Leibniz formula for derivatives

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31: 4.20 Derivatives and Differential Equations

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

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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:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

… ►

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

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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

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32: 9.2 Differential Equation

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9.2.1 $$\frac{d^{2}w}{{dz}^2}=zw.$$

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Defines:

$w$: ODE solution (locally)

Symbols:

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

Source:

Olver (1997b, (1.01), p. 392)

A&S Ref:

10.4.1 (in slightly different form)

Referenced by:

§36.8, (9.10.10), (9.10.20), (9.10.21), (9.10.8), (9.10.9), §9.10(iii), (9.11.2), §9.11(i), §9.11(iv), §9.12(i), §9.17(ii), §9.17(ii), (9.2.16), §9.2(iii), §9.2(vi), (9.8.14), (9.8.16), (9.8.18), (9.8.19)

Permalink:

http://dlmf.nist.gov/9.2.E1

Encodings:

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See also:

Annotations for §9.2(i), §9.2 and Ch.9

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§9.2(v) Connection Formulas

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9.2.16 $$\frac{dW}{dz}+W^2=z,$$

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Defines:

$W$: Riccati solution (locally)

Symbols:

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

Proof sketch:

Derive properties using (9.2.1).

Permalink:

http://dlmf.nist.gov/9.2.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(vi), §9.2 and Ch.9

► $W=(1/w)dw/dz$, where $w$ is any nontrivial solution of (9.2.1). …

33: 20.7 Identities

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§20.7(ii) Addition Formulas

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§20.7(iii) Duplication Formula

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§20.7(iv) Reduction Formulas for Products

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§20.7(vii) Derivatives of Ratios of Theta Functions

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§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

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34: 4.34 Derivatives and Differential Equations

§4.34 Derivatives and Differential Equations

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4.34.7 $$\frac{d^{2}w}{{dz}^2}-a^{2}w=0,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $a$: real or complex constant, $z$: complex variable and $w$: solution

Permalink:

http://dlmf.nist.gov/4.34.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.8 $${\left(\frac{dw}{dz}\right)}^2-a^2{w}^2=1,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $a$: real or complex constant, $z$: complex variable and $w$: solution

Permalink:

http://dlmf.nist.gov/4.34.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.9 $${\left(\frac{dw}{dz}\right)}^2-a^2{w}^2=-1,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $a$: real or complex constant, $z$: complex variable and $w$: solution

Permalink:

http://dlmf.nist.gov/4.34.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.10 $$\frac{dw}{dz}+a^2{w}^2=1,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $a$: real or complex constant, $z$: complex variable and $w$: solution

Permalink:

http://dlmf.nist.gov/4.34.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

…

35: 4.7 Derivatives and Differential Equations

§4.7 Derivatives and Differential Equations

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§4.7(i) Logarithms

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4.7.1 $$\frac{d}{dz}\ln z=\frac{1}{z},$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.46

Permalink:

http://dlmf.nist.gov/4.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(i), §4.7 and Ch.4

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4.7.5 $$\frac{dw}{dz}=\frac{f^{\prime }(z)}{f(z)}$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $f(z)$: non-vanishing analytic function

Permalink:

http://dlmf.nist.gov/4.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(i), §4.7 and Ch.4

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§4.7(ii) Exponentials and Powers

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36: 22.13 Derivatives and Differential Equations

§22.13 Derivatives and Differential Equations

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§22.13(i) Derivatives

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Table 22.13.1: Derivatives of Jacobian elliptic functions with respect to variable.

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$\frac{d}{dz}(\mathrm{sn}z)$ $=$	$\mathrm{cn}z\mathrm{dn}z$	$\frac{d}{dz}(\mathrm{dc}z)$ =	$k_{}^{\prime }{}_{}{}^2\mathrm{sc}z\mathrm{nc}z$
…

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►Note that each derivative in Table 22.13.1 is a constant multiple of the product of the corresponding copolar functions. … ►

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

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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

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37: 25.2 Definition and Expansions

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25.2.5 $${\gamma }_n=\underset{m\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^m\frac{{(\ln k)}^n}{k}-\frac{{(\ln m)}^{n+1}}{n+1}\right).$$

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Defines:

${\gamma }_n$: Stieltjes constants

Symbols:

$\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Keywords:

definition, Stieltjes constants

Sources:

Hardy (1912, p. 215); Ivić (1985, (1.12), p. 4)

Referenced by:

§25.2(ii), §25.6(ii), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E5

Encodings:

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See also:

Annotations for §25.2(ii), §25.2 and Ch.25

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§25.2(iii) Representations by the Euler–Maclaurin Formula

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25.2.8 $$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s{\int }_N^{\mathrm{\infty }}\frac{x-\lfloor x\rfloor }{x^{s+1}}dx,$$ $\mathrm{\Re }s>0$, $N=1,2,3,\mathrm{\dots }$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Source:

Apostol (1976, (25), p. 269)

Proof sketch:

Derivable from Apostol (1976, Theorem 12.21, p. 269) by setting $a=1$ and $N\mapsto N-1$.

