Hyperbolic Functions, Logarithms of Complex Numbers

cosx = ${e^{ix}+e^{-ix}}/{2}$

sinx = ${e^{ix}-e^{-ix}}/{2i}$

tanx = ${e^{ix}-e^{-ix}}/{i(e^{ix}+e^{-ix})}$

sinhx = ${e^{x}-e^{-x}}/{2}$

coshx = ${e^{x}+e^{-x}}/{2}$

tanhx = ${e^{x}-e^{-x}}/{e^{x}+e^{-x}}$

Formulae of Hyperbolic Fucntions:

$cosh^{2}x – sinh^{2}x$ = 1
$sech^{2}x$ = 1 – $tanh^{2}x$
$coth^{2}x$ = 1 + $cosech^{2}x$
sinh(x±y) = sinhx coshy ± coshx sinhy
cosh(x±y) = coshx coshy ± sinhx sinhy

$\table cosh2x,=, cosh^{2}x + sinh^{2}x; \ ,=,2cosh^{2}x-1;,=,1+2sinh^{2}x$
sinh2x = 2 sinhx coshx

tanh(x±y) = ${tanhx ± tanhy}/{1± tanhx tanhy}$
sinhx = ${2tanhx/2}/{1-tanh^{2}x/2}$

coshx = ${1+tanh^{2}x/2}/{1-tanh^{2}x/2}$
tanhx = ${2tanhx/2}/{1 + tanh^{2}x/2}$

sinh3x = 3sinhx + 4$sinh^{3}x$

cosh3x = 4$cosh^{3}x$ – 3 coshx

Separation of real & imaginary formulae

sin(x+iy) = sinx coshy + icosx sinhy

cos(x+iy) = cosx coshy – isinx sinhy
tan(x+iy) = ${sin2x}/{cos2x + cosh2y}$ + i ${sinh2y}/{cos2x + cosh2y}$

sinh(x+iy) = sinhx cosy + icoshx siny

cosh(x+iy) = coshx cosy + isinhx siny
tanh(x+iy) = ${sinh2x}/{cosh2x + cos2y}$ + i ${sin2y}/{cosh2x + cos2y}$

$\table tan2α, =, tan[(α+iβ)+(α-iβ)]; , =, {2p}/{1-p^2-q^2}$

Log of complex numbers:

$Log(x+iy) = log√{x^2 + y^2} + i(2nπ + tan^{-1}y/x)$