Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2013 / Analysis / 4

Under the assumed continuous real differentiability, the necessary and sufficient conditions are the Cauchy-Riemann equations at every point of $U$:Necessity follows by computing the complex derivative along real and imaginary increments. For sufficiency, real differentiability gives the first-order expansionThe Cauchy-Riemann equations turn the linear part into $(u_x+i{v}_x)h$, so the complex derivative exists and equals $u_x+i{v}_x$. Thus $f$ is a holomorphic function on $U$.

At any $z\ne 0$, take sufficiently small $h$ so $z+h\ne 0$. The complex difference quotient isHence the complex derivative exists at every point of the punctured plane, andThe excluded point is essential to the domain statement: the reciprocal has a simple pole there.

Write $z=x+iy$. Both moduli are nonnegative, so squaring preserves their strict inequality:ThusThe boundary is the perpendicular bisector of $0$ and $i$ in the complex plane; the strict inequality selects the half-plane containing $0$.

Use the reciprocal Möbius transformationIt is a holomorphic function on $U_1$ by part (b). For $z\ne 0$,Part (c) now gives $z\in U_1$ if and only if $|F(z)+i|>1$, so $F(U_1)\subseteq U_2$. Conversely, $0\notin U_2$ because $|0+i|=1$. For any $w\in U_2$, put $z=1/w$; the same identity gives $|z-i|>|z|$, so $z\in U_1$ and $F(z)=w$. Finally, reciprocation is injective on nonzero complex numbers. This proves bijectivity, with inverseThe inverse is also holomorphic, so this is a biholomorphism.