4 Elementary Functions Logarithm, Exponential, Powers 4.6 Power Series 4.8 Identities

§4.7 Derivatives and Differential Equations

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Permalink:: http://dlmf.nist.gov/4.7
See also:: Annotations for Ch.4

§4.7(i) Logarithms

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Keywords:: derivatives, differential equations, logarithm function
Notes:: See Levinson and Redheffer (1970, pp. 62–69).
Permalink:: http://dlmf.nist.gov/4.7.i
See also:: Annotations for §4.7 and Ch.4

4.7.1		$\frac{\mathrm{d}}{\mathrm{d}z}\ln z=\frac{1}{z},$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 4.1.46 Permalink: http://dlmf.nist.gov/4.7.E1 Encodings: TeX, pMML, png See also: Annotations for §4.7(i), §4.7 and Ch.4

4.7.2		$\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{Ln}z=\frac{1}{z},$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{Ln}\NVar{z}$ : general logarithm function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.7.E2 Encodings: TeX, pMML, png See also: Annotations for §4.7(i), §4.7 and Ch.4

4.7.3		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\ln z=(-1)^{n-1}(n-1)!z^{-n},$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : integer and $z$ : complex variable A&S Ref: 4.1.47 Permalink: http://dlmf.nist.gov/4.7.E3 Encodings: TeX, pMML, png See also: Annotations for §4.7(i), §4.7 and Ch.4

4.7.4		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\operatorname{Ln}z=(-1)^{n-1}(n-1)!z% ^{-n}.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $!$ : factorial (as in $n!$ ), $\operatorname{Ln}\NVar{z}$ : general logarithm function, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.7.E4 Encodings: TeX, pMML, png See also: Annotations for §4.7(i), §4.7 and Ch.4

For a nonvanishing analytic function $f(z)$ , the general solution of the differential equation

4.7.5		$\frac{\mathrm{d}w}{\mathrm{d}z}=\frac{f^{\prime}(z)}{f(z)}$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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: TeX, pMML, png See also: Annotations for §4.7(i), §4.7 and Ch.4

4.7.6		$w(z)=\operatorname{Ln}\left(f(z)\right)+\hbox{ constant}.$
ⓘ Defines: $w$ : solution (locally) Symbols: $\operatorname{Ln}\NVar{z}$ : general logarithm function, $z$ : complex variable and $f(z)$ : non-vanishing analytic function Permalink: http://dlmf.nist.gov/4.7.E6 Encodings: TeX, pMML, png See also: Annotations for §4.7(i), §4.7 and Ch.4

§4.7(ii) Exponentials and Powers

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Keywords:: derivatives, differential equation, exponential function, power function
Notes:: See Levinson and Redheffer (1970, pp. 53–54 and p. 69).
Permalink:: http://dlmf.nist.gov/4.7.ii
See also:: Annotations for §4.7 and Ch.4

4.7.7		$\frac{\mathrm{d}}{\mathrm{d}z}e^{z}=e^{z},$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable A&S Ref: 4.2.49 Permalink: http://dlmf.nist.gov/4.7.E7 Encodings: TeX, pMML, png See also: Annotations for §4.7(ii), §4.7 and Ch.4

4.7.8		$\frac{\mathrm{d}}{\mathrm{d}z}e^{az}=ae^{az},$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $a$ : real or complex constant and $z$ : complex variable A&S Ref: 4.2.50 (gives n-th derivative.) Permalink: http://dlmf.nist.gov/4.7.E8 Encodings: TeX, pMML, png See also: Annotations for §4.7(ii), §4.7 and Ch.4

4.7.9		$\frac{\mathrm{d}}{\mathrm{d}z}a^{z}=a^{z}\ln a,$
$a\neq 0$ .
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\ln\NVar{z}$ : principal branch of logarithm function, $a$ : real or complex constant and $z$ : complex variable A&S Ref: 4.2.51 Permalink: http://dlmf.nist.gov/4.7.E9 Encodings: TeX, pMML, png See also: Annotations for §4.7(ii), §4.7 and Ch.4

When $a^{z}$ is a general power, $\ln a$ is replaced by the branch of $\operatorname{Ln}a$ used in constructing $a^{z}$ .

4.7.10		$\frac{\mathrm{d}}{\mathrm{d}z}z^{a}=az^{a-1},$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $a$ : real or complex constant and $z$ : complex variable A&S Ref: 4.2.52 Permalink: http://dlmf.nist.gov/4.7.E10 Encodings: TeX, pMML, png See also: Annotations for §4.7(ii), §4.7 and Ch.4

4.7.11		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}z^{a}=a(a-1)(a-2)\cdots(a-n+1)z^{a-n}.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer, $a$ : real or complex constant and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.7.E11 Encodings: TeX, pMML, png See also: Annotations for §4.7(ii), §4.7 and Ch.4

The general solution of the differential equation

4.7.12		$\frac{\mathrm{d}w}{\mathrm{d}z}=f(z)w$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable and $f(z)$ : non-vanishing analytic function Permalink: http://dlmf.nist.gov/4.7.E12 Encodings: TeX, pMML, png See also: Annotations for §4.7(ii), §4.7 and Ch.4

4.7.13		$w=\exp\left(\int f(z)\,\mathrm{d}z\right)+{\rm constant}.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $z$ : complex variable and $f(z)$ : non-vanishing analytic function Permalink: http://dlmf.nist.gov/4.7.E13 Encodings: TeX, pMML, png See also: Annotations for §4.7(ii), §4.7 and Ch.4

The general solution of the differential equation

4.7.14		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=aw,$
$a\neq 0$ ,
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $a$ : real or complex constant and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.7.E14 Encodings: TeX, pMML, png See also: Annotations for §4.7(ii), §4.7 and Ch.4

4.7.15		$w=Ae^{\sqrt{a}z}+Be^{-\sqrt{a}z},$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $a$ : real or complex constant, $z$ : complex variable, $A$ : arbitrary constant and $B$ : arbitrary constant Permalink: http://dlmf.nist.gov/4.7.E15 Encodings: TeX, pMML, png See also: Annotations for §4.7(ii), §4.7 and Ch.4

where $A$ and $B$ are arbitrary constants.

For other differential equations see Kamke (1977, pp. 396–413).

4.6 Power Series 4.8 Identities

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§4.7 Derivatives and Differential Equations

Contents

§4.7(i) Logarithms

§4.7(ii) Exponentials and Powers

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