delta wing equation

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1: 31.2 Differential Equations

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

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§29.2(ii), §31.11(iii), §31.12, §31.13, §31.14(i), §31.17(i), §31.18, §31.2(v), §31.2(v), §31.2(v), §31.2(i), §31.3(i), §31.3(i), §31.3(ii), §31.3(ii), §31.3(ii), §31.3(iii), §31.3(iii), §31.4, §31.5, §31.6, §31.7(ii), §31.8, §31.8, §31.8

Permalink:

http://dlmf.nist.gov/31.2.E1

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Annotations for §31.2(i), §31.2 and Ch.31

►This equation has regular singularities at $0,1,a,\mathrm{\infty }$, with corresponding exponents $\{0,1-\gamma \}$, $\{0,1-\delta \}$, $\{0,1-ϵ\}$, $\{\alpha ,\beta \}$, respectively (§2.7(i)). … ►The parameters play different roles: $a$ is the singularity parameter; $\alpha ,\beta ,\gamma ,\delta ,ϵ$ are exponent parameters; $q$ is the accessory parameter. … ►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 $q_2=q+a\gamma (1-\delta )$; ${\alpha }_2=\alpha +1-\delta$, ${\beta }_2=\beta +1-\delta$, ${\delta }_2=2-\delta$. … ►For example, if $\tilde{z}=z/a$, then the parameters are $\tilde{a}=1/a$, $\tilde{q}=q/a$; $\tilde{\delta }=ϵ$, $\widetilde{ϵ}=\delta$. …

2: 30.2 Differential Equations

§30.2 Differential Equations

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§30.2(i) Spheroidal Differential Equation

… ► … ►The Liouville normal form of equation (30.2.1) is … ►

§30.2(iii) Special Cases

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3: 29.2 Differential Equations

§29.2 Differential Equations

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§29.2(i) Lamé’s Equation

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§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 Hypergeometric Differential Equation

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§15.10(i) Fundamental Solutions

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15.10.1 $$z(1-z)\frac{d^{2}w}{{dz}^2}+\left(c-(a+b+1)z\right)\frac{dw}{dz}-abw=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.11(i), §15.11(i), §15.17(ii), §15.19(ii), §15.2(ii), §15.5(ii), §16.13, §17.6(iv), §19.18(ii), §19.4(ii)

Permalink:

http://dlmf.nist.gov/15.10.E1

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Annotations for §15.10(i), §15.10 and Ch.15

►This is the hypergeometric differential equation. … ► …

5: 32.2 Differential Equations

… ►The six Painlevé equations ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are as follows: … ►with $\alpha$, $\beta$, $\gamma$, and $\delta$ arbitrary constants. … ►If $\gamma \delta \ne 0$ in ${\text{P}}_{\text{III}}$, then set $\gamma =1$ and $\delta =-1$, without loss of generality, by rescaling $w$ and $z$ if necessary. If $\gamma =0$ and $\alpha \delta \ne 0$ in ${\text{P}}_{\text{III}}$, then set $\alpha =1$ and $\delta =-1$, without loss of generality. … ►If $\delta \ne 0$ in ${\text{P}}_{\text{V}}$, then set $\delta =-\frac{1}{2}$, without loss of generality. …

6: 28.20 Definitions and Basic Properties

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28.20.1 $$w^{\prime \prime }-\left(a-2q\cosh \left(2z\right)\right)w=0,$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

A&S Ref:

20.1.2 (in slightly different form) 20.8.6

Referenced by:

§28.20(iii), §28.32(i), 1st item, 2nd item, item (b), §28.8(iv), §28.8(iv)

Permalink:

http://dlmf.nist.gov/28.20.E1

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Annotations for §28.20(i), §28.20 and Ch.28

