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11: 4.34 Derivatives and Differential Equations

… ►With $a\ne 0$, the general solutions of the differential equations … ►

4.34.11 $$w=A\cosh \left(az\right)+B\sinh \left(az\right),$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $a$: real or complex constant, $z$: complex variable, $w$: solution, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E11

Encodings:

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See also:

Annotations for §4.34 and Ch.4

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4.34.12 $$w=(1/a)\sinh \left(az+c\right),$$

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Symbols:

$\sinh z$: hyperbolic sine function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.13 $$w=(1/a)\cosh \left(az+c\right),$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.14 $$w=(1/a)\coth \left(az+c\right),$$

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Symbols:

$\coth z$: hyperbolic cotangent function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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12: 29.4 Graphics

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Figure 29.4.9: $a_{\nu }^0\left(k^2\right)$ as a function of $\nu$ and $k^2$. Magnify 3D Help

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Figure 29.4.10: $b_{\nu }^1\left(k^2\right)$ as a function of $\nu$ and $k^2$. Magnify 3D Help

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Figure 29.4.11: $a_{\nu }^1\left(k^2\right)$ as a function of $\nu$ and $k^2$. Magnify 3D Help

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Figure 29.4.12: $b_{\nu }^2\left(k^2\right)$ as a function of $\nu$ and $k^2$. Magnify 3D Help

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Figure 29.4.25: ${\mathrm{𝐸𝑐}}_{1.5}^0(x,k^2)$ as a function of $x$ and $k^2$. Magnify 3D Help

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13: 10.71 Integrals

… ►In the following equations $f_{\nu },g_{\nu }$ is any one of the four ordered pairs given in (10.63.1), and ${\widehat{f}}_{\nu },{\widehat{g}}_{\nu }$ is either the same ordered pair or any other ordered pair in (10.63.1). ►

10.71.1 $$\int x^{1+\nu }{f}_{\nu }dx=-\frac{x^{1+\nu }}{\sqrt{2}}(f_{\nu +1}-g_{\nu +1})=-x^{1+\nu }\left(\frac{\nu }{x}{g}_{\nu }-g_{\nu }^{\prime }\right),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $x$: real variable, $\nu$: complex parameter, $f_{\nu }$ a Kelvin function and $g_{\nu }$ a Kelvin function

A&S Ref:

9.9.21

Permalink:

http://dlmf.nist.gov/10.71.E1

Encodings:

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See also:

Annotations for §10.71(i), §10.71 and Ch.10

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10.71.3 $$\int x(f_{\nu }{\widehat{g}}_{\nu }-g_{\nu }{\widehat{f}}_{\nu })dx=\frac{x}{2\sqrt{2}}\left({\widehat{f}}_{\nu }(f_{\nu +1}+g_{\nu +1})-{\widehat{g}}_{\nu }(f_{\nu +1}-g_{\nu +1})-f_{\nu }({\widehat{f}}_{\nu +1}+{\widehat{g}}_{\nu +1})+g_{\nu }({\widehat{f}}_{\nu +1}-{\widehat{g}}_{\nu +1})\right)=\frac{1}{2}x(f_{\nu }^{\prime }{\widehat{f}}_{\nu }-f_{\nu }{\widehat{f}}_{\nu }^{\prime }+g_{\nu }^{\prime }{\widehat{g}}_{\nu }-g_{\nu }{\widehat{g}}_{\nu }^{\prime }),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $x$: real variable, $\nu$: complex parameter, $f_{\nu }$ a Kelvin function, $g_{\nu }$ a Kelvin function, ${\widehat{f}}_{\nu }$ a Kelvin function and ${\widehat{g}}_{\nu }$ a Kelvin function

A&S Ref:

9.9.23

Permalink:

http://dlmf.nist.gov/10.71.E3

Encodings:

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See also:

Annotations for §10.71(i), §10.71 and Ch.10

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10.71.4 $$\int x(f_{\nu }{\widehat{g}}_{\nu }+g_{\nu }{\widehat{f}}_{\nu })dx=\frac{1}{4}{x}^2(2f_{\nu }{\widehat{g}}_{\nu }-f_{\nu -1}{\widehat{g}}_{\nu +1}-f_{\nu +1}{\widehat{g}}_{\nu -1}+2g_{\nu }{\widehat{f}}_{\nu }-g_{\nu -1}{\widehat{f}}_{\nu +1}-g_{\nu +1}{\widehat{f}}_{\nu -1}).$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $x$: real variable, $\nu$: complex parameter, $f_{\nu }$ a Kelvin function, $g_{\nu }$ a Kelvin function, ${\widehat{f}}_{\nu }$ a Kelvin function and ${\widehat{g}}_{\nu }$ a Kelvin function

