# Ince equation

(0.001 seconds)

## 1—10 of 17 matching pages

##### 1: 28.31 Equations of Whittaker–Hill and Ince
###### §28.31(ii) Equation of Ince; Ince Polynomials
The result is the Equation of Ince: …
28.31.5 $w_{\mathit{o},s}(z)=\sum_{\ell=0}^{\infty}B_{2\ell+s}\sin(2\ell+s)z,$ $s=1,2$,
##### 3: Bibliography I
• E. L. Ince (1926) Ordinary Differential Equations. Longmans, Green and Co., London.
##### 5: 31.14 General Fuchsian Equation
###### §31.14(i) Definitions
Heun’s equation (31.2.1) corresponds to $N=3$.
###### Normal Form
An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). …
##### 6: 1.13 Differential Equations
(More generally in (1.13.5) for $n$th-order differential equations, $f(z)$ is the coefficient multiplying the $(n-1)$th-order derivative of the solution divided by the coefficient multiplying the $n$th-order derivative of the solution, see Ince (1926, §5.2).) …
##### 7: 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: …
##### 9: 29.3 Definitions and Basic Properties
For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2K$ or $4K$. … … satisfies the continued-fraction equationThe quantity $H=2a^{2m+1}_{\nu}\left(k^{2}\right)-\nu(\nu+1)k^{2}$ satisfies equation (29.3.10) with … The quantity $H=2b^{2m+1}_{\nu}\left(k^{2}\right)-\nu(\nu+1)k^{2}$ satisfies equation (29.3.10) with …
##### 10: Errata
• Section 1.13

In Equation (1.13.4), the determinant form of the two-argument Wronskian

1.13.4 $\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=\det\begin{bmatrix}w_{1}(z)&w_{2}(% z)\\ w_{1}^{\prime}(z)&w_{2}^{\prime}(z)\end{bmatrix}=w_{1}(z)w_{2}^{\prime}(z)-w_{% 2}(z)w_{1}^{\prime}(z)$

was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the $n$-argument Wronskian is given by $\mathscr{W}\left\{w_{1}(z),\ldots,w_{n}(z)\right\}=\det\left[w_{k}^{(j-1)}(z)\right]$, where $1\leq j,k\leq n$. Immediately below Equation (1.13.4), a sentence was added giving the definition of the $n$-argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for $n$th-order differential equations. A reference to Ince (1926, §5.2) was added.