| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 14.1 Functions and Variables for Logarithms |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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.
@ref{Category: Exponential and logarithm functions} · @ref{Category: Simplification flags and variables}
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]
@ref{Category: 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 ilk as well.
The logcontract command "contracts" expressions containing log.
@ref{Category: 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.
@ref{Category: Exponential and logarithm functions} · @ref{Category: Integral calculus} · @ref{Category: 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.
@ref{Category: Exponential and logarithm functions} · @ref{Category: Simplification flags and variables} · @ref{Category: Simplification functions}
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)).
@ref{Category: Exponential and logarithm functions} · @ref{Category: Simplification flags and variables}
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)).
@ref{Category: Exponential and logarithm functions}
Default value: 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, all of these simplifications will
be turned off.
@ref{Category: Exponential and logarithm functions} · @ref{Category: Simplification flags and variables}
Default value: false
If true implements the rule
log(-n) -> log(n)+%i*%pi for n a positive integer.
@ref{Category: Exponential and logarithm functions} · @ref{Category: Simplification flags and variables}
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.
@ref{Category: Exponential and logarithm functions} · @ref{Category: Simplification flags and variables} · @ref{Category: Numerical evaluation}
Default value: true
If false then no simplification of %e to a
power containing log's is done.
@ref{Category: Exponential and logarithm functions} · @ref{Category: Simplification flags and variables}
Represents the principal branch of the complex-valued natural
logarithm with -%pi < carg(x) <= +%pi .
@ref{Category: Exponential and logarithm functions} · @ref{Category: Complex variables}
| [ << ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
This document was generated by root on July, 13 2009 using texi2html 1.76.