One can use Maxima CAS to find it :

(%i1) z: x+y*%i;
(%o1) %i*y+x
(%o2) %i*y+x
(%i3) realpart(sinh(z));
(%o3) sinh(x)*cos(y)
(%i8) trigrat(sinh(x));
(%o8) (%e^−x*(%e^(2*x)−1))/2
(%i11) expand(%);
(%o11) %e^x/2−%e^−x/2

sin(Z) =

${\displaystyle Real=\sin(x)((\exp(y)+\exp(-y))/2)}$

${\displaystyle Imag=\cos(x)((\exp(y)-\exp(-y))/2)}$

cos(Z)

${\displaystyle Real=\cos(x)((\exp(y)+\exp(-y))/2)}$

${\displaystyle Imag=-\sin(x)((\exp(y)-\exp(-y))/2)}$

sinh(Z)

${\displaystyle Real=\cos(y)((\exp(x)-\exp(-x))/2)}$

${\displaystyle Imag=\sin(y)((\exp(x)+\exp(-x))/2)}$

cosh(Z)

${\displaystyle Real=\cos(y)((\exp(x)+\exp(-x))/2)}$

${\displaystyle Imag=\sin(y)((\exp(x)-\exp(-x))/2)}$

See : commons:Category:Trigonometric maps

• youtube : Spirals and Slarips by Gary Welz

• Thorn Fractal, alternatively named the "Secant Sea"
• Paul Bourke : sinjulia fractal