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1: Software Index

… ►

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

…

2: 19.18 Derivatives and Differential Equations

… ►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. … ►

19.18.11 $$\sum _{j=1}^n{z}_j{\partial }_{j}v=-av,$$

ⓘ

Symbols:

$\partial x$: partial differential of $x$ , $n$: nonnegative integer and $v$

Referenced by:

§19.18(ii), §19.18(ii)

Permalink:

http://dlmf.nist.gov/19.18.E11

Encodings:

pMML, png, TeX

See also:

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

… ►

19.18.12 $$(z_j{\partial }_j+b_j){\partial }_{l}v=(z_l{\partial }_l+b_l){\partial }_{j}v,$$

ⓘ

Symbols:

$\partial x$: partial differential of $x$ , $l$: nonnegative integer and $v$

Referenced by:

§19.18(ii), §19.18(ii)

Permalink:

http://dlmf.nist.gov/19.18.E12

Encodings:

pMML, png, TeX

See also:

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

… ►

19.18.13 $$((z_j-z_l){\partial }_j{\partial }_l+b_j{\partial }_l-b_l{\partial }_j)v=0.$$

ⓘ

Symbols:

$\partial x$: partial differential of $x$ , $l$: nonnegative integer and $v$

Permalink:

http://dlmf.nist.gov/19.18.E13

Encodings:

pMML, png, TeX

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: 16.14 Partial Differential Equations

… ►

$$x(1-x)\frac{{\partial }^2{F}_1}{{\partial x}^2}+y(1-x)\frac{{\partial }^2{F}_1}{\partial x\partial y}+\left(\gamma -(\alpha +\beta +1)x\right)\frac{\partial F_1}{\partial x}-\beta y\frac{\partial F_1}{\partial y}-\alpha \beta F_1=0,$$

►

$$y(1-y)\frac{{\partial }^2{F}_1}{{\partial y}^2}+x(1-y)\frac{{\partial }^2{F}_1}{\partial x\partial y}+\left(\gamma -(\alpha +{\beta }^{\prime }+1)y\right)\frac{\partial F_1}{\partial y}-{\beta }^{\prime }x\frac{\partial F_1}{\partial x}-\alpha {\beta }^{\prime }{F}_1=0,$$

►

$$x(1-x)\frac{{\partial }^2{F}_2}{{\partial x}^2}-xy\frac{{\partial }^2{F}_2}{\partial x\partial y}+\left(\gamma -(\alpha +\beta +1)x\right)\frac{\partial F_2}{\partial x}-\beta y\frac{\partial F_2}{\partial y}-\alpha \beta F_2=0,$$

►

$$y(1-y)\frac{{\partial }^2{F}_2}{{\partial y}^2}-xy\frac{{\partial }^2{F}_2}{\partial x\partial y}+\left({\gamma }^{\prime }-(\alpha +{\beta }^{\prime }+1)y\right)\frac{\partial F_2}{\partial y}-{\beta }^{\prime }x\frac{\partial F_2}{\partial x}-\alpha {\beta }^{\prime }{F}_2=0,$$

►

$$x(1-x)\frac{{\partial }^2{F}_3}{{\partial x}^2}+y\frac{{\partial }^2{F}_3}{\partial x\partial y}+\left(\gamma -(\alpha +\beta +1)x\right)\frac{\partial F_3}{\partial x}-\alpha \beta F_3=0,$$

…

4: 36.10 Differential Equations

… ►

36.10.1 $${\mathrm{\Phi }}_K^{\prime }(-\mathrm{i}\frac{\partial }{\partial x_1};𝐱){\mathrm{\Psi }}_K\left(𝐱\right)=0,$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, ${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $K$: codimension and $x_{\mathrm{i}}$: real parameter

Referenced by:

§36.10(i)

Permalink:

http://dlmf.nist.gov/36.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(i), §36.10 and Ch.36

… ►

36.10.3 $$\frac{{\partial }^2{\mathrm{\Psi }}_1}{{\partial x}^2}-\frac{x}{3}{\mathrm{\Psi }}_1=0.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(i), §36.10(i), §36.10 and Ch.36

