# paraxial wave equation

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##### 1: 30.1 Special Notation
βΊThe main functions treated in this chapter are the eigenvalues $\lambda^{m}_{n}\left(\gamma^{2}\right)$ and the spheroidal wave functions $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$, $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)$, $\mathit{Ps}^{m}_{n}\left(z,\gamma^{2}\right)$, $\mathit{Qs}^{m}_{n}\left(z,\gamma^{2}\right)$, and $S^{m(j)}_{n}\left(z,\gamma\right)$, $j=1,2,3,4$. …Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathsf{Ps}$, $\mathsf{Qs}$, $\mathit{Ps}$, $\mathit{Qs}$, respectively. βΊ
###### Other Notations
βΊFlammer (1957) and Abramowitz and Stegun (1964) use $\lambda_{mn}(\gamma)$ for $\lambda^{m}_{n}\left(\gamma^{2}\right)+\gamma^{2}$, $R_{mn}^{(j)}(\gamma,z)$ for $S^{m(j)}_{n}\left(z,\gamma\right)$, and …
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##### 3: 30.2 Differential Equations
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###### §30.2(i) Spheroidal Differential Equation
βΊβΊThe Liouville normal form of equation (30.2.1) is … βΊ
##### 4: 31.2 Differential Equations
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###### §31.2(i) Heun’s Equation
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31.2.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\frac{\epsilon}{z-a}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{% \alpha\beta z-q}{z(z-1)(z-a)}w=0,$ $\alpha+\beta+1=\gamma+\delta+\epsilon$.
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##### 5: 29.2 Differential Equations
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###### §29.2(ii) Other Forms
βΊEquation (29.2.10) is a special case of Heun’s equation (31.2.1).
##### 6: 15.10 Hypergeometric Differential Equation
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###### §15.10(i) Fundamental Solutions
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15.10.1 $z(1-z)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(c-(a+b+1)z\right)\frac% {\mathrm{d}w}{\mathrm{d}z}-abw=0.$
βΊThis is the hypergeometric differential equation. … βΊ
##### 7: 32.2 Differential Equations
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###### §32.2(i) Introduction
βΊThe six Painlevé equations $\mbox{P}_{\mbox{\scriptsize I}}$$\mbox{P}_{\mbox{\scriptsize VI}}$ are as follows: … βΊ
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##### 8: 28.2 Definitions and Basic Properties
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###### §28.2(i) Mathieu’s Equation
βΊ βΊThis is the characteristic equation of Mathieu’s equation (28.2.1). … βΊ
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##### 9: 28.20 Definitions and Basic Properties
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###### §28.20(i) Modified Mathieu’s Equation
βΊWhen $z$ is replaced by $\pm\mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation: βΊ βΊ
28.20.2 ${(\zeta^{2}-1)w^{\prime\prime}+\zeta w^{\prime}+\left(4q\zeta^{2}-2q-a\right)w% =0},$ $\zeta=\cosh z$.
βΊThen 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. …
##### 10: 36.10 Differential Equations
###### §36.10 Differential Equations
βΊ $K=1$, fold: (36.10.1) becomes Airy’s equation9.2(i)) … βΊ
###### §36.10(iii) Operator Equations
βΊEquation (36.10.17) is the paraxial wave equation.