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1: 4.14 Definitions and Periodicity

… ►

4.14.4 $$\tan z=\frac{\sin z}{\cos z},$$

ⓘ

Defines:

$\tan z$: tangent function

Symbols:

$\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.3

Referenced by:

§4.14, §4.45(i)

Permalink:

http://dlmf.nist.gov/4.14.E4

Encodings:

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See also:

Annotations for §4.14 and Ch.4

►

4.14.5 $$\csc z=\frac{1}{\sin z},$$

ⓘ

Defines:

$\csc z$: cosecant function

Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.4

Permalink:

http://dlmf.nist.gov/4.14.E5

Encodings:

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Annotations for §4.14 and Ch.4

… ►The functions $\sin z$ and $\cos z$ are entire. In $ℂ$ the zeros of $\sin z$ are $z=k\pi$, $k\in \mathbb{Z}$; the zeros of $\cos z$ are $z=\left(k+\frac{1}{2}\right)\pi$, $k\in \mathbb{Z}$. The functions $\tan z$, $\csc z$, $\sec z$, and $\cot z$ are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7). …

2: 10.29 Recurrence Relations and Derivatives

… ►With ${𝒵}_{\nu }\left(z\right)$ defined as in §10.25(ii), ►

${𝒵}_{\nu -1}\left(z\right)-{𝒵}_{\nu +1}\left(z\right)=(2\nu /z){𝒵}_{\nu }\left(z\right),$

… ►

${𝒵}_{\nu }^{\prime }\left(z\right)={𝒵}_{\nu -1}\left(z\right)-(\nu /z){𝒵}_{\nu }\left(z\right),$

►

${𝒵}_{\nu }^{\prime }\left(z\right)={𝒵}_{\nu +1}\left(z\right)+(\nu /z){𝒵}_{\nu }\left(z\right).$

… ►For results on modified quotients of the form $z{𝒵}_{\nu \pm 1}\left(z\right)/{𝒵}_{\nu }\left(z\right)$ see Onoe (1955) and Onoe (1956). …

3: 4.28 Definitions and Periodicity

… ►

4.28.2 $$\cosh z=\frac{{\mathrm{e}}^z+{\mathrm{e}}^{-z}}{2},$$

ⓘ

Defines:

$\cosh z$: hyperbolic cosine function

Symbols:

$\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

4.5.2

Permalink:

http://dlmf.nist.gov/4.28.E2

Encodings:

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See also:

Annotations for §4.28 and Ch.4

… ►

4.28.9 $$\cos \left(\mathrm{i}z\right)=\cosh z,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.5.8

Permalink:

http://dlmf.nist.gov/4.28.E9

Encodings:

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See also:

Annotations for §4.28, §4.28 and Ch.4

… ►

4.28.12 $$\sec \left(\mathrm{i}z\right)=\mathrm{sech}z,$$

ⓘ

Symbols:

$\mathrm{sech}z$: hyperbolic secant function, $\mathrm{i}$: imaginary unit, $\sec z$: secant function and $z$: complex variable

A&S Ref:

4.5.11

Permalink:

http://dlmf.nist.gov/4.28.E12

Encodings:

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Annotations for §4.28, §4.28 and Ch.4

… ►The functions $\sinh z$ and $\cosh z$ have period $2\pi \mathrm{i}$, and $\tanh z$ has period $\pi \mathrm{i}$. The zeros of $\sinh z$ and $\cosh z$ are $z=\mathrm{i}k\pi$ and $z=\mathrm{i}\left(k+\frac{1}{2}\right)\pi$, respectively, $k\in \mathbb{Z}$.