… ►

36.10.4 $$\frac{{\partial }^3{\mathrm{\Psi }}_2}{{\partial x}^3}-\frac{1}{2}y\frac{\partial {\mathrm{\Psi }}_2}{\partial x}-\frac{\mathrm{i}}{4}x{\mathrm{\Psi }}_2=0.$$

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(i), §36.10(i), §36.10 and Ch.36

… ►

36.10.13 $$6\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x\partial y}-2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial y}+y{\mathrm{\Psi }}^{(\mathrm{E})}=0,$$

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(iii), §36.10 and Ch.36

… ►

36.10.18 $$\mathrm{i}\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial z}=\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x\partial y}.$$

ⓘ

Symbols:

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(iv), §36.10 and Ch.36

…

5: 1.5 Calculus of Two or More Variables

… ►

${\displaystyle \frac{{\partial }^{2}f}{\partial x\partial y}}={\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \frac{\partial f}{\partial y}}\right),$

►

${\displaystyle \frac{{\partial }^{2}f}{\partial y\partial x}}={\displaystyle \frac{\partial }{\partial y}}\left({\displaystyle \frac{\partial f}{\partial x}}\right).$

►The function $f(x,y)$ is continuously differentiable if $f$, $\partial f/\partial x$, and $\partial f/\partial y$ are continuous, and twice-continuously differentiable if also ${\partial }^{2}f/{\partial x}^2$, ${\partial }^{2}f/{\partial y}^2$, ${\partial }^{2}f/\partial x\partial y$, and ${\partial }^{2}f/\partial y\partial x$ are continuous. … ►

1.5.6 $$\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}f}{\partial y\partial x}.$$

ⓘ

Symbols:

$\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/1.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(i), §1.5 and Ch.1

… ►

1.5.21 $$\frac{{\partial }^{2}f}{{\partial x}^2}\frac{{\partial }^{2}f}{{\partial y}^2}-{\left(\frac{{\partial }^{2}f}{\partial x\partial y}\right)}^2>0\text{ at }(a,b).$$

ⓘ

Symbols:

$(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/1.5.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(iii), §1.5 and Ch.1

…

6: 32.4 Isomonodromy Problems

… ►

32.4.2 $$\frac{{\partial }^2𝚿}{\partial z\partial \lambda }=\frac{{\partial }^2𝚿}{\partial \lambda \partial z},$$

ⓘ

Symbols:

$\partial x$: partial differential of $x$ and $z$: real

Permalink:

http://dlmf.nist.gov/32.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §32.4(i), §32.4 and Ch.32

… ►

32.4.3 $$\frac{\partial 𝐀}{\partial z}-\frac{\partial 𝐁}{\partial \lambda }+𝐀𝐁-𝐁𝐀=0.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $z$: real

Referenced by:

§32.4(i)

Permalink:

http://dlmf.nist.gov/32.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §32.4(i), §32.4 and Ch.32

…

7: 3.4 Differentiation

… ►

3.4.20 $$\frac{\partial u_{0,0}}{\partial x}=\frac{1}{2h}(u_{1,0}-u_{-1,0})+O\left(h^2\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.21

Permalink:

http://dlmf.nist.gov/3.4.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

… ►

3.4.25 $$\frac{{\partial }^2{u}_{0,0}}{\partial x\partial y}=\frac{1}{4h^2}(u_{1,1}-u_{1,-1}-u_{-1,1}+u_{-1,-1})+O\left(h^2\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.26

Permalink:

http://dlmf.nist.gov/3.4.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

►

3.4.26 $$\frac{{\partial }^2{u}_{0,0}}{\partial x\partial y}=-\frac{1}{2h^2}(u_{1,0}+u_{-1,0}+u_{0,1}+u_{0,-1}-2u_{0,0}-u_{1,1}-u_{-1,-1})+O\left(h^2\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.27

Permalink:

http://dlmf.nist.gov/3.4.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

… ►

3.4.31 $$\frac{{\partial }^4{u}_{0,0}}{{\partial }^{2}x{\partial }^{2}y}=\frac{1}{h^4}(u_{1,1}+u_{-1,1}+u_{1,-1}+u_{-1,-1}-2u_{1,0}-2u_{-1,0}-2u_{0,1}-2u_{0,-1}+4u_{0,0})+O\left(h^2\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.29

