partial derivatives

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31: 36.11 Leading-Order Asymptotics

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36.11.2 $${\mathrm{\Psi }}_K\left(𝐱\right)=\sqrt{2\pi }\sum _{j=1}^{j_{\max }(𝐱)}\exp \left(\mathrm{i}\left({\mathrm{\Phi }}_K(t_j(𝐱);𝐱)+\frac{1}{4}\pi {(-1)}^{j+K+1}\right)\right){\left|\frac{{\partial }^2{\mathrm{\Phi }}_K(t_j(𝐱);𝐱)}{{\partial t}^2}\right|}^{-1/2}(1+o\left(1\right)).$$

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $t$: variable, $K$: codimension and $t_j(𝐱)$: solutions

Permalink:

http://dlmf.nist.gov/36.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11 and Ch.36

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32: 9.18 Tables

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Smirnov (1960) tabulates $U_1(x,\alpha )$, $U_2(x,\alpha )$, defined by (9.13.20), (9.13.21), and also $\partial U_1(x,\alpha )/\partial x$, $\partial U_2(x,\alpha )/\partial x$, for $\alpha =1$, $x=-6(.01)10$ to 5D or 5S, and also for $\alpha =\pm \frac{1}{4}$, $\pm \frac{1}{3}$, $\pm \frac{1}{2}$, $\pm \frac{2}{3}$, $\pm \frac{3}{4}$, $\frac{5}{4}$, $\frac{4}{3}$, $\frac{3}{2}$, $\frac{5}{3}$, $\frac{7}{4}$, 2, $x=0(.01)6$; 4D.

33: 10.2 Definitions

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10.2.4 $$Y_n\left(z\right)={\frac{1}{\pi }\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}+{\frac{{(-1)}^n}{\pi }\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =-n},$$ $n=0,\pm 1,\pm 2,\mathrm{\dots }$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.15, §10.2(ii), §10.22(iv), §10.4

Permalink:

http://dlmf.nist.gov/10.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

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34: 10.27 Connection Formulas

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10.27.5 $$K_n\left(z\right)=\frac{{(-1)}^{n-1}}{2}\left({\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}+{\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =-n}\right),$$ $n=0,\pm 1,\pm 2,\mathrm{\dots }$.

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.27.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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35: 36.2 Catastrophes and Canonical Integrals

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$$\frac{{\partial }^p}{{\partial x_1}^p}{\mathrm{\Psi }}_K\left(\mathrm{𝟎}\right)=\frac{2}{K+2}\mathrm{\Gamma }\left(\frac{p+1}{K+2}\right)\cos \left(\frac{\pi }{2}\left(\frac{p+1}{K+2}+p\right)\right),$$ $K$ odd,

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$$\frac{{\partial }^{2q+1}}{{\partial x_1}^{2q+1}}{\mathrm{\Psi }}_K\left(\mathrm{𝟎}\right)=0,$$ $K$ even,

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$$\frac{{\partial }^{2q}}{{\partial x_1}^{2q}}{\mathrm{\Psi }}_K\left(\mathrm{𝟎}\right)=\frac{2}{K+2}\mathrm{\Gamma }\left(\frac{2q+1}{K+2}\right)\exp \left(\mathrm{i}\frac{\pi }{2}\left(\frac{2q+1}{K+2}+2q\right)\right),$$ $K$ even.

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36: 1.13 Differential Equations

… ►Suppose also that at (a fixed) $z_0\in D$, $w$ and $\partial w/\partial z$ are analytic functions of $u$. Then at each $z\in D$, $w$, $\partial w/\partial z$ and ${\partial }^{2}w/{\partial z}^2$ are analytic functions of $u$. …

37: 2.3 Integrals of a Real Variable

… ►Assume also that ${\partial }^{2}p(\alpha ,t)/{\partial t}^2$ and $q(\alpha ,t)$ are continuous in $\alpha$ and $t$, and for each $\alpha$ the minimum value of $p(\alpha ,t)$ in $[0,k)$ is at $t=\alpha$, at which point $\partial p(\alpha ,t)/\partial t$ vanishes, but both ${\partial }^{2}p(\alpha ,t)/{\partial t}^2$ and $q(\alpha ,t)$ are nonzero. … ►

2.3.28 $$\frac{dw}{dt}=\pm \frac{1}{{(2p(\alpha ,t)-2p(\alpha ,\alpha ))}^{1/2}}\frac{\partial p(\alpha ,t)}{\partial t}$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $p(\alpha ,t)$: function and $w$: change of variable

Permalink:

http://dlmf.nist.gov/2.3.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(v), §2.3 and Ch.2

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38: 32.6 Hamiltonian Structure

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${\displaystyle \frac{dq}{dz}}={\displaystyle \frac{\partial \mathrm{H}}{\partial p}},$

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${\displaystyle \frac{dp}{dz}}=-{\displaystyle \frac{\partial \mathrm{H}}{\partial q}},$

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39: 3.8 Nonlinear Equations

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3.8.13 $$\frac{dz}{d\alpha }=-\frac{\partial f}{\partial \alpha }/\frac{\partial f}{\partial z}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $\alpha$: parameter

Permalink:

http://dlmf.nist.gov/3.8.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §3.8(vi), §3.8 and Ch.3

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40: 1.16 Distributions

… ►The partial derivatives of distributions in ${ℝ}^n$ can be defined as in §1.16(ii). … ►

1.16.30 $$𝐃=(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1},\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_2},\mathrm{\dots },\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}).$$

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Defines:

$𝐃$: vector differential operator (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $𝐷f$: distributional derivative and $\left|x\right|$: absolute value of $x$

Referenced by:

(1.16.32), item (ii), item (ii)

Permalink:

http://dlmf.nist.gov/1.16.E30

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

Previously this equation was completely different:

$${𝐷}_{𝜶}={\mathrm{i}}^{-\left|𝜶\right|}{𝐷}^{𝜶}={\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1}\right)}^{{\alpha }_1}\mathrm{\cdots }{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}\right)}^{{\alpha }_n}.$$

The change was made to clarify the meaning of (1.16.32).

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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1.16.32 $$P(𝐃)=\sum _{𝜶}{c}_{𝜶}{𝐃}^{\alpha }=\sum _{𝜶}{c}_{𝜶}{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1}\right)}^{{\alpha }_1}\mathrm{\dots }{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}\right)}^{{\alpha }_n}.$$

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Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $𝐃$: vector differential operator, $P$: polynomial of several variables and $𝐷f$: distributional derivative

Referenced by:

(1.16.30), (1.16.33), (1.16.34), (1.16.36), (1.16.37), item (iii), item (iii)

Permalink:

http://dlmf.nist.gov/1.16.E32

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

Previously this equation was written

$$P(𝐷)=\sum _{𝜶}{c}_{𝜶}{𝐷}_{𝜶}$$

where ${𝐷}_{𝜶}$ was defined in (1.16.30). The change was made to clarify the meaning of this equation.

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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