# §31.2 Differential Equations

## §31.2(i) Heun’s Equation

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

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

 31.2.2 $w(z)=z^{-\gamma/2}(z-1)^{-\delta/2}(z-a)^{-\epsilon/2}W(z),$
 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, $z$: 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

 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 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\theta}^{2}}+\left({(2\gamma-1)\cot\theta-% (2\delta-1)\tan\theta}-\frac{\epsilon\sin\left(2\theta\right)}{a-{\sin}^{2}% \theta}\right)\frac{\mathrm{d}w}{\mathrm{d}\theta}+4\frac{\alpha\beta{\sin}^{2% }\theta-q}{a-{\sin}^{2}\theta}w=0.$

## §31.2(iv) Doubly-Periodic Forms

### Jacobi’s Elliptic Form

With the notation of §22.2 let

 31.2.7 $\displaystyle a$ $\displaystyle=k^{-2},$ $\displaystyle z$ $\displaystyle={\operatorname{sn}}^{2}\left(\zeta,k\right).$ ⓘ 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 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\zeta}^{2}}+\left((2\gamma-1)\frac{% \operatorname{cn}\zeta\operatorname{dn}\zeta}{\operatorname{sn}\zeta}-(2\delta% -1)\frac{\operatorname{sn}\zeta\operatorname{dn}\zeta}{\operatorname{cn}\zeta}% -(2\epsilon-1)k^{2}\frac{\operatorname{sn}\zeta\operatorname{cn}\zeta}{% \operatorname{dn}\zeta}\right)\frac{\mathrm{d}w}{\mathrm{d}\zeta}+4k^{2}(% \alpha\beta{\operatorname{sn}}^{2}\zeta-q)w=0.$

### Weierstrass’s Form

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

 31.2.9 $\displaystyle k^{2}$ $\displaystyle=(e_{2}-e_{3})/(e_{1}-e_{3}),$ $\displaystyle\zeta$ $\displaystyle=\mathrm{i}{K^{\prime}}+\xi(e_{1}-e_{3})^{1/2},$ $\displaystyle e_{1}$ $\displaystyle=\wp\left(\omega_{1}\right),$ $\displaystyle e_{2}$ $\displaystyle=\wp\left(\omega_{2}\right),$ $\displaystyle e_{3}$ $\displaystyle=\wp\left(\omega_{3}\right),$ $\displaystyle e_{1}+e_{2}+e_{3}$ $\displaystyle=0,$

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

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, $\omega_{j}$: 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

### $F$-Homotopic Transformations

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

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

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.