If `arg(z)=-pi/4` then the value of `arg((z^5+(bar(z))^5)/(1+z(bar(z))))^n` is

Options

A. `pi` if n is odd
B. 0, if n is even
C. `3pi//2`
D. `pi//4`

Answer

To solve the problem, we need to find the value of \( \arg\left(\frac{z^5 + \bar{z}^5}{1 + z\bar{z}}\right)^n \) given that \( \arg(z) = -\frac{\pi}{4} \).

### Step-by-Step Solution:

1. **Understanding the Argument of z**:
Given \( \arg(z) = -\frac{\pi}{4} \), we can express \( z \) in polar form:
\[ z = r \left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right) = r \left(\frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}}\right) \]
where \( r = |z| \).

2. **Finding \( \bar{z} \)**:
The conjugate of \( z \) is:
\[ \bar{z} = r \left(\cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right)\right) = r \left(\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}\right) \]

3. **Calculating \( z^5 \) and \( \bar{z}^5 \)**:
Using De Moivre's theorem:
\[ z^5 = r^5 \left(\cos\left(-\frac{5\pi}{4}\right) + i\sin\left(-\frac{5\pi}{4}\right)\right) \]
\[ \bar{z}^5 = r^5 \left(\cos\left(\frac{5\pi}{4}\right) + i\sin\left(\frac{5\pi}{4}\right)\right) \]

4. **Adding \( z^5 \) and \( \bar{z}^5 \)**:
Since \( \cos\left(-\frac{5\pi}{4}\right) = \cos\left(\frac{5\pi}{4}\right) \) and \( \sin\left(-\frac{5\pi}{4}\right) = -\sin\left(\frac{5\pi}{4}\right) \):
\[ z^5 + \bar{z}^5 = r^5 \left(\cos\left(-\frac{5\pi}{4}\right) + i\sin\left(-\frac{5\pi}{4}\right)\right) + r^5 \left(\cos\left(\frac{5\pi}{4}\right) + i\sin\left(\frac{5\pi}{4}\right)\right) \]
This simplifies to:
\[ z^5 + \bar{z}^5 = 2r^5 \cos\left(\frac{5\pi}{4}\right) = -\sqrt{2}r^5 \]

5. **Calculating \( 1 + z\bar{z} \)**:
We know:
\[ z\bar{z} = |z|^2 = r^2 \]
Therefore:
\[ 1 + z\bar{z} = 1 + r^2 \]

6. **Finding the Argument**:
Now we can substitute back into our expression:
\[ \frac{z^5 + \bar{z}^5}{1 + z\bar{z}} = \frac{-\sqrt{2}r^5}{1 + r^2} \]
The numerator is a negative real number, and the denominator is a positive real number (since \( r^2 \geq 0 \)). Thus, the entire fraction is a negative real number.

7. **Final Argument Calculation**:
The argument of a negative real number is \( \pi \). Therefore:
\[ \arg\left(\frac{z^5 + \bar{z}^5}{1 + z\bar{z}}\right) = \pi \]

8. **Raising to the Power n**:
Finally, we raise this to the power \( n \):
\[ \arg\left(\left(\frac{z^5 + \bar{z}^5}{1 + z\bar{z}}\right)^n\right) = n \cdot \pi \]

### Conclusion:
The value of \( \arg\left(\frac{z^5 + \bar{z}^5}{1 + z\bar{z}}\right)^n \) is:
\[ n \cdot \pi \]