Strip

Definition

A strip in $\mathbb {C}$ is an open connected subset in $\mathbb {C}$ bounded by two parallel lines. Some special kinds of strips:

• Vertical strip: A strip bounded by vertical lines. It is defined by saying that the real part must lie within an open interval
• Horizontal strip: A strip bounded by horizontal lines. It is defined by saying that the imaginary part must lie within an open interval

Riemann mapping

Any two strips are equivalent why translations and multiplications by complex numbers, so it suffices to give a Riemann mapping to the open unit disk, for one particular strip. consider the strip bound by the horizontal lines $y=0$ and $y=\pi i$. The Riemann mapping is given by:

$z\mapsto {\frac {e^{z}-i}{e^{z}+i}}$

The complex exponential maps the strip to the upper half-plane, while the fractional linear transformation maps the upper half-plane to the open unit disk.

Relation with other domains

• Slit plane: The strip bounded by the horizontal lines $y=\pm \pi i$ gets mapped to the slit plane under the complex exponential.
• Upper half-plane: The strip bounded by the horizontal lines $y=0$ and $y=\pi i$ gets mapped to the upper half-plane under the complex exponential.