Question

Let z, \(z_1, z_2\) be complex numbers. Then which of the following statements are True?
(A) \(e^z\) is never zero
(B) \(|e^{ix}|=1\) if x is real
(C) \(e^z = 1\) if z is an integral multiple of \(2\pi i\)
(D) \(e^{z_1} = e^{z_2}\) if and only if \(z_1 - z_2 = \frac{2\pi i n}{\sqrt{3}}\), where n is an integer
(E) \(|e^z|>e^z\) for \(z \neq 0\)
Choose the correct answer from the options given below:

• A. (A), (B) and (D) only
• B. (B), (C) and (E) only
• C. (A), (B) and (C) only
• D. (A) and (D) only

Hint:

The complex exponential function \(e^z\) has a period of \(2\pi i\). This is a fundamental property. It means \(e^{z + 2n\pi i} = e^z\) for any integer \(n\). This fact is the basis for both statements (C) and (D).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question tests fundamental properties of the complex exponential function \(e^z\).
Step 2: Detailed Explanation:

(A) \(e^z\) is never zero: Let \(z = x+iy\). Then \(e^z = e^{x+iy} = e^x e^{iy}\). The magnitude is \(|e^z| = |e^x| |e^{iy}| = e^x \cdot 1 = e^x\). Since \(e^x\) is always positive for any real \(x\), the magnitude \(|e^z|\) is never zero. Therefore, \(e^z\) can never be zero. This statement is True.
(B) \(|e^{ix}|=1\) if x is real: By Euler's formula, \(e^{ix} = \cos(x) + i\sin(x)\). The magnitude is \(|e^{ix}| = \sqrt{\cos^2(x) + \sin^2(x)} = \sqrt{1} = 1\). This statement is True.
(C) \(e^z = 1\) if z is an integral multiple of \(2\pi i\): Let \(z = 2n\pi i\) for some integer \(n\). Then \(e^z = e^{2n\pi i} = \cos(2n\pi) + i\sin(2n\pi) = 1 + i(0) = 1\). This statement is True.
(D) \(e^{z_1} = e^{z_2}\) ...: The condition \(e^{z_1} = e^{z_2}\) is equivalent to \(e^{z_1 - z_2} = 1\). From statement (C), this means that \(z_1 - z_2\) must be an integral multiple of \(2\pi i\). The statement given is \(z_1 - z_2 = \frac{2\pi i n}{\sqrt{3}}\), which is incorrect due to the \(\sqrt{3}\) factor. This statement is False.
(E) \(|e^z|>e^z\) for \(z \neq 0\): This inequality is not well-defined because \(e^z\) is generally a complex number, and inequalities (other than for magnitude) are not defined for complex numbers. Furthermore, if \(z\) is a positive real number, \(|e^z| = e^z\), so the inequality is false. This statement is False.
Step 3: Final Answer:
Statements (A), (B), and (C) are true. Therefore, option (C) is the correct choice.