# relation to Lam� equation

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##### 1: 29.2 Differential Equations
###### §29.2(ii) Other Forms
For the Weierstrass function $\wp$ see §23.2(ii). …
##### 2: 30.2 Differential Equations
###### §30.2(i) Spheroidal Differential Equation
With $\zeta=\gamma z$ Equation (30.2.1) changes toIf $\mu^{2}=\frac{1}{4}$, Equation (30.2.2) reduces to the Mathieu equation; see (28.2.1). …
##### 3: 31.2 Differential Equations
###### §31.2(i) Heun’s Equation
31.2.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\frac{\epsilon}{z-a}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{% \alpha\beta z-q}{z(z-1)(z-a)}w=0,$ $\alpha+\beta+1=\gamma+\delta+\epsilon$.
##### 4: 15.10 Hypergeometric Differential Equation
###### §15.10 Hypergeometric Differential Equation
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. … The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions. …
##### 5: 32.2 Differential Equations
###### §32.2 Differential Equations
The six Painlevé equations $\mbox{P}_{\mbox{\scriptsize I}}$$\mbox{P}_{\mbox{\scriptsize VI}}$ are as follows: … The six equations are sometimes referred to as the Painlevé transcendents, but in this chapter this term will be used only for their solutions. … An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however. … The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of $\mbox{P}_{\mbox{\scriptsize I}}$$\mbox{P}_{\mbox{\scriptsize VI}}$. …
##### 6: 28.2 Definitions and Basic Properties
A solution with the pseudoperiodic property (28.2.14) is called a Floquet solution with respect to $\nu$. … leads to a Floquet solution. …
###### §28.2(vi) Eigenfunctions
Table 28.2.2 gives the notation for the eigenfunctions corresponding to the eigenvalues in Table 28.2.1. …
##### 7: 28.20 Definitions and Basic Properties
###### §28.20(ii) Solutions $\operatorname{Ce}_{\nu}$, $\operatorname{Se}_{\nu}$, $\operatorname{Me}_{\nu}$, $\operatorname{Fe}_{n}$, $\operatorname{Ge}_{n}$
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: 6.11 Relations to Other Functions
###### Confluent Hypergeometric Function
6.11.3 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=U\left(1,1,-iz\right).$
##### 10: 19.10 Relations to Other Functions
###### §19.10(i) Theta and Elliptic Functions
For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. …
###### §19.10(ii) Elementary Functions
For relations to the Gudermannian function $\operatorname{gd}\left(x\right)$ and its inverse ${\operatorname{gd}^{-1}}\left(x\right)$4.23(viii)), see (19.6.8) and …