Permalink:

http://dlmf.nist.gov/3.4.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

… ►

3.4.32 $${\nabla }^{4}u=\frac{{\partial }^{4}u}{{\partial x}^4}+2\frac{{\partial }^{4}u}{{\partial }^{2}x{\partial }^{2}y}+\frac{{\partial }^{4}u}{{\partial y}^4}.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.32

Permalink:

http://dlmf.nist.gov/3.4.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

…

8: 10.38 Derivatives with Respect to Order

… ►

10.38.1 $$\frac{\partial I_{\pm \nu }\left(z\right)}{\partial \nu }=\pm I_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.42

Referenced by:

§10.38, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.38.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial I_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.38.2).

See also:

Annotations for §10.38 and Ch.10

… ►

10.38.3 $${{(-1)}^n\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=-K_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}{(-1)}^k\frac{{(\frac{1}{2}z)}^k{I}_k\left(z\right)}{k!(n-k)},$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: 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, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.44

Referenced by:

§10.38

Permalink:

http://dlmf.nist.gov/10.38.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

… ►

10.38.4 $${\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{K}_k\left(z\right)}{k!(n-k)}.$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $K_{\nu }\left(z\right)$: 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, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.45

Referenced by:

§10.38

Permalink:

http://dlmf.nist.gov/10.38.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

… ►

10.38.6 $${\frac{\partial I_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=-\frac{1}{\sqrt{2\pi x}}\left(E_1\left(2x\right){\mathrm{e}}^x\pm \mathrm{Ei}\left(2x\right){\mathrm{e}}^{-x}\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.2.32 and 10.2.33 (both corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

►

10.38.7 $${\frac{\partial K_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=\pm \sqrt{\frac{\pi }{2x}}{E}_1\left(2x\right){\mathrm{e}}^x.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $K_{\nu }\left(z\right)$: 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 $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.2.34 (corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

…

9: 36.4 Bifurcation Sets

… ►

36.4.1 $$\frac{\partial }{\partial t}{\mathrm{\Phi }}_K(t_j(𝐱);𝐱)=0.$$

ⓘ

Symbols:

${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $t$: variable, $K$: codimension and $t_j(𝐱)$: solutions

Referenced by:

§36.11, §36.12(i), §36.4(i), §36.5(i), §36.5(ii)

Permalink:

http://dlmf.nist.gov/36.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

… ►

36.4.3 $$\frac{{\partial }^2}{{\partial t}^2}{\mathrm{\Phi }}_K(t;𝐱)=0.$$

ⓘ

Symbols:

${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $t$: variable and $K$: codimension

Permalink:

http://dlmf.nist.gov/36.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

… ►

36.4.4

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , : elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\text{ or }\mathrm{K}$ , $t$: variable and $s$: variable

Permalink:

http://dlmf.nist.gov/36.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

…

10: 10.15 Derivatives with Respect to Order

… ►

10.15.3 $${\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=\frac{\pi }{2}{Y}_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{J}_k\left(z\right)}{k!(n-k)}.$$

ⓘ

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, $!$: factorial (as in $n!$), $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.66

Referenced by:

§10.15, §10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

… ►

10.15.4 $${\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=-\frac{\pi }{2}{J}_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{Y}_k\left(z\right)}{k!(n-k)},$$

ⓘ

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, $!$: factorial (as in $n!$), $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.67

Permalink:

http://dlmf.nist.gov/10.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

►

10.15.5 $${\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =0}=\frac{\pi }{2}{Y}_0\left(z\right),{\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =0}=-\frac{\pi }{2}{J}_0\left(z\right).$$

ⓘ

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 $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.68

Referenced by:

§10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

… ►

10.15.6 $${\frac{\partial J_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\sin x-\mathrm{Si}\left(2x\right)\cos x\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.1.41 (corrected)

Referenced by:

§10.15, §10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

►

10.15.7 $${\frac{\partial J_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =-\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\cos x+\mathrm{Si}\left(2x\right)\sin x\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.1.42 (corrected)

Referenced by:

§10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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