31 Heun Functions Properties 31.1 Special Notation 31.3 Basic Solutions

§31.2 Differential Equations

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Permalink:: http://dlmf.nist.gov/31.2
See also:: Annotations for Ch.31

§31.2(i) Heun’s Equation
§31.2(ii) Normal Form of Heun’s Equation
§31.2(iii) Trigonometric Form
§31.2(iv) Doubly-Periodic Forms
§31.2(v) Heun’s Equation Automorphisms

§31.2(i) Heun’s Equation

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Keywords:: Heun equation, Heun’s equation, accessory parameter, classification, classification of parameters, exponent parameters, parameters, singularities, singularity parameter
Notes:: See Erdélyi et al. (1955, Chapter XV) and Ronveaux (1995, Part A, p. 7).
Permalink:: http://dlmf.nist.gov/31.2.i
See also:: Annotations for §31.2 and Ch.31

31.2.1		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\frac{\epsilon}{z-a}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{% \alpha\beta z-q}{z(z-1)(z-a)}w=0,$
$\alpha+\beta+1=\gamma+\delta+\epsilon$ .
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter Referenced by: §29.2(ii), §31.11(iii), §31.12, §31.13, §31.14(i), §31.17(i), §31.18, §31.2(v), §31.2(v), §31.2(v), §31.2(i), §31.3(i), §31.3(i), §31.3(ii), §31.3(ii), §31.3(ii), §31.3(iii), §31.3(iii), §31.4, §31.5, §31.6, §31.7(ii), §31.8, §31.8, §31.8 Permalink: http://dlmf.nist.gov/31.2.E1 Encodings: TeX, pMML, png See also: Annotations for §31.2(i), §31.2 and Ch.31

This equation has regular singularities at $0,1,a,\infty$ , with corresponding exponents $\{0,1-\gamma\}$ , $\{0,1-\delta\}$ , $\{0,1-\epsilon\}$ , $\{\alpha,\beta\}$ , respectively (§2.7(i)). All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, $\mathbb{C}\cup\{\infty\}$ , can be transformed into (31.2.1).

The parameters play different roles: $a$ is the singularity parameter; $\alpha,\beta,\gamma,\delta,\epsilon$ are exponent parameters; $q$ is the accessory parameter. The total number of free parameters is six.

§31.2(ii) Normal Form of Heun’s Equation

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Keywords:: Heun’s equation, normal form
Permalink:: http://dlmf.nist.gov/31.2.ii
See also:: Annotations for §31.2 and Ch.31

31.2.2		$w(z)=z^{-\gamma/2}(z-1)^{-\delta/2}(z-a)^{-\epsilon/2}W(z),$
ⓘ Symbols: $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter and $W(z)$ : solution Permalink: http://dlmf.nist.gov/31.2.E2 Encodings: TeX, pMML, png See also: Annotations for §31.2(ii), §31.2 and Ch.31

31.2.3		$\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}=\left(\frac{A}{z}+\frac{B}{z-1}+% \frac{C}{z-a}+\frac{D}{z^{2}}+\frac{E}{(z-1)^{2}}+\frac{F}{(z-a)^{2}}\right)W,$
$A+B+C=0$ ,
ⓘ Defines: $W(z)$ : solution (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : complex parameter, $A$ , $B$ , $C$ , $D$ , $E$ and $F$ Permalink: http://dlmf.nist.gov/31.2.E3 Encodings: TeX, pMML, png See also: Annotations for §31.2(ii), §31.2 and Ch.31

