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Branch Cut

\begin{figure}\begin{center}\BoxedEPSF{SqrtReIm.epsf scaled 700}\end{center}\end{figure}

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


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

© 1996-9 Eric W. Weisstein