# §4.7(i) Logarithms

 4.7.1 $\frac{d}{dz}\mathop{\ln\/}\nolimits z=\frac{1}{z},$
 4.7.2 $\frac{d}{dz}\mathop{\mathrm{Ln}\/}\nolimits z=\frac{1}{z},$
 4.7.3 $\frac{{d}^{n}}{{dz}^{n}}\mathop{\ln\/}\nolimits z=(-1)^{n-1}(n-1)!z^{-n},$
 4.7.4 $\frac{{d}^{n}}{{dz}^{n}}\mathop{\mathrm{Ln}\/}\nolimits z=(-1)^{n-1}(n-1)!z^{-% n}.$

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

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

is

 4.7.6 $w(z)=\mathop{\mathrm{Ln}\/}\nolimits\!\left(f(z)\right)+\hbox{ constant}.$ Symbols: $\mathop{\mathrm{Ln}\/}\nolimits z$: general logarithm function, $z$: complex variable, $f(z)$: non-vanishing analytic function and $w$: solution Permalink: http://dlmf.nist.gov/4.7.E6 Encodings: TeX, pMML, png

# §4.7(ii) Exponentials and Powers

 4.7.7 $\frac{d}{dz}e^{z}=e^{z},$
 4.7.8 $\frac{d}{dz}e^{az}=ae^{az},$
 4.7.9 $\frac{d}{dz}a^{z}=a^{z}\mathop{\ln\/}\nolimits a,$ $a\neq 0$.

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

 4.7.10 $\frac{d}{dz}z^{a}=az^{a-1},$
 4.7.11 $\frac{{d}^{n}}{{dz}^{n}}z^{a}=a(a-1)(a-2)\cdots(a-n+1)z^{a-n}.$

The general solution of the differential equation

 4.7.12 $\frac{dw}{dz}=f(z)w$

is

 4.7.13 $w=\mathop{\exp\/}\nolimits\!\left(\int f(z)dz\right)+{\rm constant}.$

The general solution of the differential equation

 4.7.14 $\frac{{d}^{2}w}{{dz}^{2}}=aw,$ $a\neq 0$,

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).