§4.7 Derivatives and Differential Equations

4.7.1		$$\frac{d}{dz}\ln z=\frac{1}{z},$$
ⓘ 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: TeX, pMML, png See also: Annotations for §4.7(i), §4.7 and Ch.4

4.7.2		$$\frac{d}{dz}\mathrm{Ln}z=\frac{1}{z},$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{Ln}z$: 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		$$\frac{d^n}{{dz}^n}\ln z={(-1)}^{n-1}(n-1)!z^{-n},$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $!$: factorial (as in $n!$), $\ln z$: 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		$$\frac{d^n}{{dz}^n}\mathrm{Ln}z={(-1)}^{n-1}(n-1)!z^{-n}.$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $!$: factorial (as in $n!$), $\mathrm{Ln}z$: 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		$$\frac{dw}{dz}=\frac{f^{\prime }(z)}{f(z)}$$
ⓘ 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: TeX, pMML, png See also: Annotations for §4.7(i), §4.7 and Ch.4

is

4.7.6		$$w(z)=\mathrm{Ln}\left(f(z)\right)+\text{ constant}.$$
ⓘ Defines: $w$: solution (locally) Symbols: $\mathrm{Ln}z$: 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

4.7.7		$$\frac{d}{dz}{\mathrm{e}}^z={\mathrm{e}}^z,$$
ⓘ Symbols: $\frac{df}{dx}$: 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		$$\frac{d}{dz}{\mathrm{e}}^{az}=a{\mathrm{e}}^{az},$$
ⓘ Symbols: $\frac{df}{dx}$: 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		$$\frac{d}{dz}{a}^z=a^z\ln a,$$
$a\ne 0$.
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\ln z$: 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 $\mathrm{Ln}a$ used in constructing $a^z$.

4.7.10		$$\frac{d}{dz}{z}^a=a{z}^{a-1},$$
ⓘ Symbols: $\frac{df}{dx}$: 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		$$\frac{d^n}{{dz}^n}{z}^a=a(a-1)(a-2)\mathrm{\cdots }(a-n+1)z^{a-n}.$$
ⓘ Symbols: $\frac{df}{dx}$: 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		$$\frac{dw}{dz}=f(z)w$$
ⓘ 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.E12 Encodings: TeX, pMML, png See also: Annotations for §4.7(ii), §4.7 and Ch.4

is

4.7.13		$$w=\exp \left(\int f(z)dz\right)+\mathrm{constant}.$$
ⓘ Symbols: $dx$: differential of $x$, $\exp z$: 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		$$\frac{d^{2}w}{{dz}^2}=aw,$$
$a\ne 0$,
ⓘ Symbols: $\frac{df}{dx}$: 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

is

4.7.15		$$w=A{\mathrm{e}}^{\sqrt{a}z}+B{\mathrm{e}}^{-\sqrt{a}z},$$
ⓘ 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).