LaTeX to CAS translator

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The following LaTeX input ...

{\displaystyle q_0,}

${\displaystyle q_{0},}$

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Semantic latex: q_0

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Translation: Subscript[q, 0]

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Subscript[q, 0]

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Translation: Symbol('{q}_{0}')

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Symbol('{q}_{0}')

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Symbol('{q}_{0}')

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Translation: q[0]

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q[0]

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q[0]

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$q$

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$q_{0}=1,3,5,\ldots$

$\operatorname {Ec} _{n}^{q_{0}},\operatorname {Es} _{n}^{q_{0}+1},\operatorname {Ec} _{n}^{q_{0}-1},\operatorname {Es} _{n}^{q_{0}}$

$q-q_{0}=\mp 2{\sqrt {\frac {2}{\pi }}}\left({\frac {1+k}{1-k}}\right)^{-\kappa /k}\left({\frac {8\kappa }{1-k^{2}}}\right)^{q_{0}/2}{\frac {1}{[(q_{0}-1)/2]!}}\left[1-{\frac {3(q_{0}^{2}+1)(1+k^{2})}{2^{5}\kappa }}+\cdots \right]$

${\begin{aligned}\Lambda _{\pm }(q)\simeq {}&\Lambda (q_{0})+(q-q_{0})\left({\frac {\partial \Lambda }{\partial q}}\right)_{q_{0}}+\cdots \\[6pt]={}&\Lambda (q_{0})+(q-q_{0})\kappa \left[1-{\frac {q_{0}(1+k^{2})}{2^{2}\kappa }}-{\frac {1}{2^{6}\kappa ^{2}}}\{3(1+k^{2})^{2}(q_{0}^{2}+1)-4k^{2}(q_{0}^{2}+2q_{0}+5)\}+\cdots \right]\\[6pt]\simeq {}&\Lambda (q_{0})\mp 2\kappa {\sqrt {\frac {2}{\pi }}}\left({\frac {1+k}{1-k}}\right)^{-\kappa /k}\left({\frac {8\kappa }{1-k^{2}}}\right)^{q_{0}/2}{\frac {1}{[(q_{0}-1)/2]!}}{\Big [}1-{\frac {1}{2^{5}\kappa }}(1+k^{2})(3q_{0}^{2}+8q_{0}+3)\\[6pt]&{}+{\frac {1}{3.2^{11}\kappa ^{2}}}\{3(1+k^{2})^{2}(9q_{0}^{4}+8q_{0}^{3}-78q_{0}^{2}-88q_{0}-87)\\[6pt]&{}+128k^{2}(2q_{0}^{3}+9q_{0}^{2}+10q_{0}+15)\}-\cdots {\Big ]}.\end{aligned}}$

Description

limit of the Mathieu equation

boundary condition

ellipsoidal wave

period

prime meaning

quarter period

Lamé equation

Müller

expression

expression of the Mathieu case

Complete translation information:

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Wikitext context	{{short description\|Solutions of Lamé's equation}} In mathematics, a '''Lamé function''', or '''ellipsoidal harmonic function''', is a solution of '''Lamé's equation''', a second-order [[ordinary differential equation]]. It was introduced in the paper {{harvs\|first=Gabriel\|last= Lamé\|authorlink= Gabriel Lamé\|year=1837}}. Lamé's equation appears in the method of [[separation of variables]] applied to the [[Laplace equation]] in [[elliptic coordinates]]. In some special cases solutions can be expressed in terms of polynomials called '''Lamé polynomials'''. == The Lamé equation == Lamé's equation is :<math>\frac{d^2y}{dx^2} + (A+B\weierp(x))y = 0, </math> where ''A'' and ''B'' are constants, and <math>\wp</math> is the [[Weierstrass elliptic function]]. The most important case is when <math> B\weierp(x) = - \kappa^2 \operatorname{sn}^2x </math>, where [[sn (elliptic function)\|<math>\operatorname{sn}</math> is the elliptic sine function]], and <math> \kappa^2 = n(n+1)k^2 </math> for an integer ''n'' and <math> k</math> the elliptic modulus, in which case the solutions extend to meromorphic functions defined on the whole complex plane. For other values of ''B'' the solutions have [[branch point]]s. By changing the independent variable to <math>t</math> with <math> t=\operatorname{sn} x</math>, Lamé's equation can also be rewritten in algebraic form as : <math>\frac{d^2y}{dt^2} +\frac{1}{2}\left(\frac{1}{t-e_1}+\frac{1}{t-e_2}+\frac{1}{t-e_3}\right) \frac{dy}{dt} - \frac{A+Bt}{4(t-e_1)(t-e_2)(t-e_3)}y = 0, </math> which after a change of variable becomes a special case of [[Heun's equation]]. A more general form of Lamé's equation is the [[ellipsoidal equation]] or [[ellipsoidal wave equation]] which can be written (observe we now write <math>\Lambda</math>, not <math>A</math> as above) :<math>\frac{d^2y}{dx^2} + (\Lambda - \kappa^2 \operatorname{sn}^2x - \Omega^2k^4 \operatorname{sn}^4x)y = 0, </math> where <math> k </math> is the elliptic modulus of the Jacobian elliptic functions and <math> \kappa</math> and <math> \Omega</math> are constants. For <math>\Omega = 0</math> the equation becomes the Lamé equation with <math>\Lambda = A</math>. For <math> \Omega = 0, k = 0, \kappa = 2h, \Lambda -2h^2 = \lambda, x= z \pm \frac{\pi}{2}</math> the equation reduces to the [[Mathieu equation]] :<math> \frac{d^2y}{dz^2} + (\lambda - 2h^2\cos 2z)y = 0. </math> The Weierstrassian form of Lamé's equation is quite unsuitable for calculation (as Arscott also remarks, p. 191). The most suitable form of the equation is that in Jacobian form, as above. The algebraic and trigonometric forms are also cumbersome to use. Lamé equations arise in quantum mechanics as equations of small fluctuations about classical solutions—called [[periodic instantons]], bounces or bubbles—of Schrödinger equations for various periodic and anharmonic potentials.<ref>H. J. W. Müller-Kirsten, ''Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral'', 2nd ed. World Scientific, 2012, {{isbn\|978-981-4397-73-5}}</ref><ref>{{cite journal \| last=Liang \| first=Jiu-Qing \| last2=Müller-Kirsten \| first2=H.J.W. \| last3=Tchrakian \| first3=D.H. \| title=Solitons, bounces and sphalerons on a circle \| journal=Physics Letters B \| publisher=Elsevier BV \| volume=282 \| issue=1-2 \| year=1992 \| issn=0370-2693 \| doi=10.1016/0370-2693(92)90486-n \| pages=105–110}}</ref> ==Asymptotic expansions== Asymptotic expansions of periodic ellipsoidal wave functions, and therewith also of Lamé functions, for large values of <math>\kappa </math> have been obtained by Müller.<ref>{{cite journal \| last=W. Müller \| first=Harald J. \| title=Asymptotic Expansions of Ellipsoidal Wave Functions and their Characteristic Numbers \| journal=Mathematische Nachrichten \| publisher=Wiley \| volume=31 \| issue=1-2 \| year=1966 \| issn=0025-584X \| doi=10.1002/mana.19660310108 \| pages=89–101 \| language=de}}</ref><ref>{{cite journal \| last=Müller \| first=Harald J. W. \| title=Asymptotic Expansions of Ellipsoidal Wave Functions in Terms of Hermite Functions \| journal=Mathematische Nachrichten \| publisher=Wiley \| volume=32 \| issue=1-2 \| year=1966 \| issn=0025-584X \| doi=10.1002/mana.19660320106 \| pages=49–62 \| language=de}}</ref><ref>{{cite journal \| last=Müller \| first=Harald J. W. \| title=On Asymptotic Expansions of Ellipsoidal Wave Functions \| journal=Mathematische Nachrichten \| publisher=Wiley \| volume=32 \| issue=3-4 \| year=1966 \| issn=0025-584X \| doi=10.1002/mana.19660320305 \| pages=157–172 \| language=de}}</ref> The asymptotic expansion obtained by him for the eigenvalues <math>\Lambda </math> is, with <math> q </math> approximately an odd integer (and