§4.20 Derivatives and Differential Equations

Notes:: See Levinson and Redheffer (1970, pp. 53–60).
Keywords:: derivatives, differential equations, trigonometric functions
Permalink:: http://dlmf.nist.gov/4.20

4.20.1 $\frac{d}{dz}\mathop{\sin\/}\nolimits z=\mathop{\cos\/}\nolimits z,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 4.3.105
Permalink:: http://dlmf.nist.gov/4.20.E1
Encodings:: TeX, pMML, png

4.20.2 $\frac{d}{dz}\mathop{\cos\/}\nolimits z=-\mathop{\sin\/}\nolimits z,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 4.3.106
Permalink:: http://dlmf.nist.gov/4.20.E2
Encodings:: TeX, pMML, png

4.20.3 $\frac{d}{dz}\mathop{\tan\/}\nolimits z={\mathop{\sec\/}\nolimits^{{2}}}z,$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\sec\/}\nolimits z$ : secant function, $\mathop{\tan\/}\nolimits z$ : tangent function and $z$ : complex variable
A&S Ref:: 4.3.107
Permalink:: http://dlmf.nist.gov/4.20.E3
Encodings:: TeX, pMML, png

4.20.4 $\frac{d}{dz}\mathop{\csc\/}\nolimits z=-\mathop{\csc\/}\nolimits z\mathop{\cot\/}\nolimits z,$

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ and $z$ : complex variable
A&S Ref:: 4.3.108
Permalink:: http://dlmf.nist.gov/4.20.E4
Encodings:: TeX, pMML, png

4.20.5 $\frac{d}{dz}\mathop{\sec\/}\nolimits z=\mathop{\sec\/}\nolimits z\mathop{\tan\/}\nolimits z,$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\sec\/}\nolimits z$ : secant function, $\mathop{\tan\/}\nolimits z$ : tangent function and $z$ : complex variable
A&S Ref:: 4.3.109
Permalink:: http://dlmf.nist.gov/4.20.E5
Encodings:: TeX, pMML, png

4.20.6 $\frac{d}{dz}\mathop{\cot\/}\nolimits z=-{\mathop{\csc\/}\nolimits^{{2}}}z,$

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ and $z$ : complex variable
A&S Ref:: 4.3.110
Permalink:: http://dlmf.nist.gov/4.20.E6
Encodings:: TeX, pMML, png

4.20.7 $\frac{{d}^{n}}{{dz}^{n}}\mathop{\sin\/}\nolimits z=\mathop{\sin\/}\nolimits\!\left(z+\tfrac{1}{2}n\pi\right),$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\sin\/}\nolimits z$ : sine function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.111
Permalink:: http://dlmf.nist.gov/4.20.E7
Encodings:: TeX, pMML, png

4.20.8 $\frac{{d}^{n}}{{dz}^{n}}\mathop{\cos\/}\nolimits z=\mathop{\cos\/}\nolimits\!\left(z+\tfrac{1}{2}n\pi\right).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.112
Permalink:: http://dlmf.nist.gov/4.20.E8
Encodings:: TeX, pMML, png

With $a\neq 0$ , the general solutions of the differential equations

4.20.9 $\frac{{d}^{2}w}{{dz}^{2}}+a^{2}w=0,$

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/4.20.E9
Encodings:: TeX, pMML, png

4.20.10 $\left(\frac{dw}{dz}\right)^{2}+a^{2}w^{2}=1,$

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/4.20.E10
Encodings:: TeX, pMML, png

4.20.11 $\frac{dw}{dz}-a^{2}w^{2}=1,$

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/4.20.E11
Encodings:: TeX, pMML, png

are respectively

4.20.12 $w=A\mathop{\cos\/}\nolimits\!\left(az\right)+B\mathop{\sin\/}\nolimits\!\left(az\right),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $a$ : real or complex constant, $z$ : complex variable, $A$ : arbitrary constant and $B$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/4.20.E12
Encodings:: TeX, pMML, png

4.20.13 $w=(1/a)\mathop{\sin\/}\nolimits\!\left(az+c\right),$

Symbols:: $\mathop{\sin\/}\nolimits z$ : sine function, $a$ : real or complex constant, $z$ : complex variable and $c$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/4.20.E13
Encodings:: TeX, pMML, png

4.20.14 $w=(1/a)\mathop{\tan\/}\nolimits\!\left(az+c\right),$

Symbols:: $\mathop{\tan\/}\nolimits z$ : tangent function, $a$ : real or complex constant, $z$ : complex variable and $c$ : arbitrary constant
Permalink:: http://dlmf.nist.gov/4.20.E14
Encodings:: TeX, pMML, png

where $A,B,c$ are arbitrary constants.

For other differential equations see Kamke (1977, pp. 355–358 and 396–400).