+\lambda_n}{4}, y_{n+1}=\frac{y_n+\lambda_n}{4}, z_{n+1}=\frac{z_n+\lambda_n}{4}$$

until the desired precision is reached: if $$x$$, $$y$$ and $$z$$ are non-negative, all of the series will converge quickly to a given value, say, $$\mu$$. Therefore,

 * $$R_F\left(x,y,z\right)=R_F\left(\mu,\mu,\mu\right)=\mu^{-1/2}.$$

Evaluating $$R_C(x,y)$$ is much the same due to the relation

 * $$R_C\left(x,y\right)=R_F\left(x,y,y\right).$$

References and External links

 * B. C. Carlson, John L. Gustafson 'Asymptotic approximations for symmetric elliptic integrals' 1993 arXiv
 * B. C. Carlson 'Numerical Computation of Real Or Complex Elliptic Integrals' 1994 arXiv
 * B. C. Carlson 'Elliptic Integrals:Symmetric Integrals' in Chap. 19 of Digital Library of Mathematical Functions. Release date 2010-05-07. National Institute of Standards and Technology.
 * 'Profile: Bille C. Carlson' in Digital Library of Mathematical Functions. National Institute of Standards and Technology.
 * Fortran code from SLATEC for evaluating RF, RJ, RC,  RD,
 * Fortran code from SLATEC for evaluating RF, RJ, RC,  RD,

title: Carlson symmetric form


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