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: Conceptual Foundation: The question assesses fundamental properties of the complex exponential function, denoted as \(e^z\). Step 2: Detailed Analysis:
(A) Non-zero nature of \(e^z\): Let \(z = x+iy\), where \(x\) and \(y\) are real numbers. Then \(e^z = e^{x+iy} = e^x e^{iy}\). The magnitude of \(e^z\) is \(|e^z| = |e^x| |e^{iy}|\). Since \(|e^x| = e^x\) (as \(e^x\) is always positive) and \(|e^{iy}| = 1\) (from Euler's formula, \(e^{iy} = \cos(y) + i\sin(y)\), with magnitude \(\sqrt{\cos^2(y) + \sin^2(y)} = 1\)), we have \(|e^z| = e^x\). As \(e^x\) is strictly positive for all real \(x\), \(|e^z|\) is never zero. Consequently, \(e^z\) can never be zero. This statement is True.
(B) Magnitude of \(e^{ix}\) for real \(x\): According to Euler's formula, \(e^{ix} = \cos(x) + i\sin(x)\). The magnitude is calculated as \(|e^{ix}| = \sqrt{\cos^2(x) + \sin^2(x)}\). Using the Pythagorean identity \(\cos^2(x) + \sin^2(x) = 1\), the magnitude simplifies to \(\sqrt{1} = 1\). This statement is True.
(C) Condition for \(e^z = 1\): If \(z\) is an integral multiple of \(2\pi i\), we can write \(z = 2n\pi i\) for some integer \(n\). Substituting this into the exponential function gives \(e^z = e^{2n\pi i}\). Using Euler's formula, this becomes \(\cos(2n\pi) + i\sin(2n\pi)\). Since \(\cos(2n\pi) = 1\) and \(\sin(2n\pi) = 0\) for any integer \(n\), the expression evaluates to \(1 + i(0) = 1\). This statement is True.
(D) Equality of exponentials: The equality \(e^{z_1} = e^{z_2}\) is equivalent to \(e^{z_1} / e^{z_2} = 1\), which simplifies to \(e^{z_1 - z_2} = 1\). Based on statement (C), this equality holds if and only if \(z_1 - z_2\) is an integral multiple of \(2\pi i\). The provided condition is \(z_1 - z_2 = \frac{2\pi i n}{\sqrt{3}}\). The presence of the \(\sqrt{3}\) in the denominator means this expression is not always an integral multiple of \(2\pi i\). Therefore, the statement is False.
(E) Inequality involving magnitude: The inequality \(|e^z|>e^z\) for \(z eq 0\) is problematic for two reasons. Firstly, inequalities are generally not defined for complex numbers unless comparing their magnitudes. Secondly, if \(z\) is a positive real number (e.g., \(z=1\)), then \(|e^z| = |e^1| = e^1 = e^z\). In this case, the inequality \(e^z>e^z\) is false. Thus, the statement is False.
Step 3: Conclusion: Statements (A), (B), and (C) have been verified as true. Therefore, option (C) is the correct response.