# indicial equation

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##### 1: 30.2 Differential Equations
###### §30.2(i) Spheroidal Differential Equation
The Liouville normal form of equation (30.2.1) is …
##### 2: 31.2 Differential Equations
###### §31.2(i) Heun’s Equation
31.2.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\frac{\epsilon}{z-a}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{% \alpha\beta z-q}{z(z-1)(z-a)}w=0,$ $\alpha+\beta+1=\gamma+\delta+\epsilon$.
##### 3: 29.2 Differential Equations
###### §29.2(ii) Other Forms
Equation (29.2.10) is a special case of Heun’s equation (31.2.1).
##### 4: 15.10 Hypergeometric Differential Equation
###### §15.10(i) Fundamental Solutions
15.10.1 $z(1-z)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(c-(a+b+1)z\right)\frac% {\mathrm{d}w}{\mathrm{d}z}-abw=0.$
This is the hypergeometric differential equation. …
##### 5: 32.2 Differential Equations
###### §32.2(i) Introduction
The six Painlevé equations $\mbox{P}_{\mbox{\scriptsize I}}$$\mbox{P}_{\mbox{\scriptsize VI}}$ are as follows: …
##### 6: 28.2 Definitions and Basic Properties
###### §28.2(i) Mathieu’s Equation
This is the characteristic equation of Mathieu’s equation (28.2.1). …
##### 7: 28.20 Definitions and Basic Properties
###### §28.20(i) Modified Mathieu’s Equation
When $z$ is replaced by $\pm\mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation:
28.20.2 ${(\zeta^{2}-1)w^{\prime\prime}+\zeta w^{\prime}+\left(4q\zeta^{2}-2q-a\right)w% =0},$ $\zeta=\cosh z$.
Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to $\zeta^{\ifrac{1}{2}}e^{\pm 2\mathrm{i}h\zeta}$ as $\zeta\to\infty$ in the respective sectors $|\operatorname{ph}\left(\mp\mathrm{i}\zeta\right)|\leq\tfrac{3}{2}\pi-\delta$, $\delta$ being an arbitrary small positive constant. …
##### 8: 2.7 Differential Equations
###### §2.7 Differential Equations
An ordinary point of the differential equationLet $\alpha_{1}$, $\alpha_{2}$ denote the indices or exponents, that is, the roots of the indicial equation
###### §2.7(ii) Irregular Singularities of Rank 1
See §2.11(v) for other examples. …
##### 10: 31.13 Asymptotic Approximations
###### §31.13 Asymptotic Approximations
For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).