Hi,
Here's a stab at a translation for the two functions you sent. My German
is mediocre, but I guess it's not like I'm trying to translate poetry or
something!
I've made some notes at the bottom about things I intended to change
(rather than things I misunderstood) and also some questions that I had
when reading it. I hope this doesn't create too much extra work for you!
Rupert
(skip ahead to "NOTES" for my questions)
----------------------------------------------------------------------
Function: abs (z)
The abs function represents the mathematical absolute value function and
works for both numerical and symbolic values. If the argument, z, is a
real or complex number, abs returns the absolute value of z. If
possible, symbolic expressions using the absolute value function are
also simplified.
Maxima can differentiate, integrate and calculate limits for some
expressions containing abs. The abs_integrate package further extends
Maxima's ability to calculate integrals involving the abs function. See
(%i12) in the examples below.
When applied to a list or matrix, abs automatically distributes over
the terms. Similarly, it distributes over both sides of an
equation. To alter this behaviour, see the variable distribute_over.
Examples:
Calculation of abs for real and complex numbers, including numerical
constants and various infinities. The first example shows how abs
distributes over the elements of a list.
(%i1) abs([-4, 0, 1, 1+%i]);
(%o1) [4, 0, 1, sqrt(2)]
(%i2) abs((1+%i)*(1-%i));
(%o2) 2
(%i3) abs(%e+%i);
2
(%o3) sqrt(%e + 1)
(%i4) abs([inf, infinity, minf]);
(%o4) [inf, inf, inf]
Simplification of expressions containing abs:
(%i5) abs(x^2);
2
(%o5) x
(%i6) abs(x^3);
2
(%o6) x abs(x)
(%i7) abs(abs(x));
(%o7) abs(x)
(%i8) abs(conjugate(x));
(%o8) abs(x)
Integrating and differentiating with the abs function. Note that, if
the abs_integrate package is loaded, more integrals involving the abs
function can be performed. The last example shows the Laplace
transform of abs: see laplace.
(%i9) diff(x*abs(x),x),expand;
(%o9) 2 abs(x)
(%i10) integrate(abs(x),x);
x abs(x)
(%o10) --------
2
(%i11) integrate(x*abs(x),x);
/
[
(%o11) I x abs(x) dx
]
/
(%i12) load(abs_integrate)$
(%i13) integrate(x*abs(x),x);
2 3
x abs(x) x signum(x)
(%o13) --------- - ------------
2 6
(%i14) integrate(abs(x),x,-2,%pi);
2
%pi
(%o14) ---- + 2
2
(%i15) laplace(abs(x),x,s);
1
(%o15) --
2
s
----------------------------------------------------------------------
Function: cabs (expr)
Calculates the absolute value of an expression representing a complex
number. Unlike the function abs, the cabs function always decomposes
its argument into a real and an imaginary part. If x and y represent
real variables or expressions, the cabs function calculates the
absolute value of x + %i*y as
sqrt (y^2 + x^2).
The cabs function can use symmetry properties and identities of
complex functions to help it calculate the absolute value of an
expression. If such identities exist, they can be advertised to cabs
using function properties. The symmetries that cabs understands are:
mirror symmetry, conjugate function and complex characteristic.
cabs is a verb function and is not very suitable for symbolic
calculations. For such calculations (including integration,
differentiation and taking limits of expressions containing absolute
values), use abs.
The result of cabs can include the absolute value function, abs, and
the arc tangent, atan2.
When applied to a list or matrix, cabs automatically distributes over
the terms. Similarly, it distributes over both sides of an equation.
For further ways to compute with complex numbers, see the functions
rectform, realpart, imagpart, carg, conjugate and polarform.
Examples:
Examples with sqrt and sin.
(%i1) cabs(sqrt(1+%i*x));
2 1/4
(%o1) (x + 1)
(%i2) cabs(sin(x+%i*y));
2 2 2 2
(%o2) sqrt(cos (x) sinh (y) + sin (x) cosh (y))
The error function, erf, has mirror symmetry, which is used here in
the calculation of the absolute value with a complex argument:
(%i3) cabs(erf(x+%i*y));
2
(erf(%i y + x) - erf(%i y - x))
(%o3) sqrt(--------------------------------
4
2
(erf(%i y + x) + erf(%i y - x))
- --------------------------------)
4
Maxima knows complex identities for the Bessel functions, which allow
it to compute the absolute value for complex arguments. Here is an
example for bessel_j.
(%i4) cabs(bessel_j(1,%i));
(%o4) abs(bessel_j(1, %i))
----------------------------------------------------------------------
NOTES:
======
Some things I changed (intentionally!) from the German text: maybe
these are improvements.
(1) I pointed at the abs_integrate example when the abs_integrate
package is mentioned.
(2) I explained why one should see distribute_over.
(3) Since sqrt and sin are from the English names, I didn't clarify
what they were in the cabs examples.
Questions I have after reading this, which would be nice to add
answers for! Providing this isn't just from my poor German, maybe
answering these could improve the German manual too.
(1) Why is abs_integrate functionality not permanently loaded if it
improves Maxima's ability to integrate functions with abs() in? (I
presume the answer is that it slows down general integration, but
can someone confirm this?)
(2) Is the repeat of the information about abs_integrate necessary?
Maybe the text about it can be removed from the examples section,
since it's already explained above? (Especially if there's a
forward reference, like I added)
(3) Is there a reason that abs takes z as an argument and cabs takes
expr?
(4) I'm a bit confused about the German in the first sentence of the
second paragraph in cabs. I suspect that what I wrote isn't quite
what you meant.
(5) Is it possible for a user to set a property indicating that
his/her function f has mirror symmetry, say? Maybe this
documentation should point to the properties a user can set. Maybe
with an example? (I would suggest wording, but I don't actually
know how to do this!)
(6) Also, what exactly do these properties mean? Are they documented
elsewhere? (In which case, it would probably be good to have "See
foo, bar")
(7) Is there a reason that cabs doesn't respect distribute_over?
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