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11: 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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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 $\mathrm{i}z$.

12: 28.8 Asymptotic Expansions for Large $q$

… ►The approximations apply when the parameters $a$ and $q$ are real and large, and are uniform with respect to various regions in the $z$-plane. … ►They are uniform with respect to $a$ when $-2q\le a\le (2-\delta )q$, where $\delta$ is an arbitrary constant such that , and also with respect to $z$ in the semi-infinite strip given by $0\le \mathrm{\Re }z\le \pi$ and $\mathrm{\Im }z\ge 0$. ►The approximations are expressed in terms of Whittaker functions $W_{\kappa ,\mu }\left(z\right)$ and $M_{\kappa ,\mu }\left(z\right)$ with $\mu =\frac{1}{4}$; compare §2.8(vi). …With additional restrictions on $z$, uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii). ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions ${\mathrm{me}}_{\nu }(z,q)$ (§28.12(ii)) and modified Mathieu functions ${\mathrm{M}}_{\nu }^{(j)}(z,h)$ (§28.20(iii)). …

13: 22.13 Derivatives and Differential Equations

… ►

22.13.1 $${\left(\frac{d}{dz}\mathrm{sn}(z,k)\right)}^2=\left(1-{\mathrm{sn}}^2(z,k)\right)\left(1-k^2{\mathrm{sn}}^2(z,k)\right),$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex and $k$: modulus

Referenced by:

§22.13(iii), §22.18(iv)

Permalink:

http://dlmf.nist.gov/22.13.E1

Encodings:

pMML, png, TeX

See also:

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

►

22.13.2 $${\left(\frac{d}{dz}\mathrm{cn}(z,k)\right)}^2=\left(1-{\mathrm{cn}}^2(z,k)\right)\left(k_{}^{\prime }{}_{}{}^2+k^2{\mathrm{cn}}^2(z,k)\right),$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E2

Encodings:

pMML, png, TeX

See also:

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

►

22.13.3 $${\left(\frac{d}{dz}\mathrm{dn}(z,k)\right)}^2=\left(1-{\mathrm{dn}}^2(z,k)\right)\left({\mathrm{dn}}^2(z,k)-k_{}^{\prime }{}_{}{}^2\right).$$

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E3

Encodings:

pMML, png, TeX

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 of $f$ with respect to $x$ , $z$: 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 of $f$ with respect to $x$ , $z$: 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

…

14: 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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

Annotations for §7.4 and Ch.7

►

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

…

15: 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:

pMML, png, TeX

See also:

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:

pMML, png, TeX

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

…

16: 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 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

►

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

ⓘ

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

►

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

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sec z$: 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 of $f$ with respect to $x$ and $z$: complex variable

A&S Ref:

4.3.108

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

… ►

4.20.6 $$\frac{d}{dz}\cot z=-{\csc }^{2}z,$$

ⓘ

Symbols:

$\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

A&S Ref:

4.3.110

Permalink:

http://dlmf.nist.gov/4.20.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

…

17: 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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

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).

18: 4.6 Power Series

… ►

4.6.1 $$\ln \left(1+z\right)=z-\frac{1}{2}{z}^2+\frac{1}{3}{z}^3-\mathrm{\cdots },$$ $|z|\le 1$, $z\ne -1$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.24

Referenced by:

§4.45(i), §4.6(i), (8.15.2)

Permalink:

http://dlmf.nist.gov/4.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

… ►

4.6.3 $$\ln z=(z-1)-\frac{1}{2}{(z-1)}^2+\frac{1}{3}{(z-1)}^3-\mathrm{\cdots },$$ $|z-1|\le 1$, $z\ne 0$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.26

Permalink:

http://dlmf.nist.gov/4.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

►

4.6.4 $$\ln z=2\left(\left(\frac{z-1}{z+1}\right)+\frac{1}{3}{\left(\frac{z-1}{z+1}\right)}^3+\frac{1}{5}{\left(\frac{z-1}{z+1}\right)}^5+\mathrm{\cdots }\right),$$ $\mathrm{\Re }z\ge 0$, $z\ne 0$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.1.27

Permalink:

http://dlmf.nist.gov/4.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

… ►valid when $a$ is any real or complex constant and . If $a=0,1,2,\mathrm{\dots }$, then the series terminates and $z$ is unrestricted. …

19: 22.6 Elementary Identities

… ►

22.6.2 $$1+{\mathrm{cs}}^2(z,k)=k^2+{\mathrm{ds}}^2(z,k)={\mathrm{ns}}^2(z,k),$$

ⓘ

Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function, $z$: complex and $k$: modulus

A&S Ref:

16.9.4

Permalink:

http://dlmf.nist.gov/22.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(i), §22.6 and Ch.22

… ►

22.6.5 $$\mathrm{sn}(2z,k)=\frac{2\mathrm{sn}(z,k)\mathrm{cn}(z,k)\mathrm{dn}(z,k)}{1-k^2{\mathrm{sn}}^4(z,k)},$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $z$: complex and $k$: modulus

A&S Ref:

16.18.1

Permalink:

http://dlmf.nist.gov/22.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(ii), §22.6 and Ch.22

… ►

22.6.8 $$\mathrm{cd}(2z,k)=\frac{{\mathrm{cd}}^2(z,k)-k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^2(z,k){\mathrm{nd}}^2(z,k)}{1+k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^4(z,k)},$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(ii), §22.6 and Ch.22

►

22.6.9 $$\mathrm{sd}(2z,k)=\frac{2\mathrm{sd}(z,k)\mathrm{cd}(z,k)\mathrm{nd}(z,k)}{1+k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^4(z,k)},$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(ii), §22.6 and Ch.22

►

22.6.10 $$\mathrm{nd}(2z,k)=\frac{{\mathrm{nd}}^2(z,k)+k^2{\mathrm{sd}}^2(z,k){\mathrm{cd}}^2(z,k)}{1+k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^4(z,k)},$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(ii), §22.6 and Ch.22

…

20: 4.21 Identities

… ►

4.21.24 $$\sin \left(-z\right)=-\sin z,$$

ⓘ

Symbols:

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

A&S Ref:

4.3.13

Permalink:

http://dlmf.nist.gov/4.21.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(iii), §4.21 and Ch.4

►

4.21.25 $$\cos \left(-z\right)=\cos z,$$

ⓘ

Symbols:

$\cos z$: cosine function and $z$: complex variable

A&S Ref:

4.3.14

Permalink:

http://dlmf.nist.gov/4.21.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(iii), §4.21 and Ch.4

… ►This result is also valid when $n$ is fractional or complex, provided that $-\pi \le \mathrm{\Re }z\le \pi$. … ►If $t=\tan \left(\frac{1}{2}z\right)$, then … ►With $z=x+\mathrm{i}y$ …