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11: 10.12 Generating Function and Associated Series

§10.12 Generating Function and Associated Series

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$\cos \left(z\sin \theta \right)=J_0\left(z\right)+2{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{J}_{2k}\left(z\right)\cos \left(2k\theta \right),$

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$\sin \left(z\sin \theta \right)=2{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{J}_{2k+1}\left(z\right)\sin \left((2k+1)\theta \right),$

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$\sin z=2J_1\left(z\right)-2J_3\left(z\right)+2J_5\left(z\right)-\mathrm{\cdots },$

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$\frac{1}{2}z\sin z=4J_2\left(z\right)-16J_4\left(z\right)+36J_6\left(z\right)-\mathrm{\cdots }.$

12: 4.40 Integrals

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4.40.1 $$\int \sinh xdx=\cosh x,$$

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Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.77

Permalink:

http://dlmf.nist.gov/4.40.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

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4.40.2 $$\int \cosh xdx=\sinh x,$$

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Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.78

Permalink:

http://dlmf.nist.gov/4.40.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

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4.40.6 $$\int \coth xdx=\ln \left(\sinh x\right),$$ .

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Symbols:

$dx$: differential of $x$ , $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.82

Permalink:

http://dlmf.nist.gov/4.40.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

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4.40.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-x}\frac{\sin \left(ax\right)}{\sinh x}dx=\frac{1}{2}\pi \coth \left(\frac{1}{2}\pi a\right)-\frac{1}{a},$$ $a\ne 0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iii), §4.40 and Ch.4

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4.40.14 $$\int \mathrm{arccsch}xdx=x\mathrm{arccsch}x+\mathrm{arcsinh}x,$$ ,

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Symbols:

$dx$: differential of $x$ , $\mathrm{arccsch}z$: inverse hyperbolic cosecant function, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.6.46 (modified. The first printing had an error.)

Permalink:

http://dlmf.nist.gov/4.40.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iv), §4.40 and Ch.4

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13: 6.11 Relations to Other Functions

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6.11.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}U(1,1,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $z$: complex variable

Referenced by:

§6.11, 5th item, §6.6

Permalink:

http://dlmf.nist.gov/6.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

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6.11.3 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)=U(1,1,-\mathrm{i}z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.11, 5th item

Permalink:

http://dlmf.nist.gov/6.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

14: 4.34 Derivatives and Differential Equations

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4.34.1 $$\frac{d}{dz}\sinh z=\cosh z,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.71

Permalink:

http://dlmf.nist.gov/4.34.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.2 $$\frac{d}{dz}\cosh z=\sinh z,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.72

Permalink:

http://dlmf.nist.gov/4.34.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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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:

pMML, png, TeX

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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15: 4.20 Derivatives and Differential Equations

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4.20.1 $$\frac{d}{dz}\sin z=\cos z,$$

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Symbols:

$\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.105

Permalink:

http://dlmf.nist.gov/4.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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4.20.2 $$\frac{d}{dz}\cos z=-\sin z,$$

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Symbols:

$\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.106

Permalink:

http://dlmf.nist.gov/4.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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4.20.7 $$\frac{d^n}{{dz}^n}\sin z=\sin \left(z+\frac{1}{2}n\pi \right),$$

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Symbols:

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

A&S Ref:

4.3.111

Permalink:

http://dlmf.nist.gov/4.20.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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

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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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16: 4.31 Special Values and Limits

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4.31.1 $$\underset{z\to 0}{lim}\frac{\sinh z}{z}=1,$$

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Symbols:

$\sinh z$: hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.31 and Ch.4

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17: 23.11 Integral Representations

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23.11.2 $$\mathrm{\wp }\left(z\right)=\frac{1}{z^2}+8{\int }_0^{\mathrm{\infty }}s\left({\mathrm{e}}^{-s}{\sinh }^2\left(\frac{1}{2}zs\right)f_1(s,\tau )+{\mathrm{e}}^{\mathrm{i}\tau s}{\sin }^2\left(\frac{1}{2}zs\right)f_2(s,\tau )\right)ds,$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $𝕃$: lattice, $z$: complex, $\tau$: complex variable and $f_j(s,\tau )$: functions

Permalink:

http://dlmf.nist.gov/23.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.11 and Ch.23

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23.11.3 $$\zeta \left(z\right)=\frac{1}{z}+{\int }_0^{\mathrm{\infty }}\left({\mathrm{e}}^{-s}\left(zs-\sinh \left(zs\right)\right)f_1(s,\tau )-{\mathrm{e}}^{\mathrm{i}\tau s}\left(zs-\sin \left(zs\right)\right)f_2(s,\tau )\right)ds,$$

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $𝕃$: lattice, $z$: complex, $\tau$: complex variable and $f_j(s,\tau )$: functions

Permalink:

http://dlmf.nist.gov/23.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.11 and Ch.23

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18: 6.2 Definitions and Interrelations

… ► $\mathrm{Si}\left(z\right)$ is an odd entire function. … ►

§6.2(iii) Auxiliary Functions

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6.2.18 $$\mathrm{g}\left(z\right)=-\mathrm{Ci}\left(z\right)\cos z-\mathrm{si}\left(z\right)\sin z.$$

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Defines:

$\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals

Symbols:

$\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{si}\left(z\right)$: sine integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

5.2.7

Referenced by:

§6.4, §6.5

Permalink:

http://dlmf.nist.gov/6.2.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(iii), §6.2 and Ch.6

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6.2.19 $$\mathrm{Si}\left(z\right)=\frac{1}{2}\pi -\mathrm{f}\left(z\right)\cos z-\mathrm{g}\left(z\right)\sin z,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{Si}\left(z\right)$: sine integral and $z$: complex variable

A&S Ref:

5.2.8

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(iii), §6.2 and Ch.6

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6.2.20 $$\mathrm{Ci}\left(z\right)=\mathrm{f}\left(z\right)\sin z-\mathrm{g}\left(z\right)\cos z.$$

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Symbols:

$\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\sin z$: sine function, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

A&S Ref:

5.2.9

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.2.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(iii), §6.2 and Ch.6

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19: 14.25 Integral Representations

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14.25.1 $$P_{\nu }^{-\mu }\left(z\right)=\frac{{\left(z^2-1\right)}^{\mu /2}}{2^{\nu }\mathrm{\Gamma }\left(\mu -\nu \right)\mathrm{\Gamma }\left(\nu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\nu +1}}{{(z+\cosh t)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\mu >\mathrm{\Re }\nu >-1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.25

Permalink:

http://dlmf.nist.gov/14.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.25 and Ch.14

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14.25.2 $${𝑸}_{\nu }^{\mu }\left(z\right)=\frac{{\pi }^{1/2}{\left(z^2-1\right)}^{\mu /2}}{2^{\mu }\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{\left(z+{(z^2-1)}^{1/2}\cosh t\right)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\left(\nu +1\right)>\mathrm{\Re }\mu >-\frac{1}{2}$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.25

Permalink:

http://dlmf.nist.gov/14.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.25 and Ch.14

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20: 4.29 Graphics

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§4.29(i) Real Arguments

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Figure 4.29.1: $\sinh x$ and $\cosh x$. Magnify

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§4.29(ii) Complex Arguments

►The conformal mapping $w=\sinh z$ is obtainable from Figure 4.15.7 by rotating both the $w$-plane and the $z$-plane through an angle $\frac{1}{2}\pi$, compare (4.28.8). ►The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …