partial differentiation

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11: 19.4 Derivatives and Differential Equations

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19.4.4 $$\frac{\partial \mathrm{\Pi }({\alpha }^2,k)}{\partial k}=\frac{k}{k_{}^{\prime }{}_{}{}^2(k^2-{\alpha }^2)}(E\left(k\right)-k_{}^{\prime }{}_{}{}^2\mathrm{\Pi }({\alpha }^2,k)).$$

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Symbols:

$E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Permalink:

http://dlmf.nist.gov/19.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.5 $$\frac{\partial F(\varphi ,k)}{\partial k}=\frac{E(\varphi ,k)-k_{}^{\prime }{}_{}{}^{2}F(\varphi ,k)}{kk_{}^{\prime }{}_{}{}^2}-\frac{k\sin \varphi \cos \varphi }{k_{}^{\prime }{}_{}{}^2\sqrt{1-k^2{\sin }^2\varphi }},$$

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: 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 $x$ , $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.6 $$\frac{\partial E(\varphi ,k)}{\partial k}=\frac{E(\varphi ,k)-F(\varphi ,k)}{k},$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: 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:

§19.30(i)

Permalink:

http://dlmf.nist.gov/19.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.7 $$\frac{\partial \mathrm{\Pi }(\varphi ,{\alpha }^2,k)}{\partial k}=\frac{k}{k_{}^{\prime }{}_{}{}^2(k^2-{\alpha }^2)}\left(E(\varphi ,k)-k_{}^{\prime }{}_{}{}^2\mathrm{\Pi }(\varphi ,{\alpha }^2,k)-\frac{k^2\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}\right).$$

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Symbols:

$\cos z$: cosine function, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Permalink:

http://dlmf.nist.gov/19.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

… ►Let $D_k=\partial /\partial k$. …

12: 1.10 Functions of a Complex Variable

… ►is analytic in $D$ and its derivatives of all orders can be found by differentiating under the sign of integration. … ►The convergence of the infinite product is uniform if the sequence of partial products converges uniformly. … ►

§1.10(x) Infinite Partial Fractions

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Mittag-Leffler’s Expansion

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13: 10.40 Asymptotic Expansions for Large Argument

… ►Corresponding expansions for $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$, $I_{\nu }^{\prime }\left(z\right)$, and $K_{\nu }^{\prime }\left(z\right)$ for other ranges of $\mathrm{ph}z$ are obtainable by combining (10.34.3), (10.34.4), (10.34.6), and their differentiated forms, with (10.40.2) and (10.40.4). … ►

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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