4: 10.51 Recurrence Relations and Derivatives

… ►Let $f_n(z)$ denote any of ${\mathsf{j}}_n\left(z\right)$, ${\mathsf{y}}_n\left(z\right)$, ${\mathsf{h}}_n^{(1)}\left(z\right)$, or ${\mathsf{h}}_n^{(2)}\left(z\right)$. … ►

$${\left(\frac{1}{z}\frac{d}{dz}\right)}^m(z^{n+1}{f}_n(z))=z^{n-m+1}{f}_{n-m}(z),$$ $m=0,1,\mathrm{\dots },n$,

►

$${\left(\frac{1}{z}\frac{d}{dz}\right)}^m(z^{-n}{f}_n(z))={(-1)}^m{z}^{-n-m}{f}_{n+m}(z),$$ $m=0,1,\mathrm{\dots }.$

… ►Let $g_n(z)$ denote ${\mathsf{i}}_n^{(1)}\left(z\right)$, ${\mathsf{i}}_n^{(2)}\left(z\right)$, or ${(-1)}^n$ ${\mathsf{k}}_n\left(z\right)$. Then …

5: 10.36 Other Differential Equations

… ►The quantity ${\lambda }^2$ in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by $-{\lambda }^2$ if at the same time the symbol $𝒞$ in the given solutions is replaced by $𝒵$. … ►

10.36.1 $$z^2(z^2+{\nu }^2)w^{\prime \prime }+z(z^2+3{\nu }^2)w^{\prime }-\left({(z^2+{\nu }^2)}^2+z^2-{\nu }^2\right)w=0,$$ $w={𝒵}_{\nu }^{\prime }\left(z\right)$,

ⓘ

Symbols:

${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.36.E1

Encodings:

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See also:

Annotations for §10.36 and Ch.10

►

10.36.2 $$z^2{w}^{\prime \prime }+z(1\pm 2z)w^{\prime }+(\pm z-{\nu }^2)w=0,$$ $w={\mathrm{e}}^{\mp z}{𝒵}_{\nu }\left(z\right)$.

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.41

Permalink:

http://dlmf.nist.gov/10.36.E2

Encodings:

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Annotations for §10.36 and Ch.10

►Differential equations for products can be obtained from (10.13.9)–(10.13.11) by replacing $z$ by $\mathrm{i}z$.

6: 22.13 Derivatives and Differential Equations

… ►

22.13.4 $${\left(\frac{d}{dz}\mathrm{cd}(z,k)\right)}^2=\left(1-{\mathrm{cd}}^2(z,k)\right)\left(1-k^2{\mathrm{cd}}^2(z,k)\right),$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.13.E4

Encodings:

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See also:

Annotations for §22.13(ii), §22.13 and Ch.22

… ►

22.13.7 $${\left(\frac{d}{dz}\mathrm{dc}(z,k)\right)}^2=\left({\mathrm{dc}}^2(z,k)-1\right)\left({\mathrm{dc}}^2(z,k)-k^2\right),$$

ⓘ

Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.13.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

… ►

22.13.10 $${\left(\frac{d}{dz}\mathrm{ns}(z,k)\right)}^2=\left({\mathrm{ns}}^2(z,k)-k^2\right)\left({\mathrm{ns}}^2(z,k)-1\right),$$

ⓘ

Symbols:

$\mathrm{ns}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.13.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

… ►

22.13.19 $$\frac{d^2}{{dz}^2}\mathrm{dc}(z,k)=-(1+k^2)\mathrm{dc}(z,k)+2{\mathrm{dc}}^3(z,k),$$

ⓘ

Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.13.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(iii), §22.13 and Ch.22

… ►

22.13.22 $$\frac{d^2}{{dz}^2}\mathrm{ns}(z,k)=-(1+k^2)\mathrm{ns}(z,k)+2{\mathrm{ns}}^3(z,k),$$

ⓘ

Symbols:

$\mathrm{ns}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.13.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(iii), §22.13 and Ch.22

…

7: 7.4 Symmetry

… ►

7.4.1 $$\mathrm{erf}\left(-z\right)=-\mathrm{erf}\left(z\right),$$

ⓘ

Symbols:

$\mathrm{erf}z$: error function and $z$: complex variable

A&S Ref:

7.1.9

Permalink:

http://dlmf.nist.gov/7.4.E1

Encodings:

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See also:

Annotations for §7.4 and Ch.7

►

7.4.2 $$\mathrm{erfc}\left(-z\right)=2-\mathrm{erfc}\left(z\right),$$

ⓘ

Symbols:

$\mathrm{erfc}z$: complementary error function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.4.E2

Encodings:

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See also:

Annotations for §7.4 and Ch.7

►

7.4.3 $$w\left(-z\right)=2{\mathrm{e}}^{-z^2}-w\left(z\right).$$

ⓘ

Symbols:

