Which of the following complex functions is/are analytic on the complex plane ?

The correct option is:
$f(z)=z^2-z$

Step-by-Step Explanation:
A complex function is said to be analytic (or holomorphic) on the complex plane if it is differentiable at every point in the complex plane. An entire function is a function that is analytic everywhere in the complex plane.

Let us evaluate each of the given options to determine their analyticity:

1. Evaluating $f(z)=z^2-z$:
This function is a polynomial in terms of the complex variable $z$.
Any polynomial function of $z$, such as $P(z)=a_n{z}^n+...+a_{1}z+a_0$, is differentiable everywhere on the complex plane.
Its derivative is given by:
$f^{\prime }(z)=2z-1$
Since the derivative exists and is defined for all $z$ in the complex plane, $f(z)=z^2-z$ is analytic on the entire complex plane (entire).

2. Evaluating $f(z)=j\mathrm{Re}(z)$ and $f(z)=\mathrm{Im}(z)$:
Let $z=x+jy$, where $x$ and $y$ are real numbers.
For $f(z)=j\mathrm{Re}(z)=jx$:
Here, the real part $u(x,y)=0$ and the imaginary part $v(x,y)=x$.
Checking the Cauchy-Riemann equations:
$\frac{\partial u}{\partial x}=0,\frac{\partial v}{\partial y}=0$ (which matches)
However, $\frac{\partial u}{\partial y}=0$ and $\frac{\partial v}{\partial x}=1$. For analyticity, we must have $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$, but here $0\ne -1$. Thus, the function is not analytic anywhere.
Similarly, for $f(z)=\mathrm{Im}(z)=y$:
Here, $u(x,y)=y$ and $v(x,y)=0$.
We get $\frac{\partial u}{\partial x}=0$ and $\frac{\partial v}{\partial y}=0$.
But $\frac{\partial u}{\partial y}=1$ and $-\frac{\partial v}{\partial x}=0$. Since $1\ne 0$, the Cauchy-Riemann equations are violated, and this function is also not analytic.

3. Evaluating $f(z)=e^{|z|}$:
The absolute value function $|z|=\sqrt{x^2+y^2}$ is a real-valued function of a complex variable and is not differentiable anywhere except possibly at the origin (where it is actually not differentiable either). As a result, any function depending purely on the magnitude $|z|$ is non-analytic.

Therefore, the only function among the choices that is analytic on the entire complex plane is $f(z)=z^2-z$.