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All of the logarithmic, trigonometric, and other scientific functions are
defined for complex numbers as well as for reals.
This section describes the values
returned in cases where the general result is a family of possible values.
Calc follows section 12.5.3 of Steele’s *Common Lisp, the Language*,
second edition, in these matters. This section will describe each
function briefly; for a more detailed discussion (including some nifty
diagrams), consult Steele’s book.

Note that the branch cuts for `arctan`

and `arctanh`

were
changed between the first and second editions of Steele. Recent
versions of Calc follow the second edition.

The new branch cuts exactly match those of the HP-28/48 calculators.
They also match those of Mathematica 1.2, except that Mathematica’s
`arctan`

cut is always in the right half of the complex plane,
and its `arctanh`

cut is always in the top half of the plane.
Calc’s cuts are continuous with quadrants I and III for `arctan`

,
or II and IV for `arctanh`

.

Note: The current implementations of these functions with complex arguments
are designed with proper behavior around the branch cuts in mind, *not*
efficiency or accuracy. You may need to increase the floating precision
and wait a while to get suitable answers from them.

For ‘`sqrt(a+bi)`’: When ‘`a<0`’ and ‘`b`’ is small but positive
or zero, the result is close to the ‘`+i`’ axis. For ‘`b`’ small and
negative, the result is close to the ‘`-i`’ axis. The result always lies
in the right half of the complex plane.

For ‘`ln(a+bi)`’: The real part is defined as ‘`ln(abs(a+bi))`’.
The imaginary part is defined as ‘`arg(a+bi) = arctan2(b,a)`’.
Thus the branch cuts for `sqrt`

and `ln`

both lie on the
negative real axis.

The following table describes these branch cuts in another way.
If the real and imaginary parts of ‘`z`’ are as shown, then
the real and imaginary parts of ‘`f(z)`’ will be as shown.
Here `eps`

stands for a small positive value; each
occurrence of `eps`

may stand for a different small value.

z sqrt(z) ln(z) ---------------------------------------- +, 0 +, 0 any, 0 -, 0 0, + any, pi -, +eps +eps, + +eps, + -, -eps +eps, - +eps, -

For ‘`z1^z2`’: This is defined by ‘`exp(ln(z1)*z2)`’.
One interesting consequence of this is that ‘`(-8)^1:3`’ does
not evaluate to *-2* as you might expect, but to the complex
number ‘`(1., 1.732)`’. Both of these are valid cube roots
of *-8* (as is ‘`(1., -1.732)`’); Calc chooses a perhaps
less-obvious root for the sake of mathematical consistency.

For ‘`arcsin(z)`’: This is defined by ‘`-i*ln(i*z + sqrt(1-z^2))`’.
The branch cuts are on the real axis, less than *-1* and greater than 1.

For ‘`arccos(z)`’: This is defined by ‘`-i*ln(z + i*sqrt(1-z^2))`’,
or equivalently by ‘`pi/2 - arcsin(z)`’. The branch cuts are on
the real axis, less than *-1* and greater than 1.

For ‘`arctan(z)`’: This is defined by
‘`(ln(1+i*z) - ln(1-i*z)) / (2*i)`’. The branch cuts are on the
imaginary axis, below ‘`-i`’ and above ‘`i`’.

For ‘`arcsinh(z)`’: This is defined by ‘`ln(z + sqrt(1+z^2))`’.
The branch cuts are on the imaginary axis, below ‘`-i`’ and
above ‘`i`’.

For ‘`arccosh(z)`’: This is defined by
‘`ln(z + (z+1)*sqrt((z-1)/(z+1)))`’. The branch cut is on the
real axis less than 1.

For ‘`arctanh(z)`’: This is defined by ‘`(ln(1+z) - ln(1-z)) / 2`’.
The branch cuts are on the real axis, less than *-1* and greater than 1.

The following tables for `arcsin`

, `arccos`

, and
`arctan`

assume the current angular mode is Radians. The
hyperbolic functions operate independently of the angular mode.

z arcsin(z) arccos(z) ------------------------------------------------------- (-1..1), 0 (-pi/2..pi/2), 0 (0..pi), 0 (-1..1), +eps (-pi/2..pi/2), +eps (0..pi), -eps (-1..1), -eps (-pi/2..pi/2), -eps (0..pi), +eps <-1, 0 -pi/2, + pi, - <-1, +eps -pi/2 + eps, + pi - eps, - <-1, -eps -pi/2 + eps, - pi - eps, + >1, 0 pi/2, - 0, + >1, +eps pi/2 - eps, + +eps, - >1, -eps pi/2 - eps, - +eps, +

z arccosh(z) arctanh(z) ----------------------------------------------------- (-1..1), 0 0, (0..pi) any, 0 (-1..1), +eps +eps, (0..pi) any, +eps (-1..1), -eps +eps, (-pi..0) any, -eps <-1, 0 +, pi -, pi/2 <-1, +eps +, pi - eps -, pi/2 - eps <-1, -eps +, -pi + eps -, -pi/2 + eps >1, 0 +, 0 +, -pi/2 >1, +eps +, +eps +, pi/2 - eps >1, -eps +, -eps +, -pi/2 + eps

z arcsinh(z) arctan(z) ----------------------------------------------------- 0, (-1..1) 0, (-pi/2..pi/2) 0, any 0, <-1 -, -pi/2 -pi/2, - +eps, <-1 +, -pi/2 + eps pi/2 - eps, - -eps, <-1 -, -pi/2 + eps -pi/2 + eps, - 0, >1 +, pi/2 pi/2, + +eps, >1 +, pi/2 - eps pi/2 - eps, + -eps, >1 -, pi/2 - eps -pi/2 + eps, +

Finally, the following identities help to illustrate the relationship between the complex trigonometric and hyperbolic functions. They are valid everywhere, including on the branch cuts.

sin(i*z) = i*sinh(z) arcsin(i*z) = i*arcsinh(z) cos(i*z) = cosh(z) arcsinh(i*z) = i*arcsin(z) tan(i*z) = i*tanh(z) arctan(i*z) = i*arctanh(z) sinh(i*z) = i*sin(z) cosh(i*z) = cos(z)

The “advanced math” functions (gamma, Bessel, etc.) are also defined for general complex arguments, but their branch cuts and principal values are not rigorously specified at present.

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