Question:medium

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:

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

Updated On: Jul 29, 2026

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Match List-I with List-II and choose the correct option:

LIST-I (Function)	LIST-II (Value)
(A) $ \int_{\gamma} \frac{1}{z-a} \, dz $, where $ \gamma: \|z-a\|=r, r > 0 $	(III) $ 2i\pi $
(B) $ \int_{\gamma} \frac{z+2}{z} \, dz $, where $ \gamma: z = 2e^{it}, 0 \le t \le \pi $	(IV) $ i\pi $
(C) $ \int_{\gamma} \frac{e^{2z}}{(z-1)(z-2)} \, dz $, where $ \gamma: \|z\|=3 $	(II) $ 2i\pi(e^4 - e^2) $
(D) $ \int_{\gamma} \frac{z^2 - z + 1}{2(z-1)} \, dz $, where $ \gamma: \|z\|=2 $	(I) $ -4 + 2i\pi $

Choose the correct answer from the options given below:

LIST-I (Function)	LIST-II (Value)
(A) \( \int_{\gamma} \frac{1}{z-a} \, dz \), where \( \gamma: \|z-a\|=r, r > 0 \)	(III) \( 2i\pi \)
(B) \( \int_{\gamma} \frac{z+2}{z} \, dz \), where \( \gamma: z = 2e^{it}, 0 \le t \le \pi \)	(IV) \( i\pi \)
(C) \( \int_{\gamma} \frac{e^{2z}}{(z-1)(z-2)} \, dz \), where \( \gamma: \|z\|=3 \)	(II) \( 2i\pi(e^4 - e^2) \)
(D) \( \int_{\gamma} \frac{z^2 - z + 1}{2(z-1)} \, dz \), where \( \gamma: \|z\|=2 \)	(I) \( -4 + 2i\pi \)