Question

If \( z = 1 + i \), then the argument of \( z^2 e^{z-i} \) is

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{6} \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{\pi}{3} \)
• E. \( 0 \)

Hint:

Break complex expressions into parts and use argument addition rule carefully. Exponential of a real number always has zero argument.

Answer 1

The Correct Option is A

Concept: Argument properties:
• \( \arg(z_1 z_2) = \arg z_1 + \arg z_2 \)
• \( \arg(e^{a+ib}) = b \)

Step 1: Given value
\[ z = 1 + i \]

Step 2: Find argument of \(z\)
\[ \arg(z) = \tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4} \]

Step 3: Argument of \(z^2\)
\[ \arg(z^2) = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2} \]

Step 4: Argument of exponential term
\[ e^{z-i} = e^{(1+i)-i} = e^1 \] This is a positive real number, so: \[ \arg(e^{z-i}) = 0 \]

Step 5: Total argument
\[ \arg(z^2 e^{z-i}) = \frac{\pi}{2} + 0 = \frac{\pi}{2} \]

Step 6: Final Answer
\[ \boxed{\frac{\pi}{2}} \]