Find all roots of Rearcsinhz 0 where z is the complex numbe

Find all roots of Re(arcsinh(z)) = 0, where z is the complex number, z = x + iy.

Solution

a) Write z = x + iy and separate sin(z) into real and imaginary parts: sin(z) = sin(x + iy) = sin(x) cos(iy) + cos(x) sin(iy) = sin(x) cosh(y) + i cos(x) sinh(y) = i ? ( sin(x) cosh(y) = 0 cos(x) sinh(y) = 1 The first equation gives x = n?, n = 0,