partial derivatives

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41: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.9 $$t_j(1-t_j)\frac{{\partial }^{2}F}{{\partial t_j}^2}-\frac{1}{2}\sum _{\begin{array}{c}k=1\\ k\ne j\end{array}}^m\frac{t_k(1-t_k)}{t_j-t_k}\frac{\partial F}{\partial t_k}+\left(c-\frac{1}{2}(m-1)-\left(a+b-\frac{1}{2}(m-3)\right)t_j+\frac{1}{2}\sum _{\begin{array}{c}k=1\\ k\ne j\end{array}}^m\frac{t_j(1-t_j)}{t_j-t_k}\right)\frac{\partial F}{\partial t_j}=abF,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $a$: complex variable, $t_j$: eigenvalues of $𝐓$ , $b$: complex variable, $c$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer and $m$: positive integer

Referenced by:

§35.7(iii)

Permalink:

http://dlmf.nist.gov/35.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(iii), §35.7 and Ch.35

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42: 8.8 Recurrence Relations and Derivatives

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8.8.14 $${\frac{\partial }{\partial a}{\gamma }^{\ast }(a,z)|}_{a=0}=-E_1\left(z\right)-\ln z.$$

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Symbols:

$E_1\left(z\right)$: exponential integral, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $a$: parameter

A&S Ref:

6.5.24

Permalink:

http://dlmf.nist.gov/8.8.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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43: 19.30 Lengths of Plane Curves

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19.30.6 $$\frac{\partial s}{\partial (1/k)}=\sqrt{a^2-b^2}F(\varphi ,k)=\sqrt{a^2-b^2}{R}_F(c-1,c-k^2,c),$$ $k^2=(a^2-b^2)/(a^2+\lambda )$, $c={\csc }^2\varphi$.

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\csc z$: cosecant function, $F(\varphi ,k)$: 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, $k$: real or complex modulus and $s$: arclength

Referenced by:

§19.30(i)

Permalink:

http://dlmf.nist.gov/19.30.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.30(i), §19.30 and Ch.19

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44: 11.10 Anger–Weber Functions

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11.10.27 $${\frac{\partial }{\partial \nu }{𝐉}_{\nu }\left(z\right)|}_{\nu =0}=\frac{1}{2}\pi {𝐇}_0\left(z\right),$$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: 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 $\nu$: real or complex order

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vii), §11.10 and Ch.11

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11.10.28 $${\frac{\partial }{\partial \nu }{𝐄}_{\nu }\left(z\right)|}_{\nu =0}=\frac{1}{2}\pi J_0\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: 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 $\nu$: real or complex order

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vii), §11.10 and Ch.11

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45: 10.40 Asymptotic Expansions for Large Argument

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10.40.8 $$\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }\sim {\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}\frac{\nu {\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}\frac{{\alpha }_k(\nu )}{{(8z)}^k},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: 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 $x$ , $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_k(\nu )$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.40.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40(i), §10.40 and Ch.10

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46: 28.28 Integrals, Integral Representations, and Integral Equations

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28.28.4 $$\frac{\mathrm{i}h}{\pi }{\int }_0^{2\pi }\frac{\partial w}{\partial \alpha }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{ce}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{ce}}_n^{\prime }(\alpha ,h^2){\mathrm{Mc}}_n^{(1)}(z,h),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter

Permalink:

http://dlmf.nist.gov/28.28.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

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28.28.5 $$\frac{\mathrm{i}h}{\pi }{\int }_0^{2\pi }\frac{\partial w}{\partial \alpha }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{se}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{se}}_n^{\prime }(\alpha ,h^2){\mathrm{Ms}}_n^{(1)}(z,h).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter

Permalink:

http://dlmf.nist.gov/28.28.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

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28.28.8 $$\frac{1}{\pi }{\int }_{{ℒ}_j}2\mathrm{i}h\frac{\partial w}{\partial \alpha }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{me}}_{\nu }(t,h^2)dt={\mathrm{e}}^{\mathrm{i}\nu \pi /2}{\mathrm{me}}_{\nu }^{\prime }(\alpha ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h),$$ $j=3,4$,

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $h$: parameter, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution, $\alpha$: parameter and ${ℒ}_j$: integration paths

A&S Ref:

20.7.19 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.28.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

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47: 8.19 Generalized Exponential Integral

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8.19.15 $$\frac{{\partial }^j{E}_p\left(z\right)}{{\partial p}^j}={(-1)}^j{\int }_1^{\mathrm{\infty }}{(\ln t)}^j{t}^{-p}{\mathrm{e}}^{-zt}dt,$$ $\mathrm{\Re }z>0$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\mathrm{\Re }$: real part, $z$: complex variable, $p$: parameter and $j$: numbers

Permalink:

http://dlmf.nist.gov/8.19.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(v), §8.19(v), §8.19 and Ch.8

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48: 18.39 Applications in the Physical Sciences

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18.39.7 $$\mathscr{H}=-\frac{{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}+V(x),$$ $x\in (a,b)$ $(ℝ)$,

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Symbols:

$\in$: element of, $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $ℝ$: real line, $m$: nonnegative integer and $x$: real variable

Referenced by:

§18.39(i), §18.39(ii), Erratum (V1.2.0) Section 18.39

Permalink:

http://dlmf.nist.gov/18.39.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

… ►Here the term $-\frac{{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}$ represents the quantum kinetic energy of a single particle of mass $m$, and $V(x)$ its potential energy. … ►

18.39.9 $$\mathrm{i}\mathrm{\hslash }\frac{\partial \mathrm{\Psi }(x,t)}{\partial t}=\mathscr{H}\mathrm{\Psi }(x,t),$$

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Symbols:

$\mathrm{i}$: imaginary unit, $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $t$: real variable and $x$: real variable

Referenced by:

§18.39(i)

Permalink:

http://dlmf.nist.gov/18.39.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

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49: 13.8 Asymptotic Approximations for Large Parameters

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13.8.16 $$(k+1)c_{k+1}(z)+\sum _{s=0}^k\left(\frac{b{B}_{s+1}}{(s+1)!}+\frac{z(s+1)B_{s+2}}{(s+2)!}\right)c_{k-s}(z)=0,$$ $k=0,1,2,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.8.16), §13.8(iii)

Permalink:

http://dlmf.nist.gov/13.8.E16

Encodings:

pMML, png, TeX

Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added. In Temme (2015, Equation 10.3.28) the generating function $f(z,t)$ for $c_k(z)$ is given. It satisfies,

$$\frac{\partial f}{\partial t}=\left(b\left(\frac{1}{t}-\frac{1}{{\mathrm{e}}^t-1}\right)-z\left(\frac{1}{t^2}-\frac{{\mathrm{e}}^t}{{\left({\mathrm{e}}^t-1\right)}^2}\right)\right)f.$$

For (13.8.16) combine this differential equation with $f(z,t)={\sum }_{k=0}^{\mathrm{\infty }}{c}_k(z)t^k$ and (24.4.1).

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

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50: 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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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

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