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季格兰Ishkhanyan

From Sine to Heun: 5 New Functions for Mathematics and Physics in the Wolfram Language

2020年5月6日 -季格兰Ishkhanyan,算法R&d

Mathematicawas initially built to be a universal solver of different mathematical tasks for everything from school-level algebraic equations to complicated problems in real scientific projects. During the past 30 years of development, over250个数学函数已经在系统中实现,并在近期释放12.1版of themanbet万博appWolfram语言,我们增加了更多的人,从小学Sin功能先进Heun功能。

From Sine to Heun: 5 New Functions for Mathematics and Physics in the Wolfram Language

关于我自己有点

My name is Tigran Ishkhanyan, and I am a special functions specialist in the Algorithms R&D department at Wolfram Research, working on general problems of the theory and advanced methods of special functions. I joined Wolfram at the beginning of 2018 when I was working on my数学物理博士项目at theUniversity of Burgundy,法国,并在Institute for Physical Research,亚美尼亚。

我的博士项目有两个主要方向:改善休恩函数理论及其在量子力学中的应用,特别是在两个层面系统和相对论/非相对论波动方程的量子控制的问题。我想出了执行休恩函数到Wolfram语言的想法时,我发现这个功能尚未出台。manbet万博app

初等函数

每个高中学生熟悉简单的功能,如EXP,日志,Sin和其他人,所谓的基本功能。这些functions are well studied and we know all their properties, but from time to time we are able to implement into the Wolfram Language something completely new and insightful like theComplexPlot3D功能可能是用于教育和科学目的。

例如,下面是我们所熟悉的正弦曲线图Sin:

Plot

Plot[Sin[x], {x, -6 \[Pi], 6 \[Pi]}, PlotStyle -> Red]

And here is aPlotof the same function in the complex plane:

ComplexPlot3D

ComplexPlot3D [仙[Z],{Z,-4 \ [PI]  -  2 I,4 \ [PI] + 2 I},PlotLegends  - >自动]

特殊功能

特殊功能集团是基本的人来后,数学函数的另一个子集。特殊功能在过去的几个世纪被广泛应用于数学物理及相关问题。例如,贝塞尔函数描述夫琅禾费衍射等诸多现象的特殊功能。特别是,在振荡行为BesselJmakes it suitable for modeling the oscillations of drums:

Plot

情节[评价[表[BESSELJ [N,x]中,{N,1,3}]],{X,-10,10},灌装 - >轴]

Carl Friedrich Gauss

一般而言,贝塞尔型的功能,要么thogonal polynomials和其他人在类的分组hypergeometric functions: they are particular or limiting cases of different hypergeometric functions. The class of hypergeometric functions has a well-defined hierarchy, with theHypergeometric2F1HypergeometricPFQfunctions standing at the top of this class. The systematic treatment of these functions was first given byCarl Friedrich Gauss

从数学角度看,超几何函数的一般理论十分发达。这些功能曾在科学显著的影响(请浏览文档页面的用于应用程序的例子)的超几何函数。

先进的特殊功能

还有一批先进的特殊功能。该马蒂厄,spheroidal,Heunfunctions are more general than theHypergeometric2F1功能,所以它们是强足以解决像薛定谔方程周期势更加复杂的物理问题:

溶胶= DSolveValue

溶胶= DSolveValue [-w ''[Z] +的Cos [Z] W [Z] ==ℰ瓦特[Z],W [Z],Z]

我们有马修和球体功能,在Wolfram语言,但我们还不具备是班上威享的(作为一个特定的情况下,瘸腿的,或manbet万博app椭圆形,球形谐波)功能。我们已经实施的威享功能缺失组覆盖命名的特殊功能区域,以实现更大的完成,因为其中大部分人要么是特定或限制的休恩函数的情况。它在文学日益普及表明威享类的功能可能是下一代的特殊功能,这将成为未来科学发展的框架。(对于一些不错的参考,请查阅的书目节Heun Project。)

