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Hyperbolic functions for complex numbers

Since the exponential function can be defined for any complex argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions sinh z and cosh z are then holomorphic.

Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers: e i x = cos x + i sin x e − i x = cos x − i sin x {\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x\\e^{-ix}&=\cos x-i\sin x\end{aligned}}} so: cosh (i x) = 1 2 (e i x + e − i x) = cos x sinh (i x) = 1 2 (e i x − e − i x) = i sin x cosh (x + i y) = cosh (x) cos (y) + i sinh (x) sin (y) sinh (x + i y) = sinh (x) cos (y) + i cosh (x) sin (y) tanh (i x) = i tan x cosh x = cos (i x) sinh x = − i sin (i x) tanh x = − i tan (i x) {\displaystyle {\begin{aligned}\cosh(ix)&={\frac {1}{2}}\left(e^{ix}+e^{-ix}\right)=\cos x\\\sinh(ix)&={\frac {1}{2}}\left(e^{ix}-e^{-ix}\right)=i\sin x\\\cosh(x+iy)&=\cosh(x)\cos(y)+i\sinh(x)\sin(y)\\\sinh(x+iy)&=\sinh(x)\cos(y)+i\cosh(x)\sin(y)\\\tanh(ix)&=i\tan x\\\cosh x&=\cos(ix)\\\sinh x&=-i\sin(ix)\\\tanh x&=-i\tan(ix)\end{aligned}}}

Thus, hyperbolic functions are periodic with respect to the imaginary component, with period 2 π i {\displaystyle 2\pi i} (π i {\displaystyle \pi i} for hyperbolic tangent and cotangent).