elimination of first derivative

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1: 1.13 Differential Equations

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Elimination of First Derivative by Change of Dependent Variable

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Elimination of First Derivative by Change of Independent Variable

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1.13.17 $$\frac{d^{2}w}{{d\eta }^2}+g(z)\exp \left(2\int f(z)dz\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $z$: variable, $w(z)$: solution, $f(z)$: analytic coefficient, $\eta$: change of variable and $g(z)$: analytic coefficient

Permalink:

http://dlmf.nist.gov/1.13.E17

Encodings:

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See also:

Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

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2: 19.18 Derivatives and Differential Equations

§19.18 Derivatives and Differential Equations

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§19.18(i) Derivatives

… ►Let ${\partial }_j=\partial /\partial z_j$, and ${𝐞}_j$ be an $n$-tuple with 1 in the $j$th place and 0’s elsewhere. … ►

19.18.6 $$\left(\frac{\partial }{\partial x}+\frac{\partial }{\partial y}+\frac{\partial }{\partial z}\right)R_F(x,y,z)=\frac{-1}{2\sqrt{x}\sqrt{y}\sqrt{z}},$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Referenced by:

§19.18(ii), §19.32, Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.18.E6

Encodings:

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Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

… ►If $n=2$, then elimination of ${\partial }_{2}v$ between (19.18.11) and (19.18.12), followed by the substitution $(b_1,b_2,z_1,z_2)=(b,c-b,1-z,1)$, produces the Gauss hypergeometric equation (15.10.1). …

3: 3.5 Quadrature

… ►If $k$ in (3.5.4) is not arbitrarily large, and if odd-order derivatives of $f$ are known at the end points $a$ and $b$, then the composite trapezoidal rule can be improved by means of the Euler–Maclaurin formula (§2.10(i)). … ►As in Simpson’s rule, by combining the rule for $h$ with that for $h/2$, the first error term $c_1{h}^2$ in (3.5.9) can be eliminated. With the Romberg scheme successive terms $c_1{h}^2,c_2{h}^4,\mathrm{\dots }$, in (3.5.9) are eliminated, according to the formula … ►With $j=2$ and $k=7$, the coefficient of the derivative $f^{(16)}(\xi )$ in (3.5.13) is found to be $(0.14\mathrm{\dots })\times {10}^{-13}$. … ►For the latter $a=-1$, $b=1$, and the nodes $x_k$ are the extrema of the Chebyshev polynomial $T_n\left(x\right)$ (§3.11(ii) and §18.3). …

4: 19.25 Relations to Other Functions

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19.25.6 $$\frac{\partial F(\varphi ,k)}{\partial k}=\frac{1}{3}k{R}_D(c-1,c,c-k^2).$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i)

Permalink:

http://dlmf.nist.gov/19.25.E6

Encodings:

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See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.12 $$\frac{\partial E(\varphi ,k)}{\partial k}=-\frac{1}{3}k{R}_D(c-1,c-k^2,c).$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i)

Permalink:

http://dlmf.nist.gov/19.25.E12

Encodings:

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See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.17 $$F(\varphi ,k)=R_F(x,y,z),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.15

Permalink:

http://dlmf.nist.gov/19.25.E17

Encodings:

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See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.24 $${(z-x)}^{1/2}{R}_F(x,y,z)=F(\varphi ,k),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.14(i), §19.25(iii)

Permalink:

http://dlmf.nist.gov/19.25.E24

Encodings:

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See also:

Annotations for §19.25(iii), §19.25 and Ch.19

… ►Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of $R_F(x,y,z)$. …