31.2.4	$\displaystyle A$	$\displaystyle=-\frac{\gamma\delta}{2}-\frac{\gamma\epsilon}{2a}+\frac{q}{a},$
	$\displaystyle B$	$\displaystyle=\frac{\gamma\delta}{2}-\frac{\delta\epsilon}{2(a-1)}-\frac{q-% \alpha\beta}{a-1},$
	$\displaystyle C$	$\displaystyle=\frac{\gamma\epsilon}{2a}+\frac{\delta\epsilon}{2(a-1)}-\frac{a% \alpha\beta-q}{a(a-1)},$
	$\displaystyle D$	$\displaystyle=\tfrac{1}{2}\gamma\left(\tfrac{1}{2}\gamma-1\right),$
	$\displaystyle E$	$\displaystyle=\tfrac{1}{2}\delta\left(\tfrac{1}{2}\delta-1\right),$
	$\displaystyle F$	$\displaystyle=\tfrac{1}{2}\epsilon\left(\tfrac{1}{2}\epsilon-1\right).$
ⓘ Defines: $A$ (locally), $B$ (locally), $C$ (locally), $D$ (locally), $E$ (locally) and $F$ (locally) Symbols: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter Permalink: http://dlmf.nist.gov/31.2.E4 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png See also: Annotations for §31.2(ii), §31.2 and Ch.31

§31.2(iii) Trigonometric Form

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Keywords:: Heun’s equation, trigonometric form
Permalink:: http://dlmf.nist.gov/31.2.iii
See also:: Annotations for §31.2 and Ch.31

31.2.5		$z={\sin}^{2}\theta,$
ⓘ Symbols: $\sin\NVar{z}$ : sine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/31.2.E5 Encodings: TeX, pMML, png See also: Annotations for §31.2(iii), §31.2 and Ch.31

31.2.6		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\theta}^{2}}+\left({(2\gamma-1)\cot\theta-% (2\delta-1)\tan\theta}-\frac{\epsilon\sin\left(2\theta\right)}{a-{\sin}^{2}% \theta}\right)\frac{\mathrm{d}w}{\mathrm{d}\theta}+4\frac{\alpha\beta{\sin}^{2% }\theta-q}{a-{\sin}^{2}\theta}w=0.$
ⓘ Symbols: $\cot\NVar{z}$ : cotangent function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter Permalink: http://dlmf.nist.gov/31.2.E6 Encodings: TeX, pMML, png See also: Annotations for §31.2(iii), §31.2 and Ch.31

§31.2(iv) Doubly-Periodic Forms

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Notes:: See Ronveaux (1995, Part A, p. 26).
Permalink:: http://dlmf.nist.gov/31.2.iv
See also:: Annotations for §31.2 and Ch.31

Jacobi’s Elliptic Form

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Keywords:: Heun’s equation, Jacobi’s elliptic, Jacobi’s elliptic form, doubly-periodic forms
See also:: Annotations for §31.2(iv), §31.2 and Ch.31

With the notation of §22.2 let

31.2.7	$\displaystyle a$	$\displaystyle=k^{-2},$
31.2.7	$\displaystyle z$	$\displaystyle={\operatorname{sn}}^{2}\left(\zeta,k\right).$
ⓘ Defines: $k$ : modulus (locally) Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex variable, $a$ : complex parameter and $\zeta$ : change of variable Permalink: http://dlmf.nist.gov/31.2.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

Then (suppressing the parameter $k$ )

31.2.8		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\zeta}^{2}}+\left((2\gamma-1)\frac{% \operatorname{cn}\zeta\operatorname{dn}\zeta}{\operatorname{sn}\zeta}-(2\delta% -1)\frac{\operatorname{sn}\zeta\operatorname{dn}\zeta}{\operatorname{cn}\zeta}% -(2\epsilon-1)k^{2}\frac{\operatorname{sn}\zeta\operatorname{cn}\zeta}{% \operatorname{dn}\zeta}\right)\frac{\mathrm{d}w}{\mathrm{d}\zeta}+4k^{2}(% \alpha\beta{\operatorname{sn}}^{2}\zeta-q)w=0.$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter, $\zeta$ : change of variable and $k$ : modulus Referenced by: §31.7(ii) Permalink: http://dlmf.nist.gov/31.2.E8 Encodings: TeX, pMML, png See also: Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

