Elementary Functions of a Complex Variable

$z = 2 \pi i$;

$z = \pi i$;

$z = \pi i / 2$;

$z = -\pi i / 2$;

$z = \pi i / 4$.

$\cos (-z) = \cos z$;

$\sin (-z) = -\sin z$;

$\sin \left(z+\frac{\pi }{2}\right) = \cos z$;

$\sin (\pi +z) = -\sin z$;

$\sin (z+2 \pi ) = \sin z$;

$\cos \left(z+\frac{\pi }{2}\right) = -\sin z$;

$\sin \left(z_{1}+z_{2}\right) = \sin z_{1} \cos z_{2}+\cos z_{1} \sin z_{2}$;

$\cos \left(z_{1}+z_{2}\right) = \cos z_{1} \cos z_{2}-\sin z_{1} \sin z_{2}$.

$\operatorname {ch}(-z) = \operatorname {ch} z$;

$\operatorname {sh}(-z) = -\operatorname {sh} z$;

$\operatorname {ch}^{2} z-\operatorname {sh}^{2} z = 1$;

$\operatorname {ch}\left(z_{1}+z_{2}\right) = \operatorname {ch} z_{1} \operatorname {ch} z_{2}+\operatorname {sh} z_{1} \operatorname {sh} z_{2}$;

$\operatorname {sh}\left(z+\frac{\pi i}{2}\right) = i \operatorname {ch} z$;

$\operatorname {ch}\left(z+\frac{\pi i}{2}\right) = i \operatorname {sh} z$;

$\operatorname {sh}(z+\pi i) = -\operatorname {sh} z$;

$\operatorname {ch}(z+\pi i) = -\operatorname {ch} z$;

$\operatorname {th}(z+\pi i) = \operatorname {th} z$;

$\operatorname {ch}(z+2 \pi i) = \operatorname {ch} z$.

$\operatorname {sh} z = -i \sin (i z)$;

$\operatorname {sh}(i z) = i \sin z$;

$\cos (i z) = \operatorname {ch} z$;

$\operatorname {ch}(i z) = \cos z$;

$\operatorname {tg}(i z) = i \operatorname {th} z$;

$\operatorname {th}(i z) = i \operatorname {tg} z$;

$\operatorname {ctg}(i z) = -i \operatorname {cth} z$;

$\operatorname {cth}(i z) = -i \operatorname {ctg} z$.

$\mathrm{Re}\left(\sin z\right) = \sin x \cdot \operatorname {ch} y, \quad \mathrm{Im}\left(\sin z\right) = \cos x \cdot \operatorname {sh} y ;$

$\mathrm{Re}\left(\cos z\right) = \cos x \cdot \operatorname {ch} y, \quad \mathrm{Im}\left(\cos z\right) = -\sin x \cdot \operatorname {sh} y$;

$\mathrm{Re}\left(\operatorname {sh} z\right) = \operatorname {sh} x \cdot \cos y, \quad \mathrm{Im}\left(\operatorname {sh} z\right) = \operatorname {ch} x \cdot \sin y$;

$\mathrm{Re}\left(\operatorname {ch} z\right) = \operatorname {ch} x \cdot \cos y, \quad \mathrm{Im}\left(\operatorname {ch} z\right) = \operatorname {sh} x \cdot \sin y$;

$\mathrm{Re}\left(\operatorname {tg} z\right) = \frac{\sin 2 x}{\cos 2 x+\operatorname {ch} 2 y}, \quad \mathrm{Im}\left(\operatorname {tg} z\right) = \frac{\operatorname {sh} 2 y}{\cos 2 x+\operatorname {ch} 2 y}$;

$\mathrm{Re}\left(\operatorname {ctg} z\right) = \frac{\sin 2 x}{\operatorname {ch} 2 y-\cos 2 x}, \quad \mathrm{Im}\left(\operatorname {ctg} z\right) = \frac{-\operatorname {sh} 2 y}{\operatorname {ch} 2 y-\cos 2 x}$.

$\left|\sin z\right| = \sqrt{\operatorname {ch}^{2} y-\cos^{2} x}$;

$\left|\cos z\right| = \sqrt{\operatorname {ch}^{2} y-\sin^{2} x}$;

$\left|\operatorname {sh} z\right| = \sqrt{\operatorname {ch}^{2} x-\cos^{2} y}$;

$\left|\operatorname {ch} z\right| = \sqrt{\operatorname {ch}^{2} x-\sin^{2} y}$;

$\left|\operatorname {tg} z\right| = \sqrt{\frac{\operatorname {ch} 2 y-\cos 2 x}{\operatorname {ch} 2 y+\cos 2 x}}$;

$\left|\operatorname {cth} z\right| = \sqrt{\frac{\operatorname {ch} 2 x+\cos 2 y}{\operatorname {ch} 2 x-\cos 2 y}}$.

$\cos z$;

$\operatorname {ch} z$;

$\sin z$;

$\operatorname {tg} z$;

$\operatorname {cth} z$.

$\sin z$;

$\operatorname {sh} z$;

$\cos z$;

$\operatorname {ctg} z$;

$\operatorname {th} z$.

$\sin z$;

$\cos z$;

$\operatorname {sh} z$;

$\operatorname {ch} z$.

$\sin z = \frac{4 i}{3}$;

$\sin z = \frac{5}{3}$;

$\cos z = \frac{3 i}{4}$;

$\cos z = \frac{3+i}{4}$;

$\operatorname {tg} z = \frac{5 i}{3}$;

$\operatorname {ctg} z = -\frac{3 i}{5}$;

$\operatorname {sh} z = \frac{i}{2}$;

$\operatorname {ch} z = \frac{1}{2}$.

