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 \( re^{i\theta} \) 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: \( \mathbb{C} \)