Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2008 / 4

Write $z=x+iy$, so the real and imaginary components are $u=x^2$ and $v=y^2$. The Cauchy-Riemann equations require $u_x=v_y$, namely $2x=2y$; the other equation is $u_y=-v_x=0$. The first relation fails off the line $x=y$, so the function is not holomorphic on $\mathbb{C}$. It is complex differentiable at points of that line, but a line contains no open neighborhood on which the equations hold.

The reciprocal map is holomorphic for $z\ne 0$, and the complex exponential is entire. Its denominator vanishes exactly whenAll these points are excluded from $U$. The set $U$ is open: it is the inverse image of $\mathbb{C}\setminus \mathbb{Z}$ under the reciprocal map on $\mathbb{C}\setminus \{0\}$. A reciprocal of a nonvanishing holomorphic function is holomorphic. Therefore $f$ is holomorphic throughout $U$.

The set $U$ has two open connected components, the left and right half-planes. On each the function is constant in a neighborhood of every point, so its complex derivative is zero everywhere. Thus $f$ is holomorphic on $U$. A holomorphic function can take different constant values on different components of a disconnected domain.

At any point on the imaginary axis, approaching from the left gives limit $-1$, while the assigned value and the right-hand limit are $1$. Thus the function is discontinuous on that axis. Complex differentiability implies continuity, so $f$ is not holomorphic on $\mathbb{C}$. It is holomorphic separately in the two open half-planes.