$w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

7.1.11

Permalink:

http://dlmf.nist.gov/7.4.E3

Encodings:

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See also:

Annotations for §7.4 and Ch.7

►

7.4.4 $$F\left(-z\right)=-F\left(z\right).$$

ⓘ

Symbols:

$F\left(z\right)$: Dawson’s integral and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.4.E4

Encodings:

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Annotations for §7.4 and Ch.7

►

$C\left(-z\right)=-C\left(z\right),$

…

8: 22.10 Maclaurin Series

… ►

§22.10(i) Maclaurin Series in $z$

… ►The full expansions converge when . … ►

22.10.4 $$\mathrm{sn}(z,k)=\sin z-\frac{k^2}{4}(z-\sin z\cos z)\cos z+O\left(k^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.1

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E4

Encodings:

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Annotations for §22.10(ii), §22.10 and Ch.22

►

22.10.5 $$\mathrm{cn}(z,k)=\cos z+\frac{k^2}{4}(z-\sin z\cos z)\sin z+O\left(k^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.2

Referenced by:

§22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E5

Encodings:

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See also:

Annotations for §22.10(ii), §22.10 and Ch.22

… ►

22.10.8 $$\mathrm{cn}(z,k)=\mathrm{sech}z+\frac{k^{\prime }{}^2}{4}(z-\sinh z\cosh z)\tanh z\mathrm{sech}z+O\left(k^{\prime }{}^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.2

Permalink:

http://dlmf.nist.gov/22.10.E8

Encodings:

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See also:

Annotations for §22.10(ii), §22.10 and Ch.22

…

9: 4.20 Derivatives and Differential Equations

… ►

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

ⓘ

Symbols:

$\cos z$: cosine function, $\frac{df}{dx}$: derivative, $\sin z$: 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

►

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

ⓘ

Symbols:

$\cos z$: cosine function, $\frac{df}{dx}$: derivative, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.106

Permalink:

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

Encodings:

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See also:

Annotations for §4.20 and Ch.4

►

4.20.3 $$\frac{d}{dz}\tan z={\sec }^{2}z,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative, $\sec z$: secant function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.107

Permalink:

http://dlmf.nist.gov/4.20.E3

Encodings:

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See also:

Annotations for §4.20 and Ch.4

►

4.20.4 $$\frac{d}{dz}\csc z=-\csc z\cot z,$$

ⓘ

Symbols:

$\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{df}{dx}$: derivative and $z$: complex variable

A&S Ref:

4.3.108

Permalink:

http://dlmf.nist.gov/4.20.E4

Encodings:

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See also:

Annotations for §4.20 and Ch.4

►

4.20.5 $$\frac{d}{dz}\sec z=\sec z\tan z,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative, $\sec z$: secant function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.109

Permalink:

http://dlmf.nist.gov/4.20.E5

Encodings:

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See also:

Annotations for §4.20 and Ch.4

…

10: 4.33 Maclaurin Series and Laurent Series

… ►

4.33.1 $$\sinh z=z+\frac{z^3}{3!}+\frac{z^5}{5!}+\mathrm{\cdots },$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.62

Permalink:

http://dlmf.nist.gov/4.33.E1

Encodings:

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See also:

Annotations for §4.33 and Ch.4

►

4.33.2 $$\cosh z=1+\frac{z^2}{2!}+\frac{z^4}{4!}+\mathrm{\cdots }.$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function and $z$: complex variable

A&S Ref:

4.5.63

Permalink:

http://dlmf.nist.gov/4.33.E2

Encodings:

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See also:

Annotations for §4.33 and Ch.4

►

4.33.3 $$\tanh z=z-\frac{z^3}{3}+\frac{2}{15}{z}^5-\frac{17}{315}{z}^7+\mathrm{\cdots }+\frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ .

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\tanh z$: hyperbolic tangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.64

Permalink:

http://dlmf.nist.gov/4.33.E3

Encodings:

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See also:

Annotations for §4.33 and Ch.4

… ►For expansions that correspond to (4.19.4)–(4.19.9), change $z$ to $\mathrm{i}z$ and use (4.28.8)–(4.28.13).