`arg(bar(z))+arg(-z)={{:(pi",","if arg (z) "lt 0),(-pi",", "if arg (z) "gt 0):},"where" -pi lt arg(z) le pi`.
If `arg(z) gt 0`, then arg (-z)-arg(z) is equal to

Options

A. `-pi`
B. `-(pi)/2`
C. `pi/2`
D. `pi`

Answer

To solve the problem, we need to find the value of \( \text{arg}(-z) - \text{arg}(z) \) given that \( \text{arg}(z) > 0 \).

### Step-by-Step Solution:

1. **Understanding the Argument of z**:
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. The argument of \( z \) is given by:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{y}{x}\right) \]
Given that \( \text{arg}(z) > 0 \), this implies that \( y > 0 \) and \( x > 0 \) (since \( z \) is in the first quadrant).

**Hint**: Remember that the argument of a complex number is the angle it makes with the positive x-axis.

2. **Finding the Argument of -z**:
The complex number \( -z \) can be expressed as:
\[ -z = -x - iy \]
The argument of \( -z \) is:
\[ \text{arg}(-z) = \tan^{-1}\left(\frac{-y}{-x}\right) = \tan^{-1}\left(\frac{y}{x}\right) + \pi = \text{arg}(z) + \pi \]
This is because \( -z \) lies in the third quadrant.

**Hint**: When you multiply a complex number by -1, the argument increases by \( \pi \).

3. **Calculating arg(-z) - arg(z)**:
Now we can find:
\[ \text{arg}(-z) - \text{arg}(z) = (\text{arg}(z) + \pi) - \text{arg}(z) \]
Simplifying this gives:
\[ \text{arg}(-z) - \text{arg}(z) = \pi \]

**Hint**: When subtracting the same terms, they cancel out, leaving you with the additional term.

4. **Final Answer**:
Therefore, the value of \( \text{arg}(-z) - \text{arg}(z) \) is:
\[ \pi \]

### Summary:
The final answer is \( \pi \).