Question

Which of the following are correct
A. $(e^z)^n = e^{nz}$, $(n = 0, \pm 1, \pm 2, \dots)$ B. Let $f(z) = u(x,y) + i v(x,y)$ be analytic on some domain $D$. Then $T(x,y) = e^{u(x,y)} \cos v(x,y)$ is harmonic in $D$. C. $e^z \neq 0$ for all $z \in \mathbb{C}$. D. The principal value of $(i)^i$ is $\exp\left(\frac{\pi}{2}\right)$
E. $\cos z = \frac{e^{iz} - e^{-iz}}{2}$ Choose the correct answer from the options given below:

• A. A, B, C, D Only
• B. A, B, C, E Only
• C. A, B, C Only
• D. A, D Only

Hint:

Always double check sign conventions: $i^i = e^{-\pi/2}$, $\cos z = \frac{e^{iz}+e^{-iz}}{2}$, and $\sin z = \frac{e^{iz}-e^{-iz}}{2i}$.

Answer 1

The Correct Option is C

Step 1 : Concept:
This question tests algebraic properties of exponential and trigonometric functions in complex analysis, harmonic functions, and principal values of complex powers.

Step 2 : Key Formulas and Approach:
1. Complex exponential properties: $(e^z)^n = e^{nz}$ for integer $n$, and $|e^z| = e^{\text{Re}(z)} \neq 0$.
2. If $g(z)$ is analytic, its real part $\text{Re}(g(z))$ is automatically harmonic.
3. Principal value of $a^b$: $a^b = \exp(b \text{Log } a)$, where $\text{Log } a = \ln|a| + i \text{Arg}(a)$.
4. Euler's formulas: $\cos z = \frac{e^{iz} + e^{-iz}}{2}$ and $\sin z = \frac{e^{iz} - e^{-iz}}{2i}$.

Step 3 : Step-by-step Explanation:

• Statement A:
For any complex number $z$ and integer $n \in \mathbb{Z}$, the exponent law $(e^z)^n = e^{nz}$ holds rigorously. Statement A is correct.

• Statement B:
Since $f(z) = u + iv$ is analytic on $D$, the composite function $g(z) = e^{f(z)}$ is also analytic on $D$.
Expanding $g(z)$: \[ g(z) = e^{u + iv} = e^u e^{iv} = e^u (\cos v + i \sin v) = e^u \cos v + i e^u \sin v \] The real part of an analytic function is always harmonic. Here, $\text{Re}(g(z)) = e^{u(x,y)} \cos v(x,y) = T(x,y)$. Thus, $T(x,y)$ is harmonic in $D$. Statement B is correct.

• Statement C:
For $z = x + iy$, $|e^z| = e^x > 0$ for all real $x$. Since the magnitude is strictly positive, $e^z \neq 0$ for any $z \in \mathbb{C}$. Statement C is correct.

• Statement D:
The principal value of $i^i$ is calculated using $\text{Log}(i) = \ln|i| + i \text{Arg}(i) = 0 + i \frac{\pi}{2} = i \frac{\pi}{2}$: \[ i^i = \exp(i \text{Log } i) = \exp\left(i \cdot i \frac{\pi}{2}\right) = \exp\left(-\frac{\pi}{2}\right) \] The statement gives $\exp\left(\frac{\pi}{2}\right)$ (positive sign), which is incorrect. Statement D is false.

• Statement E:
By definition, $\cos z = \frac{e^{iz} + e^{-iz}}{2}$. The given formula has a minus sign, which corresponds to $i \sin z$, not $\cos z$. Statement E is false.

Step 4 : Final Answer:
Statements A, B, and C are correct. Therefore, option (C) is the correct answer.