$\left|\operatorname {ch} z\right| \leqslant \operatorname {ch} R$;

$\left|\operatorname {sh} z\right| \leqslant \operatorname {sh} R$;

$\left|\cos z\right| \leqslant \operatorname {ch} R$;

$\left|\sin z\right| \leqslant \operatorname {sh} R$.

The function $e^{z}$ tends to infinity as $x$ tends to $+\infty$, and this tendency is uniform in $y$.

The function $e^{z}$ tends to zero as $x \rightarrow -\infty$, and this tendency is uniform in $y$.

The functions $\sin z$ and $\cos z$ tend to infinity as $y \rightarrow \pm \infty$, and this tendency is uniform in $x$.

The function $e^{z^{2}}$ tends to infinity as $z \rightarrow \infty$ in any angle of the form

and in any angle of the form $\left|\arg z\right| \leqslant \alpha$, provided only that $\alpha < \frac{\pi }{4}$.

The function $e^{z^{2}}$ tends to zero as $z \rightarrow \infty$ in any angle of the form

$\frac{2 e^{-2 y}}{1+e^{-2 y}} \leqslant \left|\operatorname {tg} z-i\right| \leqslant \frac{2 e^{-2 y}}{1-e^{-2 y}} \quad (y > 0)$;

$\frac{2 e^{2 y}}{1+e^{2 y}} \leqslant \left|\operatorname {tg} z+i\right| \leqslant \frac{2 e^{2 y}}{1-e^{-2 y}} \quad (y < 0)$;

$\frac{2 e^{-2 y}}{1+e^{-2 y}} \leqslant \left|\operatorname {ctg} z+i\right| \leqslant \frac{2 e^{-2 y}}{1-e^{-2 y}} \quad (y > 0)$;

$\frac{2 e^{2 y}}{1+e^{2 y}} \leqslant \left|\operatorname {ctg} z-i\right| \leqslant \frac{2 e^{2 y}}{1-e^{2 y}} \quad (y < 0)$.

$(\operatorname {sh} z)^{\prime } = \operatorname {ch} z$;

$(\operatorname {ch} z)^{\prime } = \operatorname {sh} z$;

$(\sin z)^{\prime } = \cos z$;

$(\cos z)^{\prime } = -\sin z$.

$e^{\operatorname {sh} z}$;

$\cos \left(2 e^{z}\right)$;

$\sin z \operatorname {sh} z+i \cos z \operatorname {ch} z$;

$\frac{e^{z}}{z}$;

$\frac{z}{e^{z}}$;

$\frac{\sin z}{1+z^{2}}$.

$\operatorname {tg} z$;

$\operatorname {ctg} z$;

$\frac{e^{z}+2}{e^{z}-2}$;

$\frac{1}{\operatorname {tg} z+\operatorname {ctg} z}$;

$\left(e^{z}+e^{-z}\right)^{-3}$;

$\frac{\sin z}{\sin z-\cos z}$.

$e^{a z}$;

$\operatorname {ch} a z$;

$\operatorname {sh} a z$;

$\cos a z$;

$\sin a z$;

$e^{a z} \cos b z$;

$z e^{a z}$;

$z^{2} \operatorname {ch} a z$;

$z \cos a z$.

$\frac{1}{4}\left(e^{z}+e^{-z}+2 \cos z\right) = \sum_{n = 0}^{\infty } \frac{z^{4 n}}{(4 n)!}$;

$\frac{1}{3}\left(e^{z}+2 e^{-z / 2} \cos \frac{z \sqrt{3}}{2}\right) = \sum_{n = 0}^{\infty } \frac{z^{3 n}}{(3 n)!}$.

$\sin^{2} z$;

$\cos^{3} z$;

$\sin^{4} z+\cos^{4} z$;

$\cos^{2} z+\operatorname {ch}^{2} z$;

$e^{2} \sin z$;

$\operatorname {ch} z \cdot \cos z$.

$\operatorname {tg} z$;

$\operatorname {ctg} z$;

$\operatorname {th} z$;

$e^{1 / \cos z}$;

$e^{-\operatorname {tg} z}$;

$\sin (\operatorname {th} z)$;

$\cos \left(e^{z^{2}}\right)$;

$\operatorname {th}\left(e^{z}\right)$;

$\operatorname {ctg}(\operatorname {ch} z)$.

$e^{z_{1}+z_{2}} = e^{z_{1}} \cdot e^{z_{2}}$;

$\cos^{2} z+\sin^{2} z = 1$;

$\sin 2 z = 2 \sin z \cos z$;

$\operatorname {ch} 2 z = \operatorname {ch}^{2} z+\operatorname {sh}^{2} z$;

$\sin \left(z_{1}+z_{2}\right) = \sin z_{1} \cos z_{2}+\cos z_{1} \sin z_{2}$;

$\operatorname {ch}\left(z_{1}+z_{2}\right) = \operatorname {ch} z_{1} \operatorname {ch} z_{2}+\operatorname {sh} z_{1} \operatorname {sh} z_{2}$;

$\cos z_{1}+\cos z_{2} = 2 \cos \frac{z_{1}+z_{2}}{2} \cos \frac{z_{1}-z_{2}}{2}$;

$\operatorname {sh} z_{1}+\operatorname {sh} z_{2} = 2 \operatorname {sh} \frac{z_{1}+z_{2}}{2} \operatorname {ch} \frac{z_{1}-z_{2}}{2}$.