Branch Cut

A line in the Complex Plane across which a Function is discontinuous.

function	branch cut(s)
$\cos^{-1} z$	$(-\infty,-1)$ and $(1,\infty)$
$\cosh^{-1}$	$(-\infty,1)$
$\cot^{-1} z$	$(-i,i)$
$\coth^{-1}$	$[-1,1]$
$\csc^{-1} z$	$(-1,1)$
$\mathop{\rm csch}\nolimits ^{-1}$	$(-i,i)$
$\ln z$	$(-\infty,0]$
$\sec^{-1} z$	$(-1,1)$
$\mathop{\rm sech}\nolimits ^{-1}$	$(\infty,0]$ and $(1,\infty)$
$\sin^{-1} z$	$(-\infty,-1)$ and $(1,\infty)$
$\sinh^{-1}$	$(-i\infty,-i)$ and $(i,i\infty)$
$\sqrt{z}$	$(-\infty, 0)$
$\tan^{-1} z$	$(-i\infty,-i)$ and $(i,i\infty)$
$\tanh^{-1}$	$(-\infty,-1]$ and $[1,\infty)$
$z^n, n\notin \Bbb{Z}$	$(-\infty, 0)$ for $\Re[n]\leq 0$; $(-\infty,0]$ for $\Re[n] >0$

See also Branch Point

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 399-401, 1953.