The Heun Biconfluent function

はじめる前に
新機能一覧
Maple ワークシートの作成
Mapleワークシートを共有
Maple ウィンドウのカスタマイズ
Maple システムのカスタマイズ
コネクティビティ
Mathematics
数学
- テンソル解析
- パッケージ
- ファンクションアドバイザ
- ベキ級数
- ベクトル解析
- 不活性関数
- 代数
- 因数分解と方程式解法
- 基本数学
- 基本的な注意事項
- 変分法
- 変換
  - 関数
  - 関数のクラス
  - 特殊関数への変換
  - 関数のルール
- 幾何
- 微分代数方程式
- 微分幾何学
- 微分方程式
- 微積分
- 数
- 数値計算
- 数学関数
  - MathematicalFunctions パッケージ
  - ファンクションアドバイザ
  - 関数の定義
  - abs
  - AiryBi
  - AiryBiZeros
  - arctanh
  - argument
  - bernoulli コマンド
  - BesselYZeros
  - Beta
  - binomial
  - ChebyshevT
  - ChebyshevU
  - CoulombF
  - csgn
  - CylinderV
  - dawson
  - dilog
  - Dirac
  - Ei
  - EllipticE
  - EllipticK
  - EllipticModulus
  - EllipticNome
  - EllipticPi
  - erfi
  - euler コマンド
  - exp
  - factorial
  - FresnelS
  - GaussAGM
  - GegenbauerC
  - HankelH2
  - harmonic
  - Heaviside
  - HermiteH
  - Heun 関数
  - HeunBPrime
  - HeunCPrime
  - HeunDPrime
  - HeunGPrime
  - HeunTPrime
  - hypergeom
  - ilog2
  - IncompleteBellB
  - InverseJacobiSN
  - JacobiP
  - JacobiSN
  - JacobiTheta4
  - JacobiZeta
  - KelvinKer
  - KummerU
  - LaguerreL
  - LambertW
  - LegendreQ
  - LerchPhi
  - Li
  - lnGAMMA
  - log10
  - LommelS2
  - MathieuSPrime
  - MeijerG
  - min コマンド
  - ModifiedMeijerG
  - pochhammer
  - polylog
  - Psi
  - Re
  - RiemannTheta
  - sign
  - signum
  - SphericalY
  - Ssi
  - Stirling1
  - Stirling2
  - StruveL
  - surd
  - tanh
  - trunc
  - WeberE
  - WeierstrassZeta
  - WhittakerW
  - Wrightomega
  - Zeta
  - マシュー関数の例
  - 共役
  - 平方根
  - 極
- 数論
- 最適化
- 特殊関数
  - Integrals
  - MathematicalFunctions
  - ファンクションアドバイザ
  - 例題
  - AiryBi
  - AiryBiZeros
  - bernoulli コマンド
  - BesselYZeros
  - Beta
  - ChebyshevT
  - ChebyshevU
  - CoulombF
  - CylinderV
  - dilog
  - Dirac
  - EllipticModulus
  - erfi
  - euler コマンド
  - Fresnelg
  - GaussAGM
  - GegenbauerC
  - HankelH2
  - harmonic
  - Heaviside
  - HermiteH
  - Heun 関数
  - HeunBPrime
  - HeunCPrime
  - HeunDPrime
  - HeunGPrime
  - HeunTPrime
  - hypergeom
  - InverseJacobiSN
  - JacobiP
  - JacobiSN
  - JacobiTheta4
  - JacobiZeta
  - KelvinKer
  - KummerU
  - LaguerreL
  - LambertW
  - LegendreQ
  - LerchPhi
  - lnGAMMA
  - LommelS2
  - MathieuSEPrime
  - MeijerG
  - ModifiedMeijerG
  - pochhammer
  - polylog
  - Psi
  - RiemannTheta
  - SphericalY
  - StruveL
  - WeberE
  - WeierstrassZeta
  - WhittakerW
  - Wrightomega
  - Zeta
  - マシューの例
- 統計
- 線形代数
- 群論
- 評価
- 論理
- 過去のパッケージ
- 金融
- 離散数学
- piecewise examples
- グラフ電卓
- 区分関数
物理
プログラミング
グラフィックス
Student パッケージ
Science and Engineering
科学・エンジニアリング
アプリケーションと例題
リファレンス
システム
MapleSim
MapleSim ツールボックス
MapleSim ヘルプ
Tasks
Toolboxes
エラーメッセージガイド
マニュアル
数学アプリ

HeunB - The Heun Biconfluent function

HeunBPrime - The derivative of the Heun Biconfluent function

	Calling Sequence
	HeunB(, , , , ) HeunBPrime(, , , , )

Parameters

	-	algebraic expression
	-	algebraic expression
	-	algebraic expression
	-	algebraic expression
z	-	algebraic expression

Description

•	The HeunB function is the solution of the Heun Biconfluent equation. Following the first reference (at the end), the equation and the conditions at the origin satisfied by HeunB are

>	FunctionAdvisor(definition, HeunB);

$HeunB(alpha, beta, gamma, delta, z) = DESol({diff(_Y(z), `$`(z, 2))-(-alpha+beta*z+2*z^2-1)*(diff(_Y(z), z))/z-(1/2)*((-2*gamma+2*alpha+4)*z+delta+beta+beta*alpha)*_Y(z)/z}, {_Y(z)}, {_Y(0) = 1, (D(_Y))(0) = (delta+beta*alpha+beta)/(2*alpha+2)})$

(1)

•	The HeunB(, , , , z) function is a local (Frobenius) solution to Heun's Biconfluent equation, computed as a power series expansion around the origin, a regular singular point. Because the next singularity is located at , this series converges in the whole complex plane.

