The document presents a problem involving the roots of the equation re(arcsinh(z)) = 0 for the complex number z = x + iy. It explains how to separate sin(z) into real and imaginary components to analyze the equations that arise. The solution starts by setting up equations based on the real and imaginary parts derived from the sine function.

1. 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,