§31.2 Differential Equations

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§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

Notes:: See Erdélyi et al. (1955, Chapter XV) and Ronveaux (1995, Part A, p. 7).
Keywords:: Heun equation, Heun’s equation, singularities
Permalink:: http://dlmf.nist.gov/31.2.i

31.2.1 $\frac{{d}^{2}w}{{dz}^{2}}+\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-a}\right)\frac{dw}{dz}+\frac{\alpha\beta z-q}{z(z-1)(z-a)}w=0,$ $\alpha+\beta+1=\gamma+\delta+\epsilon$ .

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, $\Complex\cup\{\infty\}$ , 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

Keywords:: Heun’s equation
Permalink:: http://dlmf.nist.gov/31.2.ii

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
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31.2.3 $\frac{{d}^{2}W}{{dz}^{2}}=\left(\frac{A}{z}+\frac{B}{z-1}+\frac{C}{z-a}+\frac{D}{z^{2}}+\frac{E}{(z-1)^{2}}+\frac{F}{(z-a)^{2}}\right)W,$ $A+B+C=0$ ,

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : complex parameter, $W(z)$ : solution, $A$ , $B$ , $C$ , $D$ , $E$ and $F$
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31.2.4

$A=-\frac{\gamma\delta}{2}-\frac{\gamma\epsilon}{2a}+\frac{q}{a},$

$B=\frac{\gamma\delta}{2}-\frac{\delta\epsilon}{2(a-1)}-\frac{q-\alpha\beta}{a-1},$

$C=\frac{\gamma\epsilon}{2a}+\frac{\delta\epsilon}{2(a-1)}-\frac{a\alpha\beta-q}{a(a-1)},$

$D=\tfrac{1}{2}\gamma\left(\tfrac{1}{2}\gamma-1\right),$

$E=\tfrac{1}{2}\delta\left(\tfrac{1}{2}\delta-1\right),$

$F=\tfrac{1}{2}\epsilon\left(\tfrac{1}{2}\epsilon-1\right).$

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, $\beta$ : real or complex parameter, $A$ , $B$ , $C$ , $D$ , $E$ and $F$
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§31.2(iii) Trigonometric Form

Keywords:: Heun’s equation
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31.2.5 $z={\mathop{\sin\/}\nolimits^{{2}}}\theta,$

Symbols:: $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
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31.2.6 $\frac{{d}^{2}w}{{d\theta}^{2}}+\left({(2\gamma-1)\mathop{\cot\/}\nolimits\theta-(2\delta-1)\mathop{\tan\/}\nolimits\theta}-\frac{\epsilon\mathop{\sin\/}\nolimits\!\left(2\theta\right)}{a-{\mathop{\sin\/}\nolimits^{{2}}}\theta}\right)\frac{dw}{d\theta}+4\frac{\alpha\beta{\mathop{\sin\/}\nolimits^{{2}}}\theta-q}{a-{\mathop{\sin\/}\nolimits^{{2}}}\theta}w=0.$

Symbols:: $\mathop{\cot\/}\nolimits z$ : cotangent function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : 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
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§31.2(iv) Doubly-Periodic Forms

Notes:: See Ronveaux (1995, Part A, p. 26).
Permalink:: http://dlmf.nist.gov/31.2.iv

¶ Jacobi’s Elliptic Form

Keywords:: Heun’s equation

With the notation of §22.2 let

31.2.7

$a=k^{{-2}},$

$z={\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(\zeta,k\right).$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $z$ : complex variable, $a$ : complex parameter, $\zeta$ : change of variable and $k$ : modulus
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Then (suppressing the parameter $k$ )

31.2.8 $\frac{{d}^{2}w}{{d\zeta}^{2}}+\left((2\gamma-1)\frac{\mathop{\mathrm{cn}\/}\nolimits\zeta\mathop{\mathrm{dn}\/}\nolimits\zeta}{\mathop{\mathrm{sn}\/}\nolimits\zeta}-(2\delta-1)\frac{\mathop{\mathrm{sn}\/}\nolimits\zeta\mathop{\mathrm{dn}\/}\nolimits\zeta}{\mathop{\mathrm{cn}\/}\nolimits\zeta}-(2\epsilon-1)k^{2}\frac{\mathop{\mathrm{sn}\/}\nolimits\zeta\mathop{\mathrm{cn}\/}\nolimits\zeta}{\mathop{\mathrm{dn}\/}\nolimits\zeta}\right)\frac{dw}{d\zeta}+4k^{2}(\alpha\beta{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\zeta-q)w=0.$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : 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
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¶ Weierstrass’s Form

