Fractals/trigonometric

How to compute it

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) =

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

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

cos(Z)

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

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

sinh(Z)

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

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

cosh(Z)

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

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

Images

See : commons:Category:Trigonometric maps

This image shows the Julia set of acomplex function of the form f(z)=a*sin(z), where a is a suitably chosen number in the interval (0,1).

Videos

• youtube : Spirals and Slarips by Gary Welz

See also

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