Natural Domain of Complex Functions

In this exercise, we will find the domain of definition for four different complex functions. The domain of a function is the set of all complex numbers for which the function's expression is well-defined.

Let's start with part A. The function is f of z equals one over, z squared plus one.

This function is undefined when the denominator is zero. So, we solve the equation z squared plus one equals zero.

Subtracting one from both sides, we get z squared equals negative one. Taking the square root, we find that z equals plus or minus i.

Therefore, the domain consists of all complex numbers except for i and negative i.

Now for part B. The function is the principal argument of one over z.

The principal argument is defined for all non-zero complex numbers. However, we have a fraction inside the argument, which means z itself cannot be zero to avoid division by zero.

Since one over z can never be zero for any finite z, the only restriction is that z does not equal zero.

Moving on to part C. We have f of z equals z over, z plus the conjugate of z.