Keywords:: Heun’s equation

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

31.2.9

$k^{2}=(e_{2}-e_{3})/(e_{1}-e_{3}),$

$\zeta=i\mathop{{K^{{\prime}}}\/}\nolimits+\xi(e_{1}-e_{3})^{{1/2}},$

$e_{1}=\mathop{\wp\/}\nolimits\!\left(\omega _{1}\right),$

$e_{2}=\mathop{\wp\/}\nolimits\!\left(\omega _{2}\right),$

$e_{3}=\mathop{\wp\/}\nolimits\!\left(\omega _{3}\right),$

$e_{1}+e_{2}+e_{3}=0,$

where $2\omega _{1}$ and $2\omega _{3}$ with $\imagpart{(\omega _{3}/\omega _{1})}>0$ are generators of the lattice $\mathbb{L}$ for $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ . Then

31.2.10 $w(\xi)=\left(\mathop{\wp\/}\nolimits\!\left(\xi\right)-e_{3}\right)^{{(1-2\gamma)/4}}\left(\mathop{\wp\/}\nolimits\!\left(\xi\right)-e_{2}\right)^{{(1-2\delta)/4}}\*\left(\mathop{\wp\/}\nolimits\!\left(\xi\right)-e_{1}\right)^{{(1-2\epsilon)/4}}W(\xi),$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\mathbb{L}$ : lattice and $W(\xi)$ : solution
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where $W(\xi)$ satisfies

31.2.11 $\ifrac{{d}^{2}W}{{d\xi}^{2}}+\left(H+b_{0}\mathop{\wp\/}\nolimits\!\left(\xi\right)+b_{1}\mathop{\wp\/}\nolimits\!\left(\xi+\omega _{1}\right)+b_{2}\mathop{\wp\/}\nolimits\!\left(\xi+\omega _{2}\right)+b_{3}\mathop{\wp\/}\nolimits\!\left(\xi+\omega _{3}\right)\right)W=0,$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\omega _{j}$ : lattice generators, $\mathbb{L}$ : lattice, $W(\xi)$ : solution, $b_{j}$ : coefficients and $H$
Permalink:: http://dlmf.nist.gov/31.2.E11
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with

31.2.12

$b_{0}=4\alpha\beta-(\gamma+\delta+\epsilon-\tfrac{1}{2})(\gamma+\delta+\epsilon-\tfrac{3}{2}),$

$b_{1}=-(\epsilon-\tfrac{1}{2})(\epsilon-\tfrac{3}{2}),$

$b_{2}=-(\delta-\tfrac{1}{2})(\delta-\tfrac{3}{2}),$

$b_{3}=-(\gamma-\tfrac{1}{2})(\gamma-\tfrac{3}{2}),$

$H=e_{1}(\gamma+\delta-1)^{2}+e_{2}(\gamma+\epsilon-1)^{2}+e_{3}(\delta+\epsilon-1)^{2}-4\alpha\beta e_{3}-4q(e_{2}-e_{3}).$

Symbols:: $\gamma$ : 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, $b_{j}$ : coefficients and $H$
Permalink:: http://dlmf.nist.gov/31.2.E12
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§31.2(v) Heun’s Equation Automorphisms

Keywords:: Heun’s equation
Referenced by:: §31.3(iii), §31.8
Permalink:: http://dlmf.nist.gov/31.2.v

¶ $F$ -Homotopic Transformations

Notes:: See Ronveaux (1995, Part A, p. 16).
Keywords:: Heun’s equation

$w(z)=z^{{1-\gamma}}w_{1}(z)$ 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

Keywords:: Heun’s equation

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

Keywords:: Heun’s equation

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.