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14.1 Functions and Variables for Logarithms |
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Default value: false
When true
, r
some rational number, and
x
some expression, %e^(r*log(x))
will be simplified into x^r
. It
should be noted that the radcan
command also does this transformation,
and more complicated transformations of this ilk as well.
The logcontract
command "contracts" expressions containing log
.
Represents the polylogarithm function of order s and argument z, defined by the infinite series
inf ==== k \ z Li (z) = > -- s / s ==== k k = 1
li [1]
is - log (1 - z)
.
li [2]
and li [3]
are the dilogarithm and trilogarithm functions, respectively.
When the order is 1, the polylogarithm simplifies to - log (1 - z)
,
which in turn simplifies to a numerical value
if z is a real or complex floating point number or the numer
evaluation flag is present.
When the order is 2 or 3,
the polylogarithm simplifies to a numerical value
if z is a real floating point number
or the numer
evaluation flag is present.
Examples:
(%i1) assume (x > 0); (%o1) [x > 0] (%i2) integrate ((log (1 - t)) / t, t, 0, x); (%o2) - li (x) 2 (%i3) li [2] (7); (%o3) li (7) 2 (%i4) li [2] (7), numer; (%o4) 1.24827317833392 - 6.113257021832577 %i (%i5) li [3] (7); (%o5) li (7) 3 (%i6) li [2] (7), numer; (%o6) 1.24827317833392 - 6.113257021832577 %i (%i7) L : makelist (i / 4.0, i, 0, 8); (%o7) [0.0, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0] (%i8) map (lambda ([x], li [2] (x)), L); (%o8) [0, .2676526384986274, .5822405249432515, .9784693966661848, 1.64493407, 2.190177004178597 - .7010261407036192 %i, 2.374395264042415 - 1.273806203464065 %i, 2.448686757245154 - 1.758084846201883 %i, 2.467401098097648 - 2.177586087815347 %i] (%i9) map (lambda ([x], li [3] (x)), L); (%o9) [0, .2584613953442624, 0.537213192678042, .8444258046482203, 1.2020569, 1.642866878950322 - .07821473130035025 %i, 2.060877505514697 - .2582419849982037 %i, 2.433418896388322 - .4919260182322965 %i, 2.762071904015935 - .7546938285978846 %i]
Categories: Exponential and logarithm functions
Represents the natural (base e) logarithm of x.
Maxima does not have a built-in function for the base 10 logarithm or other
bases. log10(x) := log(x) / log(10)
is a useful definition.
Simplification and evaluation of logarithms is governed by several global flags:
logexpand
causes log(a^b)
to become b*log(a)
. If it is
set to all
, log(a*b)
will also simplify to log(a)+log(b)
.
If it is set to super
, then log(a/b)
will also simplify to
log(a)-log(b)
for rational numbers a/b
, a#1
.
(log(1/b)
, for b
integer, always simplifies.) If it is set to
false
, all of these simplifications will be turned off.
logsimp
if false
then no simplification of %e
to a power containing
log
's is done.
lognumer
if true
then negative floating point arguments to log
will always
be converted to their absolute value before the log
is taken. If
numer
is also true
, then negative integer arguments to log
will also be converted to their absolute value.
lognegint
if true
implements the rule log(-n)
-> log(n)+%i*%pi
for
n
a positive integer.
%e_to_numlog
when true
, r
some rational number, and x
some expression,
%e^(r*log(x))
will be simplified into x^r
. It should be noted
that the radcan
command also does this transformation, and more
complicated transformations of this as well. The logcontract
command
"contracts" expressions containing log
.
Categories: Exponential and logarithm functions
Default value: false
When doing indefinite integration where
logs are generated, e.g. integrate(1/x,x)
, the answer is given in
terms of log(abs(...))
if logabs
is true
, but in terms of log(...)
if
logabs
is false
. For definite integration, the logabs:true
setting is
used, because here "evaluation" of the indefinite integral at the
endpoints is often needed.
Categories: Exponential and logarithm functions · Integral calculus · Global flags
When the global variable logarc
is true
,
inverse circular and hyperbolic functions are replaced by
equivalent logarithmic functions.
The default value of logarc
is false
.
The function logarc(expr)
carries out that replacement for
an expression expr
without setting the global variable logarc
.
Default value: false
Controls which coefficients are
contracted when using logcontract
. It may be set to the name of a
predicate function of one argument. E.g. if you like to generate
SQRTs, you can do logconcoeffp:'logconfun$
logconfun(m):=featurep(m,integer) or ratnump(m)$
. Then
logcontract(1/2*log(x));
will give log(sqrt(x))
.
Recursively scans the expression expr, transforming
subexpressions of the form a1*log(b1) + a2*log(b2) + c
into
log(ratsimp(b1^a1 * b2^a2)) + c
(%i1) 2*(a*log(x) + 2*a*log(y))$ (%i2) logcontract(%); 2 4 (%o2) a log(x y )
If you do declare(n,integer);
then logcontract(2*a*n*log(x));
gives
a*log(x^(2*n))
. The coefficients that "contract" in this manner are
those such as the 2 and the n
here which satisfy
featurep(coeff,integer)
. The user can control which coefficients are
contracted by setting the option logconcoeffp
to the name of a
predicate function of one argument. E.g. if you like to generate
SQRTs, you can do logconcoeffp:'logconfun$
logconfun(m):=featurep(m,integer) or ratnump(m)$
. Then
logcontract(1/2*log(x));
will give log(sqrt(x))
.
Categories: Exponential and logarithm functions
Default value: false
If true
causes log(a^b)
to become b*log(a)
. If it is set
to all
, log(a*b)
will also simplify to log(a)+log(b)
. If
it is set to super
, then log(a/b)
will also simplify to
log(a)-log(b)
for rational numbers a/b
, a#1
.
(log(1/b)
, for integer b
, always simplifies.) If it is set to
false
, that is the default value, all of these simplifications will be
turned off.
Default value: false
If true
implements the rule
log(-n)
-> log(n)+%i*%pi
for n
a positive integer.
Default value: false
If true
then negative floating point
arguments to log
will always be converted to their absolute value
before the log
is taken. If numer
is also true
, then negative integer
arguments to log
will also be converted to their absolute value.
Default value: true
If false
then no simplification of %e
to a
power containing log
's is done.
Represents the principal branch of the complex-valued natural
logarithm with -%pi
< carg(x)
<= +%pi
.
Categories: Exponential and logarithm functions · Complex variables
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