•

The Biconfluent Heun Equation (BHE) above is obtained from the Confluent Heun Equation (CHE) through a confluence process, that is, a process where two singularities coalesce, performed by redefining parameters and taking limits. In this case one regular singularity of the CHE is coalesced with its irregular singularity at . The resulting Heun Biconfluent equation, thus, has one regular singularity at the origin, one irregular one at , and includes as a particular case the 1F1 hypergeometric confluent equation

>	DEtools[hyperode]( hypergeom([a],[c],z), y(z) ) = 0;

a*y(z)+(z-c)*(diff(y(z), z))-z*(diff(y(z), `$`(z, 2))) = 0

(2)

So besides the standard hypergeometric solution of this equation, a solution expressed in terms of HeunB functions can also be constructed, and in this way HeunB contains as particular cases all the hypergeometric functions of the 1F1 class. Some of these specializations are listed at the end of the Examples section.

•

A special case happens when in HeunB(, , , , z) the third parameter satisfies , where is a positive integer. In this case the th coefficient in the series expansion is a polynomial of degree in . When is a root of this polynomial, the th and subsequent coefficients cancel and the series truncates, resulting in a polynomial form of degree for HeunB.

Examples

Heun's Biconfluent equation,

BHE := diff(y(z), z, z) = (2*z^2-1-alpha+beta*z)*(diff(y(z), z))/z+((-2*gamma+2*alpha+4)*z+delta+beta+beta*alpha)*y(z)/(2*z)

BHE := diff(y(z), `$`(z, 2)) = (2*z^2-1-alpha+beta*z)*(diff(y(z), z))/z+(1/2)*((-2*gamma+2*alpha+4)*z+delta+beta+beta*alpha)*y(z)/z

(3)

can be transformed into another version of itself, that is, an equation with one regular and one irregular singularity respectively located at 0 and through transformations of the form

z = kappa*t, y(z) = z^((1/2)*(epsilon-1)*alpha)*exp((1/2)*(z+beta)*(1-kappa^2)*z)*u(z)

(4)

where are new variables, and . Under this transformation, the HeunB parameters transform according to -> , -> , -> and -> . These transformations form a group and imply on a number of identities, among which you have

(5)

A relation between HeunB and the confluent 1F1 hypergeometric function is

[[HeunB(alpha, beta, gamma, delta, z) = hypergeom([1/2-(1/4)*alpha-(1/4)*gamma], [1-(1/2)*alpha], z^2)/z^alpha, And(alpha::negint, beta = 0, delta = 0)], [HeunB(alpha, beta, gamma, delta, z) = hypergeom([1/2+(1/4)*alpha-(1/4)*gamma], [1+(1/2)*alpha], z^2), And(alpha::(Not(negint)), beta = 0, delta = 0)]]

(6)

When, in HeunB(,,,,z), , with a positive integer, the th coefficient in the series expansion is a polynomial in of order . If is a root of that polynomial, that th coefficient and the subsequent ones are zero. The series then truncates and HeunB reduces to a polynomial. For example, this is the necessary condition for a polynomial form

(7)

Considering the first non-trivial case, for , the function is

(8)

So the coefficient of in the series expansion is

Q := series(1+((delta+beta*alpha+beta)/(2*alpha+2))*z+((1/8)*(beta^2*alpha^2+(-8+2*delta*beta+4*beta^2)*alpha-8+delta^2+3*beta^2+4*delta*beta)/((alpha+1)*(alpha+2)))*z^2+O(z^3),z,3)

(9)

c2 := (1/8)*(beta^2*alpha^2+(-8+2*delta*beta+4*beta^2)*alpha-8+delta^2+3*beta^2+4*delta*beta)/((alpha+1)*(alpha+2))

(10)

solving for , requesting from solve to return using RootOf, you have

(11)

(12)

substituting in we have

(13)

When the function admits a polynomial form, as is the case of by construction, to obtain the actual polynomial of degree (in this case ) use

1+(RootOf(_Z^2+(4*beta+2*beta*alpha)*_Z-8*alpha-8+4*beta^2*alpha+beta^2*alpha^2+3*beta^2)+beta*alpha+beta)*z/(2*alpha+2)

(14)

(15)

	See Also
	FunctionAdvisor, Heun, HeunC, HeunD, HeunG, HeunT, hypergeom

References

Decarreau, A.; Dumont-Lepage, M.C.; Maroni, P.; Robert, A.; and Ronveaux, A. "Formes Canoniques de Equations confluentes de l'equation de Heun." Annales de la Societe Scientifique de Bruxelles. Vol. 92 I-II, (1978): 53-78.