to be determined more precisely by boundary conditions – see below), : <math> \begin{align} \Lambda(q) = {} & q\kappa - \frac{1}{2^3}(1+k^2)(q^2+1) - \frac{q}{2^6\kappa}\{(1+k^2)^2(q^2+3) - 4k^2(q^2+5)\} \\[6pt] & {} -\frac{1}{2^{10}\kappa^2} \Big\{(1+k^2)^3(5q^4+34q^2+9) - 4k^2(1+k^2)(5q^4+34q^2+9) \\[6pt] & {} - 384\Omega^2k^4(q^2+1)\Big\} - \cdots , \end{align} </math> (another (fifth) term not given here has been calculated by Müller, the first three terms have also been obtained by Ince<ref>{{cite journal \| last=Ince \| first=E. L. \| title=VII—Further Investigations into the Periodic Lamé Functions \| journal=Proceedings of the Royal Society of Edinburgh \| publisher=Cambridge University Press (CUP) \| volume=60 \| issue=1 \| year=1940 \| issn=0370-1646 \| doi=10.1017/s0370164600020071 \| pages=83–99}}</ref>). Observe terms are alternately even and odd in <math>q</math> and <math>\kappa</math> (as in the corresponding calculations for [[Mathieu function]]s, and [[oblate spheroidal wave functions]] and [[prolate spheroidal wave functions]]). With the following boundary conditions (in which <math> K(k)</math> is the quarter period given by a complete elliptic integral) : <math> \operatorname{Ec}(2K) = \operatorname{Ec}(0) = 0,\;\; \operatorname{Es}(2K) = \operatorname{Es}(0) = 0, </math> as well as (the [[Notation_for_differentiation#Lagrange's_notation\|prime]] meaning derivative) : <math> (\operatorname{Ec})^'_{2K} = (\operatorname{Ec})^'_0 = 0, \;\; (\operatorname{Es})^'_{2K} = (\operatorname{Es})^'_0 = 0, </math> defining respectively the ellipsoidal wave functions : <math>\operatorname{Ec}^{q_0}_n, \operatorname{Es}^{q_0+1}_n, \operatorname{Ec}^{q_0-1}_n, \operatorname{Es}^{q_0}_n</math> of periods <math>4K, 2K, 2K, 4K, </math> and for <math>q_0=1,3,5, \ldots</math> one obtains : <math>q-q_0 = \mp 2\sqrt{\frac{2}{\pi}} \left( \frac{1+k}{1-k}\right)^{-\kappa/k} \left( \frac{8\kappa}{1-k^2}\right)^{q_0/2}\frac{1}{[(q_0-1)/2]!} \left[ 1 - \frac{3(q^2_0+1)(1+k^2)}{2^5\kappa} + \cdots \right]. </math> Here the upper sign refers to the solutions <math>\operatorname{Ec}</math> and the lower to the solutions <math>\operatorname{Es}</math>. Finally expanding <math>\Lambda(q)</math> about <math>q_0,</math> one obtains : <math> \begin{align} \Lambda_{\pm}(q) \simeq {} & \Lambda(q_0) + (q-q_0)\left(\frac{\partial \Lambda}{\partial q} \right)_{q_0} + \cdots \\[6pt] = {} & \Lambda(q_0) +(q-q_0)\kappa \left[1 - \frac{q_0(1+k^2)}{2^2\kappa} - \frac{1}{2^6\kappa^2}\{3(1+k^2)^2(q^2_0+1)-4k^2(q^2_0+2q_0+5)\}+ \cdots\right] \\[6pt] \simeq {} & \Lambda(q_0) \mp 2\kappa\sqrt{\frac{2}{\pi}} \left( \frac{1+k}{1-k} \right)^{-\kappa/k} \left( \frac{8\kappa}{1-k^2}\right)^{q_0/2} \frac{1}{[(q_0-1)/2]!} \Big[ 1 - \frac{1}{2^5\kappa}(1+k^2)(3q^2_0+8q_0+3) \\[6pt] & {} + \frac{1}{3.2^{11}\kappa^2}\{3(1+k^2)^2(9q^4_0+8q^3_0-78q^2_0-88q_0-87) \\[6pt] & {} + 128k^2(2q^3_0+9q^2_0+10q_0+15)\} - \cdots\Big]. \end{align} </math> In the limit of the Mathieu equation (to which the Lamé equation can be reduced) these expressions reduce to the corresponding expressions of the Mathieu case (as shown by Müller). ==Notes== {{Reflist}} ==References== {{Citation \| last = Arscott \| first = F. M. \| title = Periodic Differential Equations \| place = Oxford \| publisher = [[Pergamon Press]] \| pages = 191–236 \| year = 1964 }}. {{Citation \| last = Erdélyi \| first = Arthur \| authorlink = Arthur Erdélyi \| last2 = Magnus \| first2 = Wilhelm \| author2-link = Wilhelm Magnus \| last3 = Oberhettinger \| first3 = Fritz \| last4 = Tricomi \| first4 = Francesco G. \| author4-link = Francesco Tricomi \| title = Higher transcendental functions \| series = Bateman Manuscript Project \| volume = Vol. III \| place = New York–Toronto–London \| publisher = [[McGraw-Hill]] \| pages = XVII + 292 \| year = 1955 \| url = http://apps.nrbook.com/bateman/Vol3.pdf \| mr = 0066496 \| zbl = 0064.06302 }}. {{Citation \| first = G. \| last = Lamé \| author-link = Gabriel Lamé \| title = Sur les surfaces isothermes dans les corps homogènes en équilibre de température \| journal = [[Journal de mathématiques pures et appliquées]] \| volume = 2 \| year = 1837 \| pages = 147–188 \| url = http://gallica.bnf.fr/ark:/12148/bpt6k163818/f155.image }}. Available at [[Gallica]]. {{springer\|id=L/l057400\|first=N. Kh.\|last= Rozov\|title=Lamé equation}} {{springer\|id=L/l057410\|first=N. Kh.\|last= Rozov\|title=Lamé function}} {{dlmf\|id=29\|first=H. \|last=Volkmer}} * {{Citation \| first = Harald J. W. \| last = Müller-Kirsten \|title = Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. \|publisher = World Scientific \|year = 2012 }} {{DEFAULTSORT:Lame function}} [[Category:Special functions]]
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