# §4.20 Derivatives and Differential Equations

 4.20.1 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sin z$ $\displaystyle=\cos z,$ 4.20.2 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cos z$ $\displaystyle=-\sin z,$ 4.20.3 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\tan z$ $\displaystyle={\sec}^{2}z,$ 4.20.4 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\csc z$ $\displaystyle=-\csc z\cot z,$ 4.20.5 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sec z$ $\displaystyle=\sec z\tan z,$ 4.20.6 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cot z$ $\displaystyle=-{\csc}^{2}z,$ 4.20.7 $\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\sin z$ $\displaystyle=\sin\left(z+\tfrac{1}{2}n\pi\right),$ 4.20.8 $\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\cos z$ $\displaystyle=\cos\left(z+\tfrac{1}{2}n\pi\right).$

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

 4.20.9 $\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+a^{2}w$ $\displaystyle=0,$ 4.20.10 $\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}+a^{2}w^{2}$ $\displaystyle=1,$ 4.20.11 $\displaystyle\frac{\mathrm{d}w}{\mathrm{d}z}-a^{2}w^{2}$ $\displaystyle=1,$

are respectively

 4.20.12 $\displaystyle w$ $\displaystyle=A\cos\left(az\right)+B\sin\left(az\right),$ 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).