Weierstrass’s Form

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Keywords:: Heun’s equation, Weierstrass’s, Weierstrass’s form, doubly-periodic forms
See also:: Annotations for §31.2(iv), §31.2 and Ch.31

With the notation of §§19.2(ii) and 23.2 let

31.2.9	$\displaystyle k^{2}$	$\displaystyle=(e_{2}-e_{3})/(e_{1}-e_{3}),$
	$\displaystyle\zeta$	$\displaystyle=\mathrm{i}{K^{\prime}}+\xi(e_{1}-e_{3})^{1/2},$
	$\displaystyle e_{1}$	$\displaystyle=\wp\left(\omega_{1}\right),$
	$\displaystyle e_{2}$	$\displaystyle=\wp\left(\omega_{2}\right),$
	$\displaystyle e_{3}$	$\displaystyle=\wp\left(\omega_{3}\right),$
	$\displaystyle e_{1}+e_{2}+e_{3}=0,$
ⓘ Defines: $\zeta$ : change of variable (locally) Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\omega_{j}$ : nonzero complex number, $\mathbb{L}$ : lattice and $k$ : modulus Permalink: http://dlmf.nist.gov/31.2.E9 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png See also: Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

where $2\omega_{1}$ and $2\omega_{3}$ with $\Im\left(\omega_{3}/\omega_{1}\right)>0$ are generators of the lattice $\mathbb{L}$ for $\wp\left(z|\mathbb{L}\right)$ . Then

31.2.10		$w(\xi)=\left(\wp\left(\xi\right)-e_{3}\right)^{(1-2\gamma)/4}\left(\wp\left(% \xi\right)-e_{2}\right)^{(1-2\delta)/4}\*\left(\wp\left(\xi\right)-e_{1}\right% )^{(1-2\epsilon)/4}W(\xi),$
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\mathbb{L}$ : lattice and $W(\xi)$ : solution Permalink: http://dlmf.nist.gov/31.2.E10 Encodings: TeX, pMML, png See also: Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

where $W(\xi)$ satisfies

31.2.11		$\ifrac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}+\left(H+b_{0}\wp\left(\xi\right% )+b_{1}\wp\left(\xi+\omega_{1}\right)+b_{2}\wp\left(\xi+\omega_{2}\right)+b_{3% }\wp\left(\xi+\omega_{3}\right)\right)W=0,$
ⓘ Defines: $W(\xi)$ : solution (locally) Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\omega_{j}$ : nonzero complex number, $\mathbb{L}$ : lattice, $b_{j}$ : coefficients and $H$ Permalink: http://dlmf.nist.gov/31.2.E11 Encodings: TeX, pMML, png See also: Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

with

31.2.12	$\displaystyle b_{0}$	$\displaystyle=4\alpha\beta-(\gamma+\delta+\epsilon-\tfrac{1}{2})(\gamma+\delta% +\epsilon-\tfrac{3}{2}),$
	$\displaystyle b_{1}$	$\displaystyle=-(\epsilon-\tfrac{1}{2})(\epsilon-\tfrac{3}{2}),$
	$\displaystyle b_{2}$	$\displaystyle=-(\delta-\tfrac{1}{2})(\delta-\tfrac{3}{2}),$
	$\displaystyle b_{3}$	$\displaystyle=-(\gamma-\tfrac{1}{2})(\gamma-\tfrac{3}{2}),$
	$\displaystyle H$	$\displaystyle=e_{1}(\gamma+\delta-1)^{2}+e_{2}(\gamma+\epsilon-1)^{2}+e_{3}(% \delta+\epsilon-1)^{2}-4\alpha\beta e_{3}-4q(e_{2}-e_{3}).$
ⓘ Defines: $b_{j}$ : coefficients (locally) and $H$ (locally) Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $j$ : nonnegative integer, $q$ : real or complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter and $\mathbb{L}$ : lattice Permalink: http://dlmf.nist.gov/31.2.E12 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

§31.2(v) Heun’s Equation Automorphisms

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Keywords:: Heun’s equation, automorphisms
Referenced by:: §31.3(iii), §31.8
Permalink:: http://dlmf.nist.gov/31.2.v
See also:: Annotations for §31.2 and Ch.31

