4 Elementary Functions Hyperbolic Functions 4.33 Maclaurin Series and Laurent Series 4.35 Identities

§4.34 Derivatives and Differential Equations

ⓘ

Keywords:: derivatives, differential equations, hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.34
See also:: Annotations for Ch.4

4.34.1		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sinh z$	$\displaystyle=\cosh z,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable A&S Ref: 4.5.71 Permalink: http://dlmf.nist.gov/4.34.E1 Encodings: TeX, pMML, png See also: Annotations for §4.34 and Ch.4
4.34.2		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cosh z$	$\displaystyle=\sinh z,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable A&S Ref: 4.5.72 Permalink: http://dlmf.nist.gov/4.34.E2 Encodings: TeX, pMML, png See also: Annotations for §4.34 and Ch.4
4.34.3		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\tanh z$	$\displaystyle={\operatorname{sech}}^{2}z,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\tanh\NVar{z}$ : hyperbolic tangent function and $z$ : complex variable A&S Ref: 4.5.73 Permalink: http://dlmf.nist.gov/4.34.E3 Encodings: TeX, pMML, png See also: Annotations for §4.34 and Ch.4
4.34.4		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{csch}z$	$\displaystyle=-\operatorname{csch}z\coth z,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\coth\NVar{z}$ : hyperbolic cotangent function and $z$ : complex variable A&S Ref: 4.5.74 Permalink: http://dlmf.nist.gov/4.34.E4 Encodings: TeX, pMML, png See also: Annotations for §4.34 and Ch.4
4.34.5		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sech}z$	$\displaystyle=-\operatorname{sech}z\tanh z,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\tanh\NVar{z}$ : hyperbolic tangent function and $z$ : complex variable A&S Ref: 4.5.75 Permalink: http://dlmf.nist.gov/4.34.E5 Encodings: TeX, pMML, png See also: Annotations for §4.34 and Ch.4
4.34.6		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\coth z$	$\displaystyle=-{\operatorname{csch}}^{2}z.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\coth\NVar{z}$ : hyperbolic cotangent function and $z$ : complex variable A&S Ref: 4.5.76 Permalink: http://dlmf.nist.gov/4.34.E6 Encodings: TeX, pMML, png See also: Annotations for §4.34 and Ch.4

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

4.34.7		$\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-a^{2}w$	$\displaystyle=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $a$ : real or complex constant, $z$ : complex variable and $w$ : solution Permalink: http://dlmf.nist.gov/4.34.E7 Encodings: TeX, pMML, png See also: Annotations for §4.34 and Ch.4
4.34.8		$\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}$	$\displaystyle=1,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $a$ : real or complex constant, $z$ : complex variable and $w$ : solution Permalink: http://dlmf.nist.gov/4.34.E8 Encodings: TeX, pMML, png See also: Annotations for §4.34 and Ch.4
4.34.9		$\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}$	$\displaystyle=-1,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $a$ : real or complex constant, $z$ : complex variable and $w$ : solution Permalink: http://dlmf.nist.gov/4.34.E9 Encodings: TeX, pMML, png See also: Annotations for §4.34 and Ch.4
4.34.10		$\displaystyle\frac{\mathrm{d}w}{\mathrm{d}z}+a^{2}w^{2}$	$\displaystyle=1,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $a$ : real or complex constant, $z$ : complex variable and $w$ : solution Permalink: http://dlmf.nist.gov/4.34.E10 Encodings: TeX, pMML, png See also: Annotations for §4.34 and Ch.4

are respectively

4.34.11		$\displaystyle w$	$\displaystyle=A\cosh\left(az\right)+B\sinh\left(az\right),$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $a$ : real or complex constant, $z$ : complex variable, $w$ : solution, $A$ : arbitrary constant and $B$ : arbitrary constant Permalink: http://dlmf.nist.gov/4.34.E11 Encodings: TeX, pMML, png See also: Annotations for §4.34 and Ch.4
4.34.12		$\displaystyle w$	$\displaystyle=(1/a)\sinh\left(az+c\right),$
ⓘ Symbols: $\sinh\NVar{z}$ : hyperbolic sine function, $a$ : real or complex constant, $z$ : complex variable, $w$ : solution and $c$ : arbitrary constant Permalink: http://dlmf.nist.gov/4.34.E12 Encodings: TeX, pMML, png See also: Annotations for §4.34 and Ch.4
4.34.13		$\displaystyle w$	$\displaystyle=(1/a)\cosh\left(az+c\right),$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $a$ : real or complex constant, $z$ : complex variable, $w$ : solution and $c$ : arbitrary constant Permalink: http://dlmf.nist.gov/4.34.E13 Encodings: TeX, pMML, png See also: Annotations for §4.34 and Ch.4
4.34.14		$\displaystyle w$	$\displaystyle=(1/a)\coth\left(az+c\right),$
ⓘ Symbols: $\coth\NVar{z}$ : hyperbolic cotangent function, $a$ : real or complex constant, $z$ : complex variable, $w$ : solution and $c$ : arbitrary constant Permalink: http://dlmf.nist.gov/4.34.E14 Encodings: TeX, pMML, png See also: Annotations for §4.34 and Ch.4

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

For other differential equations see Kamke (1977, pp. 289–400).

4.33 Maclaurin Series and Laurent Series 4.35 Identities

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