Dawson function

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:

$R_{F}(x,y,z)={\tfrac {1}{2}}\int _{0}^{\infty }{\frac {dt}{\sqrt {(t+x)(t+y)(t+z)}}}$

$R_{J}(x,y,z,p)={\tfrac {3}{2}}\int _{0}^{\infty }{\frac {dt}{(t+p){\sqrt {(t+x)(t+y)(t+z)}}}}$

$R_{C}(x,y)=R_{F}(x,y,y)={\tfrac {1}{2}}\int _{0}^{\infty }{\frac {dt}{(t+y){\sqrt {(t+x)}}}}$

$R_{D}(x,y,z)=R_{J}(x,y,z,z)={\tfrac {3}{2}}\int _{0}^{\infty }{\frac {dt}{(t+z)\,{\sqrt {(t+x)(t+y)(t+z)}}}}$

Since $\scriptstyle {R_{C}}$ and $\scriptstyle {R_{D}}$ are special cases of $\scriptstyle {R_{F}}$ and $\scriptstyle {R_{J}}$, all elliptic integrals can ultimately be evaluated in terms of just $\scriptstyle {R_{F}}$ and $\scriptstyle {R_{J}}$.

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

The Carlson elliptic integrals are named after Bille C. Carlson.

Relation to the Legendre forms

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 $0\leq \phi \leq 2\pi$ and $0\leq k^{2}\sin ^{2}\phi \leq 1$)

Complete elliptic integrals

Complete elliptic integrals can be calculated by substituting φ = Template:Fracπ:

$K(k)=R_{F}\left(0,1-k^{2},1\right)$

$E(k)=R_{F}\left(0,1-k^{2},1\right)-{\tfrac {1}{3}}k^{2}R_{D}\left(0,1-k^{2},1\right)$

$\Pi (n,k)=R_{F}\left(0,1-k^{2},1\right)+{\tfrac {1}{3}}nR_{J}\left(0,1-k^{2},1,1-n\right)$

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.

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

Similarly, when at least two of the first three arguments of $R_{J}$ are the same,

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

Properties

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 $\lambda ={\sqrt {xy}}+{\sqrt {yz}}+{\sqrt {zx}}$.

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

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

Series Expansion

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

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

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

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

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

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

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

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

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

The elementary symmetric polynomials in $\scriptstyle {\Delta x}$, $\scriptstyle {\Delta y}$, $\scriptstyle {\Delta z}$, $\scriptstyle {\Delta p}$ and (again) $\scriptstyle {\Delta p}$ 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 $\scriptstyle {E_{2}}$, $\scriptstyle {E_{3}}$ and $\scriptstyle {E_{4}}$ using the fact that $\scriptstyle {E_{1}=0}$. Expressing the integrand in terms of these polynomials, performing a multidimensional Taylor expansion and integrating term-by-term as before...

${\begin{aligned}R_{J}(x,y,z,p)&={\frac {3}{2A^{\frac {3}{2}}}}\int _{0}^{\infty }{\frac {1}{\sqrt {(t+1)^{5}-(t+1)^{4}E_{1}+(t+1)^{3}E_{2}-(t+1)^{2}E_{3}+(t+1)E_{4}-E_{5}}}}dt\\&={\frac {3}{2A^{\frac {3}{2}}}}\int _{0}^{\infty }\left({\frac {1}{(t+1)^{\frac {5}{2}}}}-{\frac {E_{2}}{2(t+1)^{\frac {9}{2}}}}+{\frac {E_{3}}{2(t+1)^{\frac {11}{2}}}}+{\frac {3E_{2}^{2}-4E_{4}}{8(t+1)^{\frac {13}{2}}}}+{\frac {2E_{5}-3E_{2}E_{3}}{4(t+1)^{\frac {15}{2}}}}+O(E_{1})+O(\Delta ^{6})\right)dt\\&={\frac {1}{A^{\frac {3}{2}}}}\left(1-{\frac {3}{14}}E_{2}+{\frac {1}{6}}E_{3}+{\frac {9}{88}}E_{2}^{2}-{\frac {3}{22}}E_{4}-{\frac {9}{52}}E_{2}E_{3}+{\frac {3}{26}}E_{5}+O(E_{1})+O(\Delta ^{6})\right)\end{aligned}}$

As with $\scriptstyle {R_{J}}$, by expanding about the mean value of the arguments, more than half the terms (those involving $\scriptstyle {E_{1}}$) 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 $\scriptstyle {R_{C}}$, or the fourth argument, p, of $\scriptstyle {R_{J}}$ 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

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

where

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

which must be greater than zero for $\scriptstyle {R_{J}(x,y,z,q)}$ 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

$\lambda _{n}={\sqrt {x_{n}y_{n}}}+{\sqrt {y_{n}z_{n}}}+{\sqrt {z_{n}x_{n}}},$

$x_{n+1}={\frac {x_{n}+\lambda _{n}}{4}},y_{n+1}={\frac {y_{n}+\lambda _{n}}{4}},z_{n+1}={\frac {z_{n}+\lambda _{n}}{4}}$

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.

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

• Fortran code from SLATEC for evaluating RF, RJ, RC, RD,