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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

4.34.1 $\frac{d}{dz}\mathop{\sinh\/}\nolimits z=\mathop{\cosh\/}\nolimits z,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and : complex variable
A&S Ref:: 4.5.71
Permalink:: http://dlmf.nist.gov/4.34.E1
Encodings:: TeX, pMML, png

4.34.2 $\frac{d}{dz}\mathop{\cosh\/}\nolimits z=\mathop{\sinh\/}\nolimits z,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and : complex variable
A&S Ref:: 4.5.72
Permalink:: http://dlmf.nist.gov/4.34.E2
Encodings:: TeX, pMML, png

4.34.3 $\frac{d}{dz}\mathop{\tanh\/}\nolimits z={\mathop{\mathrm{sech}\/}\nolimits^{{2% }}}z,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function and : complex variable
A&S Ref:: 4.5.73
Permalink:: http://dlmf.nist.gov/4.34.E3
Encodings:: TeX, pMML, png

4.34.4 $\frac{d}{dz}\mathop{\mathrm{csch}\/}\nolimits z=-\mathop{\mathrm{csch}\/}% \nolimits z\mathop{\coth\/}\nolimits z,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\mathop{\mathrm{csch}\/}\nolimits z$ : hyperbolic cosecant function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function and : complex variable
A&S Ref:: 4.5.74
Permalink:: http://dlmf.nist.gov/4.34.E4
Encodings:: TeX, pMML, png

4.34.5 $\frac{d}{dz}\mathop{\mathrm{sech}\/}\nolimits z=-\mathop{\mathrm{sech}\/}% \nolimits z\mathop{\tanh\/}\nolimits z,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function and : complex variable
A&S Ref:: 4.5.75
Permalink:: http://dlmf.nist.gov/4.34.E5
Encodings:: TeX, pMML, png

4.34.6 $\frac{d}{dz}\mathop{\coth\/}\nolimits z=-{\mathop{\mathrm{csch}\/}\nolimits^{{% 2}}}z.$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\mathop{\mathrm{csch}\/}\nolimits z$ : hyperbolic cosecant function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function and : complex variable
A&S Ref:: 4.5.76
Permalink:: http://dlmf.nist.gov/4.34.E6
Encodings:: TeX, pMML, png

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

4.34.7 $\frac{{d}^{2}w}{{dz}^{2}}-a^{2}w=0,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : real or complex constant, : complex variable and : solution
Permalink:: http://dlmf.nist.gov/4.34.E7
Encodings:: TeX, pMML, png

4.34.8 $\left(\frac{dw}{dz}\right)^{2}-a^{2}w^{2}=1,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : real or complex constant, : complex variable and : solution
Permalink:: http://dlmf.nist.gov/4.34.E8
Encodings:: TeX, pMML, png

4.34.9 $\left(\frac{dw}{dz}\right)^{2}-a^{2}w^{2}=-1,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : real or complex constant, : complex variable and : solution
Permalink:: http://dlmf.nist.gov/4.34.E9
Encodings:: TeX, pMML, png

4.34.10 $\frac{dw}{dz}+a^{2}w^{2}=1,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : real or complex constant, : complex variable and : solution
Permalink:: http://dlmf.nist.gov/4.34.E10
Encodings:: TeX, pMML, png

are respectively

4.34.11 $w=A\mathop{\cosh\/}\nolimits\!\left(az\right)+B\mathop{\sinh\/}\nolimits\!% \left(az\right),$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, : real or complex constant, : complex variable, : solution, : arbitrary constant and : arbitrary constant
Permalink:: http://dlmf.nist.gov/4.34.E11
Encodings:: TeX, pMML, png

4.34.12 $w=(1/a)\mathop{\sinh\/}\nolimits\!\left(az+c\right),$

Symbols:: $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, : real or complex constant, : complex variable, : solution and : arbitrary constant
Permalink:: http://dlmf.nist.gov/4.34.E12
Encodings:: TeX, pMML, png

4.34.13 $w=(1/a)\mathop{\cosh\/}\nolimits\!\left(az+c\right),$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, : real or complex constant, : complex variable, : solution and : arbitrary constant
Permalink:: http://dlmf.nist.gov/4.34.E13
Encodings:: TeX, pMML, png

4.34.14 $w=(1/a)\mathop{\coth\/}\nolimits\!\left(az+c\right),$

Symbols:: $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, : real or complex constant, : complex variable, : solution and : arbitrary constant
Permalink:: http://dlmf.nist.gov/4.34.E14
Encodings:: TeX, pMML, png

where are arbitrary constants.

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

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