4 Elementary FunctionsHyperbolic Functions4.33 Maclaurin Series and Laurent Series4.35 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		${\displaystyle \frac{d}{dz}}\sinh z$	$=\cosh z,$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\cosh z$: hyperbolic cosine function, $\sinh z$: 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		${\displaystyle \frac{d}{dz}}\cosh z$	$=\sinh z,$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\cosh z$: hyperbolic cosine function, $\sinh z$: 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		${\displaystyle \frac{d}{dz}}\tanh z$	$={\mathrm{sech}}^{2}z,$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{sech}z$: hyperbolic secant function, $\tanh z$: 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		${\displaystyle \frac{d}{dz}}\mathrm{csch}z$	$=-\mathrm{csch}z\coth z,$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{csch}z$: hyperbolic cosecant function, $\coth z$: 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		${\displaystyle \frac{d}{dz}}\mathrm{sech}z$	$=-\mathrm{sech}z\tanh z,$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{sech}z$: hyperbolic secant function, $\tanh z$: 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		${\displaystyle \frac{d}{dz}}\coth z$	$=-{\mathrm{csch}}^{2}z.$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{csch}z$: hyperbolic cosecant function, $\coth z$: 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 $a\ne 0$, the general solutions of the differential equations

4.34.7		${\displaystyle \frac{d^{2}w}{{dz}^2}}-a^{2}w$	$=0,$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $a$: real or complex constant, $z$: 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		${\left({\displaystyle \frac{dw}{dz}}\right)}^2-a^2{w}^2$	$=1,$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $a$: real or complex constant, $z$: 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		${\left({\displaystyle \frac{dw}{dz}}\right)}^2-a^2{w}^2$	$=-1,$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $a$: real or complex constant, $z$: 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		${\displaystyle \frac{dw}{dz}}+a^2{w}^2$	$=1,$
ⓘ Symbols: $\frac{df}{dx}$: 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		$w$	$=A\cosh \left(az\right)+B\sinh \left(az\right),$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function, $\sinh z$: 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		$w$	$=(1/a)\sinh \left(az+c\right),$
ⓘ Symbols: $\sinh z$: 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		$w$	$=(1/a)\cosh \left(az+c\right),$
ⓘ Symbols: $\cosh z$: 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		$w$	$=(1/a)\coth \left(az+c\right),$
ⓘ Symbols: $\coth z$: 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 Series4.35 Identities

© 2010–2024 NIST / Disclaimer / Feedback; Version 1.2.3; Release date 2024-12-15.

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.

Alternative Proxies:

Alternative Proxy

pFad Proxy

pFad v3 Proxy

pFad v4 Proxy