A&S Ref:

9.9.24

Permalink:

http://dlmf.nist.gov/10.71.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.71(i), §10.71 and Ch.10

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10.71.6 $$\int x{f}_{\nu }{g}_{\nu }dx=\frac{1}{4}{x}^2\left(2f_{\nu }{g}_{\nu }-f_{\nu -1}{g}_{\nu +1}-f_{\nu +1}{g}_{\nu -1}\right),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $x$: real variable, $\nu$: complex parameter, $f_{\nu }$ a Kelvin function and $g_{\nu }$ a Kelvin function

A&S Ref:

9.9.26

Permalink:

http://dlmf.nist.gov/10.71.E6

Encodings:

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See also:

Annotations for §10.71(i), §10.71 and Ch.10

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14: Software Index

… ► ►►►►►►►

	Open Source													With Book						Commercial
…
4.48(iv) Lambert $W$-Function	✓					✓			✓			✓	✓								✓	✓	a
…
9.20(vi) Scorer Functions	✓											a	✓
…
10.77(ix) Integrals of Bessel Functions	✓										✓	a	✓	✓					✓		✓	✓	a
10.77(x) Zeros of Bessel Functions					✓							a	✓	✓	✓				✓		✓	✓	a
…
28.36(iii) Mathieu Functions														✓					✓	✓	✓	✓	a		Van Buren
…

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15: 23.18 Modular Transformations

… ►

23.18.3 $$\lambda \left(𝒜\tau \right)=\lambda \left(\tau \right),$$

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Symbols:

$\lambda \left(\tau \right)$: elliptic modular function, $\tau$: complex variable and $𝒜$: bilinear transformation

Permalink:

http://dlmf.nist.gov/23.18.E3

Encodings:

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See also:

Annotations for §23.18, §23.18 and Ch.23

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23.18.5 $$\eta \left(𝒜\tau \right)=\epsilon (𝒜){\left(-\mathrm{i}(c\tau +d)\right)}^{1/2}\eta \left(\tau \right),$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\mathrm{i}$: imaginary unit, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer and $\epsilon (𝒜)$: function

Permalink:

http://dlmf.nist.gov/23.18.E5

Encodings:

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See also:

Annotations for §23.18, §23.18 and Ch.23

►where the square root has its principal value and ►

23.18.6 $$\epsilon (𝒜)=\exp \left(\pi \mathrm{i}\left(\frac{a+d}{12c}+s(-d,c)\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $𝒜$: bilinear transformation, $a$: integer, $c$: integer, $d$: integer and $\epsilon (𝒜)$: function

Permalink:

http://dlmf.nist.gov/23.18.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

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16: 8.16 Generalizations

… ►For a generalization of the incomplete gamma function, including asymptotic approximations, see Chaudhry and Zubair (1994, 2001) and Chaudhry et al. (1996). …

17: 4.20 Derivatives and Differential Equations

… ►

4.20.8 $$\frac{d^n}{{dz}^n}\cos z=\cos \left(z+\frac{1}{2}n\pi \right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: integer and $z$: complex variable

A&S Ref:

4.3.112

Permalink:

http://dlmf.nist.gov/4.20.E8

Encodings:

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See also:

Annotations for §4.20 and Ch.4

►With $a\ne 0$, the general solutions of the differential equations … ►

4.20.12 $$w=A\cos \left(az\right)+B\sin \left(az\right),$$

ⓘ

Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $a$: real or complex constant, $z$: complex variable, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.20.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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4.20.13 $$w=(1/a)\sin \left(az+c\right),$$

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Symbols:

$\sin z$: sine function, $a$: real or complex constant, $z$: complex variable and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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4.20.14 $$w=(1/a)\tan \left(az+c\right),$$

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Symbols:

$\tan z$: tangent function, $a$: real or complex constant, $z$: complex variable and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.20.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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18: Viewing DLMF Interactive 3D Graphics

… ►Users can render a 3D scene and interactively rotate, scale, and otherwise explore a function surface. …

19: Roderick S. C. Wong

… ►He is also the coauthor of Special Functions: A Graduate Text with Richard Beals, that will be published by Cambridge University Press in 2010. … ►Wong served as a Validator for the original release and publication in May 2010 of the NIST Digital Library of Mathematical Functions and the NIST Handbook of Mathematical Functions. …

20: 5 Gamma Function

Chapter 5 Gamma Function

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