… ►Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ${\zeta }^{1/2}{\mathrm{e}}^{\pm 2\mathrm{i}h\zeta }$ as $\zeta \to \mathrm{\infty }$ in the respective sectors $|\mathrm{ph}\left(\mp \mathrm{i}\zeta \right)|\le \frac{3}{2}\pi -\delta$, $\delta$ being an arbitrary small positive constant. …as $\mathrm{\Re }z\to +\mathrm{\infty }$ with $-\pi +\delta \le \mathrm{\Im }z\le 2\pi -\delta$, and …as $\mathrm{\Re }z\to +\mathrm{\infty }$ with $-2\pi +\delta \le \mathrm{\Im }z\le \pi -\delta$. …as $\mathrm{\Re }z\to +\mathrm{\infty }$ with $|\mathrm{\Im }z|\le \pi -\delta$. …

7: 28.2 Definitions and Basic Properties

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§28.2(i) Mathieu’s Equation

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28.2.1 $$w^{\prime \prime }+(a-2q\cos \left(2z\right))w=0.$$

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Symbols:

$\cos z$: cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

A&S Ref:

20.3.1 20.1.1 (in slightly different form)

Referenced by:

§1.18(vii), Figure 28.17.1, Figure 28.17.1, §28.17, §28.17, §28.2(ii), §28.2(ii), §28.2(iii), §28.2(iv), §28.20(i), §28.29(i), §28.32(i), §28.32(i), 1st item, 2nd item, §28.33(iii), item (a), item (c), item (c), §28.5(i), §28.8(iv), §28.8(iv), §28.8(iv), §30.2(iii)

Permalink:

http://dlmf.nist.gov/28.2.E1

Encodings:

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See also:

Annotations for §28.2(i), §28.2 and Ch.28

… ►This is the characteristic equation of Mathieu’s equation (28.2.1). … ►

§28.2(iv) Floquet Solutions

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8: 29.19 Physical Applications

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§29.19(ii) Lamé Polynomials

►Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …Hargrave (1978) studies high frequency solutions of the delta wing equation. …Roper (1951) solves the linearized supersonic flow equations. Clarkson (1991) solves nonlinear evolution equations. …

9: 1.17 Integral and Series Representations of the Dirac Delta

§1.17 Integral and Series Representations of the Dirac Delta

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§1.17(i) Delta Sequences

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Sine and Cosine Functions

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Coulomb Functions (§33.14(iv))

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Airy Functions (§9.2)

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10: 18.25 Wilson Class: Definitions

… ►For the Wilson class OP’s $p_n(x)$ with $x=\lambda (y)$: if the $y$-orthogonality set is $\{0,1,\mathrm{\dots },N\}$, then the role of the differentiation operator $d/dx$ in the Jacobi, Laguerre, and Hermite cases is played by the operator ${\mathrm{\Delta }}_y$ followed by division by ${\mathrm{\Delta }}_y(\lambda (y))$, or by the operator ${\nabla }_y$ followed by division by ${\nabla }_y(\lambda (y))$. Alternatively if the $y$-orthogonality interval is $(0,\mathrm{\infty })$, then the role of $d/dx$ is played by the operator ${\delta }_y$ followed by division by ${\delta }_y(\lambda (y))$. … ►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials $W_n(x;a,b,c,d)$, continuous dual Hahn polynomials $S_n(x;a,b,c)$, Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta )$, and dual Hahn polynomials $R_n(x;\gamma ,\delta ,N)$. … ►The first four sets imply $\gamma +\delta >-2$, and the last four imply . … ►

18.25.9 $$\sum _{y=0}^N{p}_n(y(y+\gamma +\delta +1))p_m(y(y+\gamma +\delta +1))\frac{\gamma +\delta +1+2y}{\gamma +\delta +1+y}{\omega }_y=h_n{\delta }_{n,m}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant, $p_n(x)$: polynomial of degree $n$ , $m$: nonnegative integer, $n$: nonnegative integer and $h_n$

Referenced by:

§18.38(iii)

Permalink:

http://dlmf.nist.gov/18.25.E9

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See also:

Annotations for §18.25(iii), §18.25 and Ch.18

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