§4.7 Derivatives and Differential Equations

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

Contents

§4.7(i) Logarithms
§4.7(ii) Exponentials and Powers

§4.7(i) Logarithms

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

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\ln\/}\nolimits z$ : 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

4.7.2 $\frac{d}{dz}\mathop{\mathrm{Ln}\/}\nolimits z=\frac{1}{z},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{Ln}\/}\nolimits z$ : general logarithm function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.7.E2
Encodings:: TeX, pMML, png

4.7.3 $\frac{{d}^{n}}{{dz}^{n}}\mathop{\ln\/}\nolimits z=(-1)^{{n-1}}(n-1)!z^{{-n}},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : 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

4.7.4 $\frac{{d}^{n}}{{dz}^{n}}\mathop{\mathrm{Ln}\/}\nolimits z=(-1)^{{n-1}}(n-1)!z^{{-n}}.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $!$ : $n!$ : factorial, $\mathop{\mathrm{Ln}\/}\nolimits z$ : general logarithm function, $n$ : integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.7.E4
Encodings:: TeX, pMML, png

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

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

Notes:: See Levinson and Redheffer (1970, pp. 53–54 and p. 69).
Keywords:: derivatives, exponential function, power function
Permalink:: http://dlmf.nist.gov/4.7.ii

4.7.7 $\frac{d}{dz}e^{z}=e^{z},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $e$ : base of exponential function and $z$ : complex variable
A&S Ref:: 4.2.49
Permalink:: http://dlmf.nist.gov/4.7.E7
Encodings:: TeX, pMML, png

4.7.8 $\frac{d}{dz}e^{{az}}=ae^{{az}},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $e$ : base of exponential function, $a$ : 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

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\ln\/}\nolimits z$ : 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

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 $\frac{d}{dz}z^{a}=az^{{a-1}},$

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

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

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

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

is

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

Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : 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

The general solution of the differential equation

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

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

is

4.7.15 $w=Ae^{{\sqrt{a}z}}+Be^{{-\sqrt{a}z}},$

Symbols:: $e$ : base of exponential function, $a$ : 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

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

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