Identity $(e^z)^n = e^{nz}$

This property is correct for any complex number $z$ and any integer $n$. Let $z = x + iy$. We know that $e^z = e^x e^{iy}$. Using the properties of exponents and De Moivre's theorem: $(e^z)^n = (e^x e^{iy})^n = (e^x)^n (e^{iy})^n = e^{nx} (\cos(ny) + i \sin(ny)) = e^{nx} e^{iny} = e^{n(x+iy)} = e^{nz}$ Therefore, statement A is correct.

Harmonic Function $T(x, y) = e^u \cos v$

If $f(z) = u(x, y) + i v(x, y)$ is analytic in a domain $D$, then the function $g(z) = e^{f(z)}$ is also analytic in $D$. We can write $g(z)$ as: $g(z) = e^{u+iv} = e^u e^{iv} = e^u (\cos v + i \sin v)$ Let $g(z) = U(x, y) + i V(x, y)$. Then the real part is $U(x, y) = e^u \cos v$. Since $g(z)$ is analytic, its real part $U(x, y)$ must be a harmonic function. Thus, $T(x, y) = e^{u(x, y)} \cos v(x, y)$ is harmonic in $D$. Statement B is correct.

$e^z$ Non-Zero Property

For any complex number $z = x + iy$, the exponential function is $e^z = e^{x+iy} = e^x e^{iy}$. The magnitude of $e^z$ is $|e^z| = |e^x| |e^{iy}|$. Since $|e^{iy}| = |\cos y + i \sin y| = 1$, we have: $|e^z| = e^x$ The real exponential $e^x$ is always positive for any real $x$. Therefore, $|e^z| = e^x > 0$ for all $z \in \mathbb{C}$. This confirms that $e^z$ is never zero. Statement C is correct.

$(i)^i$ Principal Value

The general formula for $a^b$ in complex numbers is $a^b = e^{b \log a}$. For $(i)^i$, we have $a=i$ and $b=i$. The complex logarithm is $\log a = \ln|a| + i(\arg a + 2k\pi)$. For $a=i$, $|i|=1$ and the principal argument is $\arg i = \frac{\pi}{2}$. So, $\log i = \ln 1 + i(\frac{\pi}{2} + 2k\pi) = i(\frac{\pi}{2} + 2k\pi)$. Substituting into the formula: $(i)^i = e^{i \log i} = e^{i \left[ i(\frac{\pi}{2} + 2k\pi) \right]} = e^{-\left(\frac{\pi}{2} + 2k\pi\right)}$ The principal value occurs when $k=0$, giving $e^{-\frac{\pi}{2}}$. Statement D is correct.

$\cos z$ Definition

The correct definition of the complex cosine function is: $\cos z = \frac{e^{iz} + e^{-iz}}{2}$ The expression given in statement E, $\frac{e^{iz} - e^{-iz}}{2}$, is the definition of the complex sine function ($\sin z$). Therefore, statement E is incorrect.

Conclusion on Statements A, B, C, D, E

Statements A, B, C, and D are mathematically correct properties or calculations in complex analysis. Statement E presents an incorrect definition.