Standard semicircle, rectangle, keyhole, indentation, and sector contours

All contours on this page are positively oriented unless a local direction is stated explicitly.

• The upper semicircle contour of radius $R$ is the union of the real segment $[-R,R]$ and the arc ${\gamma }_R^{+}(t)=R{e}^{it}$ for $0\le t\le \pi$.
• The lower semicircle contour of radius $R$ traverses the real segment from $R$ to $-R$ and then the lower arc ${\gamma }_R^{-}(t)=R{e}^{-it}$ with $t$ decreasing from $\pi$ to $0$.
• The rectangle contour $\mathrm{\partial }[-R,R]\times [-T,T]$ is the positively oriented boundary of the Euclidean rectangle with vertices $\pm R\pm iT$.
• The keyhole contour about the positive real axis with radii $0<\epsilon <R$ is the positively oriented boundary of the slit annulus $\{z:\epsilon <\mathrm{\mid }z\mathrm{\mid }<R,z\mathrm{\notin }[0,\mathrm{\infty })\}$: it consists of the upper lip $x+i0$ from $\epsilon$ to $R$, the outer circle run counterclockwise, the lower lip $x-i0$ from $R$ to $\epsilon$, and the inner circle run clockwise.
• An upper indentation arc around a real point $a$ of radius $\epsilon$ is the clockwise semicircle $a+\epsilon e^{it}$ for $\pi \ge t\ge 0$; a lower indentation arc is the counterclockwise semicircle $a+\epsilon e^{it}$ for $\pi \le t\le 2\pi$.
• The sector contour with opening angles $\alpha <\beta$ and radii $0<\epsilon <R$ is the positively oriented boundary of $\{\epsilon <\mathrm{\mid }z\mathrm{\mid }<R,\alpha <\arg z<\beta \}$.

When a keyhole or sector contour is used with a complex power, this page states the branch explicitly. In the default keyhole convention the slit is the positive real axis and $z^{\alpha -1}=\exp ((\alpha -1)\mathrm{Log}z)$ is taken with $\mathrm{Arg}z\in (0,2\pi )$; the principal branch from Complex logarithms, the principal logarithm, and principal and multivalued complex powers and The principal logarithm is the normalised holomorphic branch on the slit plane is used only when the slit is the negative real axis.