发展观

有发展的数学函数的两个主要方向在Wolfram语言:对于已经在制度和实施的新功能,包括新的功能,方法和计算技术的功能改进文档。manbet万博app

In the first direction, we have recently standardized and significantly improved the documentation pages for the 250+ mathematical functions based on a large collection of more than 5,000 examples so that documentation pages now look like small, well-structured handbooks:

BesselJ

在介绍新功能的方向,我们已经实施了强大asymptotic tools喜欢渐近,渐近DSolveValueAsymptoticIntegrate。对于12.1版本,我们引入了是目前最普遍的特殊功能的10种新威享功能。

I will take a short detour and discuss the relation between mathematical functions and differential equations, since this provides the foundation for my approach to the Heun and other special functions.

通过微分方程生成的函数

许多经典的小学和特殊功能微分方程的特解。万博体育app怎么样事实上,很多的这些功能,试图解决在物理学,天文学等领域出现微分方程首次推出。因此,它们可以被看作是由相关联的微分方程的产生。

例如,exponential functionis generated by a simple first-order differential equation:

DSolveValue

DSolveValue[{w'[z] == w[z], w[0] == 1}, w[z], z]

类似地,下面的线性二阶微分方程来生成勒让德多项式:

DSolveValue

DSolveValue[ w''[z] + (2 z )/(z^2 - 1) w'[z] - (n (n + 1))/(z^2 - 1) w[z] == 0, w[z], z]

我直接与微分方程,而不是他们的特定解决方案的工作的想法的忠实粉丝;万博体育app怎么样这种方法是更有利的,因为微分方程被认为是大数据结构, and we are able to mine a lot of additional information about mathematical functions from their generating differential equations.

现在,线性微分方程的分类是紧密结合的结构连接它们的奇点要么singular points that might either be定期要么irregular:这些是在复平面上,其中所述差分方程的系数发散的点。

For the famous Bessel differential equation:

BesselEq = W”

BesselEq = W ''[Z] + 1 / Z W'[Z] +(Z ^ 2  -  N ^ 2)/ Z ^ 2瓦特[Z];

...定义贝塞尔函数:

DSolveValue

DSolveValue [BesselEq == 0,W [Z],Z]

...点is a regular singular point.

We may generate the solution of a linear differential equation at regular singular points using the尼乌斯方法,即,随着系数服从递推关系唯一地由差分方程定义生成无限术语扩展的幂级数方法。强大的渐近DSolveValue函数给出确切,这些解决方案弗罗贝纽斯:万博体育app怎么样

渐近DSolveValue

渐近DSolveValue[BesselEq == 0, w[z], {z, 0, 4}]

在这里,第一尼乌斯溶液(在奇点常规one)被称为BesselJ而第二个(单数一个)被称为BesselY。Interestingly, this is a rather common situation in the theory of special functions. Of course there are exceptions to this rule, but usually special functions are Frobenius solutions of their generating equations at some regular singular points. For the Gauss hypergeometric equation that is the most general differential equation with three regular singular points located at,:

HypergeometricEq = w

HypergeometricEq = w''[z] + (c/z + (1 + a + b - c)/(z - 1)) w'[z] + (a b)/(z (z - 1)) w[z];

其中一个解决方案弗罗贝纽斯(正规的)被称为万博体育app怎么样Hypergeometric2F1并且是在物理学中最有名的功能之一:

DSolveValue

DSolveValue[HypergeometricEq == 0, w[z], z]

Naturally, the second solution in this output (i.e. the singular one with the pre-factor power function) is the second Frobenius solution of the Gauss hypergeometric equation.

Hypergeometric2F1函数是一个无限的系列;这一系列的系数服从形式的两递推关系:

系列

系列[Hypergeometric2F1 [A,B,C,X],{X,0,3}]

… and there is an exact closed-form expression for then膨胀的个系数。这是所有的超几何函数的一个共同特点。

但一个重要的要点是,对于先进的特殊功能(如休恩函数),至少有三个方面的他们尼乌斯扩张服从递推关系的系数。有这些功能没有一般的封闭形式表达。我们不知道他们的明确形式,显然是被迫工作,有一个更奇点的生成公式。这种额外的定期奇点通向解决方案的显著并发症。万博体育app怎么样

At last, after this brief diversion into the theory of special functions, we are ready to proceed and present the Heun functions.