A&S Ref:

23.2.9

Referenced by:

(25.2.9)

Permalink:

http://dlmf.nist.gov/25.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

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25.2.9 $$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-\frac{1}{2}{N}^{-s}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}{N}^{1-s-2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_N^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$; $n,N=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Proof sketch:

Derivable from (25.2.8) by repeated integration by parts.

Referenced by:

§25.18(i), (25.2.10)

Permalink:

http://dlmf.nist.gov/25.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

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25.2.10 $$\zeta \left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_1^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$, $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler–Maclaurin formula

Proof sketch:

Derivable from (25.2.9) with $N=1$.

A&S Ref:

23.2.3

Permalink:

http://dlmf.nist.gov/25.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

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38: 4.13 Lambert $W$-Function

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4.13.4 $$\frac{dW}{dz}=\frac{{\mathrm{e}}^{-W}}{1+W}=\frac{W}{z(1+W)}.$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Referenced by:

§4.45(iii)

Permalink:

http://dlmf.nist.gov/4.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.13 and Ch.4

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4.13.4_1 $$\frac{d^{n}W}{{dz}^n}=\frac{{\mathrm{e}}^{-nW}{p}_{n-1}(W)}{{\left(1+W\right)}^{2n-1}},$$ $n=1,2,3,\mathrm{\dots }$,

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: integer and $z$: complex variable

Notes:

See Corless et al. (1997, formula (7)).

Referenced by:

§4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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4.13.5_1 $${\left(\frac{W_0\left(z\right)}{z}\right)}^a={\mathrm{e}}^{-a{W}_0\left(z\right)}=\sum _{n=0}^{\mathrm{\infty }}\frac{a{\left(n+a\right)}^{n-1}}{n!}{\left(-z\right)}^n,$$ , $a\in ℂ$.

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $ℂ$: complex plane, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $a$: real or complex constant and $z$: complex variable

Notes:

See Corless et al. (1996, formula (2.36)).

Referenced by:

§4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E5_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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4.13.5_2 $$\frac{1}{1+W_0\left(-z\right)}=\sum _{n=0}^{\mathrm{\infty }}\frac{n^n}{n!}{z}^n,$$ .

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer and $z$: complex variable

Notes:

See Corless et al. (1997, formula (14)).

Referenced by:

§4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E5_2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.13 and Ch.4

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4.13.9_1 $$W_0\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}{d}_n{\left(\mathrm{e}z+1\right)}^{n/2},$$ , ,

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $n$: integer and $z$: complex variable

Proof sketch:

Substitute (4.13.9_1) into $z(1+W)\frac{dW}{dz}=W$.

Referenced by:

(4.13.9_1), §4.13, §4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E9_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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39: 15.11 Riemann’s Differential Equation

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15.11.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{1-a_1-a_2}{z-\alpha }+\frac{1-b_1-b_2}{z-\beta }+\frac{1-c_1-c_2}{z-\gamma }\right)\frac{dw}{dz}+\left(\frac{(\alpha -\beta )(\alpha -\gamma )a_1{a}_2}{z-\alpha }+\frac{(\beta -\alpha )(\beta -\gamma )b_1{b}_2}{z-\beta }+\frac{(\gamma -\alpha )(\gamma -\beta )c_1{c}_2}{z-\gamma }\right)\frac{w}{(z-\alpha )(z-\beta )(z-\gamma )}=0,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\alpha$: point, $\beta$: point, $\gamma$: point and $w$: solution

Referenced by:

§15.11(i), §15.11(i), §15.17(v)

Permalink:

http://dlmf.nist.gov/15.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(i), §15.11 and Ch.15

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§15.11(ii) Transformation Formulas

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40: 7.18 Repeated Integrals of the Complementary Error Function

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§7.18(iii) Properties

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7.18.3 $$\frac{d}{dz}{\mathrm{i}}^n\mathrm{erfc}\left(z\right)=-{\mathrm{i}}^{n-1}\mathrm{erfc}\left(z\right),$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.8

Permalink:

http://dlmf.nist.gov/7.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iii), §7.18 and Ch.7

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7.18.4 $$\frac{d^n}{{dz}^n}\left({\mathrm{e}}^{z^2}\mathrm{erfc}z\right)={(-1)}^n{2}^{n}n!{\mathrm{e}}^{z^2}{\mathrm{i}}^n\mathrm{erfc}\left(z\right),$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$\mathrm{erfc}z$: complementary error function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.9

Permalink:

http://dlmf.nist.gov/7.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iii), §7.18 and Ch.7

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7.18.5 $$\frac{d^{2}W}{{dz}^2}+2z\frac{dW}{dz}-2nW=0,$$ $W(z)=A{\mathrm{i}}^n\mathrm{erfc}\left(z\right)+B{\mathrm{i}}^n\mathrm{erfc}\left(-z\right)$,

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.2

Permalink:

http://dlmf.nist.gov/7.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iii), §7.18 and Ch.7

… ►The confluent hypergeometric function on the right-hand side of (7.18.10) is multivalued and in the sectors one has to use the analytic continuation formula (13.2.12). …