# §4.7 Derivatives and Differential Equations

## §4.7(i) Logarithms

 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, $\ln\NVar{z}$: 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, $\operatorname{Ln}\NVar{z}$: 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 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\ln z=(-1)^{n-1}(n-1)!z^{-n},$
 4.7.4 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\operatorname{Ln}z=(-1)^{n-1}(n-1)!z% ^{-n}.$

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, $z$: 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 $w(z)=\operatorname{Ln}\left(f(z)\right)+\hbox{ constant}.$ ⓘ 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

 4.7.7 $\frac{\mathrm{d}}{\mathrm{d}z}e^{z}=e^{z},$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\mathrm{e}$: 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, $\mathrm{e}$: 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, $\ln\NVar{z}$: 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, $a$: 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, $n$: 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, $z$: 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 $w=\exp\left(\int f(z)\mathrm{d}z\right)+{\rm constant}.$

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, $a$: 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 $w=Ae^{\sqrt{a}z}+Be^{-\sqrt{a}z},$

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

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