休恩函数

HEUN的一般微分方程是一个二阶线性微分方程与位于四个普通奇异点,,在复平面上:

HeunEq = W”

HeunEq = w”[z] +(\[伽马]/ z +\[Delta]/(z - 1) + ( 1 + \[Alpha] + \[Beta] - \[Gamma] - \[Delta])/(z - a)) w'[ z] + (\[Alpha] \[Beta] z - q)/(z (z - 1) (z - a)) w[z];

Karl Heun

该general Heun equation is a generalization of the Gauss hypergeometric equation with one more additional regular singular point located at(which is complex), so this equation is a direct generalization of the hypergeometric one with just one more regular singular point. This equation was第一次写于1889年通过Karl Heun, who was a German mathematician.

只有一本书one chapter在里面数学函数的数字图书馆,加上不同的属性和这些一般特殊功能的应用程序约有300家的文章。的休恩函数理论不发达,很多重要的问题仍然是开放的,但正在积极调查。

该general Heun equation has six parameters. Four of them () are the characteristic exponents of Frobenius solutions at different singular points:

渐近DSolveValue

AsymptoticDSolveValue [HeunEq == 0,W [Z],Z  - >∞] // FullSimplify

该parameterstands for the third regular singular point, while the parameter内前者到作为附件或频谱参数是不在超几何函数的情况下可用的一个极其重要的参数。

In analogy with the hypergeometric equation, the regular Frobenius solution of the general Heun equation at a regular singular origin is called。它具有1在复杂的起源和分支切割的不连续的值飞机从运行DirectedInfinity:

DSolveValue

DSolveValue[HeunEq == 0, w[z], z]

用于该参数的值的范围如下所示的休恩函数的曲线图:

{a,

{A,\α,\β,\γ,\ [德尔塔]} = {4 + I,+ -0.6 0.9 I,-0.7 I,-0.18  -  0.03 I,0.3 + 0.6 I};

Plot

情节[评价[表[阿布斯[乡[A,Q,\α,\β,\γ,\ [Δ,Z]],{Q,-20,-3,1}]],{Z,-3/10 9/10},PlotStyle  - >表[{色相[I / 20],厚度[0.002]},{I,20}],PlotRange  - >全部,帧 - >真,轴 - > FALSE]

被简化为Hypergeometric2F1以下设置的参数:

Special case of HeunG

这里的一个小而重要的要点是,虽然的休恩函数的封闭形式是未知的,这些功能不同的功能可能会从微分方程显露出来。例如,变换群的function has 192 members (in total, 192 different local solutions for the general Heun equation, written in terms of a single功能)。

不同于超几何函数,其衍生物是具有偏移参数超几何函数,的休恩函数衍生物是解决更复杂的微分方程更复杂类的特殊功能。这些衍生物在12.1版实现为独立的功能。的衍生物is乡Prime:

D[HeunG

D[HeunG[a, q, \[Alpha], \[Beta], \[Gamma], \[Delta], z], z]

此双功能可以被用于计算中的较高衍生物使用微分方程来消除的高阶导数高于一:

D[HeunG

d [乡[A,Q,\α,\β,\γ,\ [Δ,Z],{Z,2}] //简化

另一个特点是休恩函数是不定积分不能在小学或其他特殊功能来表示:

Integrate

整合[乡[A,Q,\α,\β,\γ,\ [Δ,Z],Z]

Hypergeometric2F1function,has confluent cases when one or more of the regular singular points in the general Heun equation coalesce, generating equations with a different structure of singularities. We recall thatHypergeometric2F1has one confluent case: theHypergeometric1F1function.有四个融合修改叫HeunC,HeunD,HeunBHeunT解决单,双,双 - 和三 - 汇合威享方程,分别。

HeunC有一个宝贵的重要性,因为它概括了MathieuCMathieuS功能,以及像其他人BesselIHypergeometric2F1功能:

Simpler Special Function

一个值得注意的例子是HeunCsolves the generalized spheroidal equation in its general form without specification of the parameter:

溶胶= DSolveValue

溶胶= DSolveValue [(1  -  z ^ 2)W ''[Z]  -  2 Z W'[Z] +(\ [λ1 + \ [伽玛] ^ 2(1  -  z ^ 2) - 平方公尺/(1  -  z ^ 2))W [Z] == 0,W [Z],Z,假设 - > {\ [伽玛]> 0,M> 0}]

Plot

情节[阿布斯[溶胶/。{米 - > 4/3,\ [伽玛]  - > 7/2} /。{C [1]  - > 1/3,C [2]  - > 1/3} /。\ [λ1  - > {-2,-1,0,1,2}] //评估,{Z,-3/4,3/4}]

HeunD在普通的点双汇合威享方程的标准系列溶液:

Plot3D

Plot3D[Abs[ HeunD[q, 0.2 + I, -0.6 + 0.9 I, -0.7 I, 0.3 + 0.6 I, z]], {q, -20, 2}, {z, 1/2, 2}, ColorFunction -> Function[{q, z, HD}, Hue[HD]], PlotRange -> All]

HeunB功能解决了双向融合威享公式:

sol = DSolve

sol = DSolve[ y''[z] + (\[Gamma]/z + \[Delta] + \[Epsilon] z) y'[ z] + (\[Alpha] z - q)/z y[z] == 0, y[z], z]

它围绕以下近似:

terms = Normal@Table

terms=Normal@Table[Series[HeunB[1/31,9/10,1/10,1/10,3/2,z],{z,0,m}],{m,1,5,2}]

这里是近似的一个情节:

Plot

情节[{HeunB [1/31,9/10,1/10,1/10,3/2,Z],术语},{Z,-6,3},PlotRange  - > {-4,8},PlotLegends  - > { “HeunB [q,\α,\γ,\δ,\ [δ,ε,\ Z]”, “第一近似”, “第二近似”, “第三近似”}]

HeunBis truly useful, as different problems of classical and quantum physics are solved using this function. For example, the whole family of doubly anharmonic oscillator potentials (or, in fact, an arbitrary potential up to sixth-order polynomial form):

V[x_]

V[x_] := \[Mu] x^2 + \[Lambda] x^4 + \[Eta] x^6 Plot[V[x] /. {\[Mu] -> -7, \[Lambda] -> -5, \[Eta] -> 1}, {x, -3, 3}]

…是解决了的HeunBfunction:

DSolve

DSolve[-w ''[z] + V[z] w[z] == ℰ w[z], w[z], z]

......而normalizable束缚态的问题仍然没有解决。

最后汇合威享功能,HeunTfunction, which might be considered as a generalization of theAiryfunctions, is the solution of the tri-confluent Heun equation:

DSolve

DSolve[ y''[ z] + (\[Gamma] + \[Delta] z + \[Epsilon] z^2) y'[ z] + (\[Alpha] z - q) y[z] == 0, y[z], z]

HeunTsolves the classical anharmonic oscillator problem (in fact, the quartic potential):

sol = DSolve

溶胶= DSolve [U ''[Z] +(下标[\ [λ1,1] +下标[\ [λ1,2] Z 1 2 +下标[\ [λ1,4] Z 1 4)U [Z] == 0,U [Z],Z]

我们能够使用模拟振荡器的动态HeunT功能:

{Subscript

{下标[\ [λ1,1]中,下标[\ [λ1,2]中,下标[\ [λ1,4]} = {1,1/2,1/4};情节[{U [Z] /。溶胶/。{C [1]  - > 1,C [2]  - > 1}},{Z,0,9/2}]

Surprisingly (or not?) the “primes” of the Heun functions are independent actors and have important applications in science.

沃尔夫勒manbet万博app姆语言也有MeijerGsuperfunction,提供了强大的工具集和多种功能:

MeijerG

MeijerG[{{}, {}}, {{v}, {-v}}, z]

Unfortunately, theMeijerGrepresentations of special functions are limited to the hypergeometric class of functions and are not applicable in the Heun case (as well as Mathieu and spheroidal cases).