$F$ -Homotopic Transformations

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Keywords:: $F$ -homotopic transformations, Heun’s equation, automorphisms
Notes:: See Ronveaux (1995, Part A, p. 16).
See also:: Annotations for §31.2(v), §31.2 and Ch.31

$w(z)=z^{1-\gamma}w_{1}(z)$ satisfies (31.2.1) if $w_{1}$ is a solution of (31.2.1) with transformed parameters $q_{1}=q+(a\delta+\epsilon)(1-\gamma)$ ; $\alpha_{1}=\alpha+1-\gamma$ , $\beta_{1}=\beta+1-\gamma$ , $\gamma_{1}=2-\gamma$ . Next, $w(z)=(z-1)^{1-\delta}w_{2}(z)$ satisfies (31.2.1) if $w_{2}$ is a solution of (31.2.1) with transformed parameters $q_{2}=q+a\gamma(1-\delta)$ ; $\alpha_{2}=\alpha+1-\delta$ , $\beta_{2}=\beta+1-\delta$ , $\delta_{2}=2-\delta$ . Lastly, $w(z)=(z-a)^{1-\epsilon}w_{3}(z)$ satisfies (31.2.1) if $w_{3}$ is a solution of (31.2.1) with transformed parameters $q_{3}=q+\gamma(1-\epsilon)$ ; $\alpha_{3}=\alpha+1-\epsilon$ , $\beta_{3}=\beta+1-\epsilon$ , $\epsilon_{3}=2-\epsilon$ . By composing these three steps, there result $2^{3}=8$ possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1).

Homographic Transformations

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Keywords:: Heun’s equation, automorphisms, homographic transformations
See also:: Annotations for §31.2(v), §31.2 and Ch.31

There are $4!=24$ homographies $\tilde{z}(z)=(Az+B)/(Cz+D)$ that take $0,1,a,\infty$ to some permutation of $0,1,a^{\prime},\infty$ , where $a^{\prime}$ may differ from $a$ . If $\tilde{z}=\tilde{z}(z)$ is one of the $3!=6$ homographies that map $\infty$ to $\infty$ , then $w(z)=\tilde{w}(\tilde{z})$ satisfies (31.2.1) if $\tilde{w}(\tilde{z})$ is a solution of (31.2.1) with $z$ replaced by $\tilde{z}$ and appropriately transformed parameters. For example, if $\tilde{z}=z/a$ , then the parameters are $\tilde{a}=1/a$ , $\tilde{q}=q/a$ ; $\tilde{\delta}=\epsilon$ , $\tilde{\epsilon}=\delta$ . If $\tilde{z}=\tilde{z}(z)$ is one of the $4!-3!=18$ homographies that do not map $\infty$ to $\infty$ , then an appropriate prefactor must be included on the right-hand side. For example, $w(z)=(1-z)^{-\alpha}\tilde{w}(z/(z-1))$ , which arises from $\tilde{z}=z/(z-1)$ , satisfies (31.2.1) if $\tilde{w}(\tilde{z})$ is a solution of (31.2.1) with $z$ replaced by $\tilde{z}$ and transformed parameters $\tilde{a}=a/(a-1)$ , $\tilde{q}=-(q-a\alpha\gamma)/(a-1)$ ; $\tilde{\beta}=\alpha+1-\delta$ , $\tilde{\delta}=\alpha+1-\beta$ .

Composite Transformations

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Keywords:: Heun’s equation, automorphisms, composite
See also:: Annotations for §31.2(v), §31.2 and Ch.31

There are $8\cdot 24=192$ automorphisms of equation (31.2.1) by compositions of $F$ -homotopic and homographic transformations. Each is a substitution of dependent and/or independent variables that preserves the form of (31.2.1). Except for the identity automorphism, each alters the parameters.

31.1 Special Notation 31.3 Basic Solutions

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