
Description


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The trigonometric and hyperbolic functions are not invertible over the entire complex plane, or for many of them even over the real line, so it is necessary to define a principal branch for each such inverse function. This is done by restricting the forward function to a principal domain on which it is invertible, and taking that domain as the range of the inverse function.

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This process necessarily results in discontinuities in the inverse functions, which can be taken to be along line segments (called branch cuts) in the real or imaginary axes. There is choice involved with this process, and the choices can have far reaching mathematical consequences.

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In addition to choosing the locations of the branch cuts, it is necessary to define the closure of the functions with respect to these branch cuts, meaning the continuity properties of the functions along paths which approach a branch cut from one side or the other. In the absence of any clearer choice, Maple uses the convention of counterclockwise continuity (abbreviated CCC), meaning that the function is continuous along a path which encircles a branch point in a counterclockwise direction and which terminates on the corresponding branch cut. The alternative, clockwise continuity (abbreviated CC) is used whenever imposing CCC would lead to the loss of a useful identity or other such mathematical difficulty.

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Maple implements a signed zero in its software floatingpoint computation domain. This means that in this domain, branch cuts can be closed from both sides of the cut. For example, the results of

>

ln(2.+0.*I), ln(2.0.*I);

${0.6931471806}{+}{3.141592654}{}{\mathrm{I}}{,}{0.6931471806}{}{3.141592654}{}{\mathrm{I}}$
 (1) 

are not the same. This allows the identity $\mathrm{ln}\left(\frac{1}{z}\right)=\mathrm{ln}\left(z\right)$ to be preserved throughout the complex plane:

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ln(2.0.*I) = ln(.5+0.*I);

${0.6931471806}{}{3.141592654}{}{\mathrm{I}}{=}{0.6931471806}{}{3.141592654}{}{\mathrm{I}}$
 (2) 

By convention in Maple, a floatingpoint number with no imaginary part is interpreted as having imaginary part equal to +0.0 when determining values on branch cuts. Similarly, a purely imaginary number is interpreted as having real part equal to +0.0 in similar contexts. Exact rational real numbers are also interpreted as having +0 imaginary part, so

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arccosh(1) = arccosh(1.+0.*I);

${\mathrm{I}}{}{\mathrm{\pi}}{=}{3.141592654}{}{\mathrm{I}}$
 (3) 
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As a further notational convention, in the tables below a branch cut will be shown as having a closed endpoint at a branch point if the corresponding function is finitevalued at that branch point. Otherwise, an open interval is indicated.

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You can use the branches function to obtain graphical representations of the ranges and closures of the principal branches of these functions as they are defined in Maple.



Branch cuts of the inverse trigonometric functions


Function

Branch Cuts

Closure

arcsin

$\left(\infty ,1\right]$, $\left[1,\infty \right)$

CCC @ 1, 1

arccos

$\left(\infty ,1\right]$, $\left[1,\infty \right)$

CCC @ 1, 1

arctan

$\left(i\infty ,i\right)$, $\left(i,i\infty \right)$

CCC @ $i$, $i$

arccsc

$\left[1,0\right)$, $\left(0,1\right]$

CCC @ 1, 1

arcsec

$\left[1,0\right)$, $\left(0,1\right]$

CCC @ 1, 1

arccot

$\left(i\infty ,i\right)$, $\left(i,i\infty \right)$

CCC @ $i$, $i$





Branch cuts of the inverse hyperbolic functions


arcsinh

$\left(i\infty ,i\right]$, $\left[i,i\infty \right)$

CCC @ $i$, $i$

arccosh

$\left(\infty ,1\right]$, $\left(\infty ,1\right]$

CCC @ 1, 1

arctanh

$\left(\infty ,1\right)$, $\left(1,\infty \right)$

CCC @ 1, 1

arccsch

$\left[i,0\right)$, $\left(0,i\right]$

CCC @ $i$, $i$

arcsech

$\left(\infty ,0\right)$, $\left[1,\infty \right)$

CCC @ 1, CC @ 0

arccoth

$\left[1,1\right]$

CC @ 1, CCC @ 1





References



Kahan, W., "Branch cuts for complex elementary functions, or, Much ado about nothing's sign bit." Proceedings of the joint IMA/SIAM conference on The State of the Art in Numerical Analysis, University of Birmingham, A. Iserles & M.J.D. Powell, eds., Clarendon Press, Oxford. (1987): 165210.