这些和a lot of other interesting examples on the properties and applications of the Heun functions are noted in the文档页面

在物理休恩函数

休恩函数有一系列的现代物理中的应用和功能强大足以产生一个显著组来自量子力学,黑洞理论,共形场论和其他悬而未决问题的解决方案。万博体育app怎么样他们被在真实的物理问题,以极快的速度成功应用:在过去十年中,涉及到的休恩函数理论出版物的数量相比增加了两倍和所有其他出版物,直到2010年,根据arXiv

具体而言,的休恩函数的强大设备允许的新无限类量子控制和工程的不同的问题使用相对论和非相对论波动方程积电位的推导(请参阅recent paperby A. M. Ishkhanyan for different examples).

休恩函数出现在克尔德西特黑洞理论和更复杂的几何形状可用于分析(上发表学术论文R. S. Borissov和P. P. FizievH. Suzuki, E. Takasugi and H. Umetsu讨论这些问题)。

在威享类方程的Painlevé之间的关系transcendents导致基于威享方程解的分析二维共形场理论的新成果(见的论文万博体育app怎么样B. C.达库尼亚和J. P.卡瓦尔坎特F.阿泰和E. Langmann).

上述例子以及其他表明威享功能是解决当代物理学完全不同的问题,在一些重要和流行。

Closing Words

At Wolfram, we are in a constant search for fresh ideas and methods that make the Wolfram Language one of the most famous, popular, powerful and user-friendly tools for scientists working in different areas of contemporary science.

不时,数学工具集进行更新,以适应新的问题和挑战。二十世纪的量子力学是密切相关的超几何级的功能,但该组的解具有这些特殊功能的问题,在很大程度上是用尽,因此需要新一代的功能。这就是为什么在Wolfram语言的12.1版,我们实施了威享功能和计划,manbet万博app不断提高先进的特殊功能的覆盖面,以满足在未来更复杂的科学挑战。

可以完全访问最新的Wolfram语言的功能与manbet万博app数学12.1要么manbet万博app沃尔弗拉姆|一审判。

Leave a Comment

6 Comments


乔治·伍德罗III

I am very happy to see continued development of *maths* in Mathematica. Mathematica was originally billed as “A system for Doing Mathematics by Computer”. While the range of what Wolfram Language can do has expanded greatly, at the core is mathematics.

发贴者伍德罗·乔治三世2020可7日在上午8:40
    季格兰Ishkhanyan

    谢谢你的鼓励的话!威享功能是现代科学很受欢迎,所以他们的执行是我们的一个重要任务。

    Posted by Tigran Ishkhanyan May 8, 2020 at 9:04 am
M.Iwaniuk

List important function for physicists and mathematicians:
聚赫尔维茨Zeta函数。
GeneralizedPolylog功能。
MultiPolylog function.
MultiZeta function.
该AppellF2功能。
该AppellF3功能。
该AppellF4 function.
SOPHOMORE’S DREAM FUNCTION
Lommel Function.
MacRobert’s E-Function.
Lauricella Functions
申佩德Fériet功能。
福克斯H-功能。
Horn Function.
莱特功能。
Leaky aquifer function.
瘸功能。
瘸-Wangerin functions.
Epstein Zeta Function.
Q-β函数。

发贴者M.Iwaniuk 2020年5月9日在上午10:12
    季格兰Ishkhanyan

    Thank you for you comment! We constantly update the functionality of Wolfram Language, some of these functions are currently in development being prepared for future Mathematica versions.

    发贴者季格兰Ishkhanyan 2020可15日上午06时06分
SYD拉蒂

非常感谢您的优秀的补充,以数学。我很欣赏结构良好的博客文章和翔实的背景资料。祝您今后的工作。

发贴者西德妮拉蒂2020年5月9日在下午3时36分
    季格兰Ishkhanyan

    Thank you!

    发贴者季格兰Ishkhanyan 2020可11日,在下午2:14


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