find the domain and range of complex function f(z)=z+Argument(1/z)

To determine the domain and range of the complex function $f(z)=z+\text{Argument}\left(\frac{1}{z}\right)$, we first need to analyze the components of the function. The function involves both the variable $z$ and the argument of a complex number, which can introduce some interesting considerations.

Understanding the Domain

The domain of a function consists of all the input values for which the function is defined. In this case, we need to consider the term $\frac{1}{z}$. The expression $\frac{1}{z}$ is undefined when $z=0$. Therefore, the function $f(z)$ is also undefined at this point.

Thus, the domain of $f(z)$ can be expressed as:

• All complex numbers except $z=0$.

In mathematical notation, we can write this as:

Domain: $\mathbb{C}\setminus \{0\}$

Exploring the Range

Next, let’s delve into the range of the function. The range consists of all possible output values of $f(z)$. To analyze this, we can rewrite the function using the polar form of complex numbers.

For any non-zero complex number $z$, we can express it in polar form as:

z = re^{i\theta}, where $r>0$ and $\theta$ is the argument of $z$.

Now, substituting this into our function:

f(z) = re^{i\theta} + \text{Argument}\left(\frac{1}{re^{i\theta}}\right)

The argument of $\frac{1}{z}$ can be simplified as:

Argument\left(\frac{1}{re^{i\theta}}\right) = -\theta

Thus, we can rewrite the function as:

f(z) = re^{i\theta} - \theta

Analyzing the Output

Now, let’s break down the output of this function. The term $r{e}^{i\theta }$ represents a point in the complex plane, while $-\theta$ is a real number that shifts the imaginary part of $f(z)$.

As $z$ varies over all non-zero complex numbers, $r$ can take any positive value, and $\theta$ can take any angle from $-\pi$ to $\pi$. Therefore, the function can produce a wide range of complex numbers.

However, since $-\theta$ can also take any real value, the imaginary part of $f(z)$ can be adjusted by the angle $\theta$. This means that the function can cover all possible complex values except for the point where $z=0$ is involved.

Final Thoughts on the Range

In conclusion, the range of $f(z)$ is the entire complex plane, as the function can produce any complex number through appropriate choices of $z$ in its domain.

Range: $\mathbb{C}$

To summarize:

• Domain: $\mathbb{C}\setminus \{0\}$
• Range: $\mathbb{C}$