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4 Elementary Functions Trigonometric Functions 4.19 Maclaurin Series and Laurent Series 4.21 Identities

§4.20 Derivatives and Differential Equations

ⓘ

Keywords:: derivatives, differential equations, trigonometric functions
Notes:: See Levinson and Redheffer (1970, pp. 53–60).
Permalink:: http://dlmf.nist.gov/4.20
See also:: Annotations for Ch.4

4.20.1		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sin z$	$\displaystyle=\cos z,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.105 Permalink: http://dlmf.nist.gov/4.20.E1 Encodings: TeX, pMML, png See also: Annotations for §4.20 and Ch.4
4.20.2		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cos z$	$\displaystyle=-\sin z,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.106 Permalink: http://dlmf.nist.gov/4.20.E2 Encodings: TeX, pMML, png See also: Annotations for §4.20 and Ch.4
4.20.3		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\tan z$	$\displaystyle={\sec}^{2}z,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\sec\NVar{z}$ : secant function, $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.107 Permalink: http://dlmf.nist.gov/4.20.E3 Encodings: TeX, pMML, png See also: Annotations for §4.20 and Ch.4
4.20.4		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\csc z$	$\displaystyle=-\csc z\cot z,$
ⓘ Symbols: $\csc\NVar{z}$ : cosecant function, $\cot\NVar{z}$ : cotangent function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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 See also: Annotations for §4.20 and Ch.4
4.20.5		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sec z$	$\displaystyle=\sec z\tan z,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\sec\NVar{z}$ : secant function, $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.109 Permalink: http://dlmf.nist.gov/4.20.E5 Encodings: TeX, pMML, png See also: Annotations for §4.20 and Ch.4
4.20.6		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cot z$	$\displaystyle=-{\csc}^{2}z,$
ⓘ Symbols: $\csc\NVar{z}$ : cosecant function, $\cot\NVar{z}$ : cotangent function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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 See also: Annotations for §4.20 and Ch.4
4.20.7		$\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\sin z$	$\displaystyle=\sin\left(z+\tfrac{1}{2}n\pi\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\sin\NVar{z}$ : 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 See also: Annotations for §4.20 and Ch.4
4.20.8		$\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\cos z$	$\displaystyle=\cos\left(z+\tfrac{1}{2}n\pi\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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 See also: Annotations for §4.20 and Ch.4

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

4.20.9		$\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+a^{2}w$	$\displaystyle=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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 See also: Annotations for §4.20 and Ch.4
4.20.10		$\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}+a^{2}w^{2}$	$\displaystyle=1,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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 See also: Annotations for §4.20 and Ch.4
4.20.11		$\displaystyle\frac{\mathrm{d}w}{\mathrm{d}z}-a^{2}w^{2}$	$\displaystyle=1,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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 See also: Annotations for §4.20 and Ch.4

are respectively

4.20.12		$\displaystyle w$	$\displaystyle=A\cos\left(az\right)+B\sin\left(az\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : 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 See also: Annotations for §4.20 and Ch.4
4.20.13		$\displaystyle w$	$\displaystyle=(1/a)\sin\left(az+c\right),$
ⓘ Symbols: $\sin\NVar{z}$ : 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 See also: Annotations for §4.20 and Ch.4
4.20.14		$\displaystyle w$	$\displaystyle=(1/a)\tan\left(az+c\right),$
ⓘ Symbols: $\tan\NVar{z}$ : 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 See also: Annotations for §4.20 and Ch.4

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

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

4.19 Maclaurin Series and Laurent Series 4.21 Identities

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