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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.
Default value: true
When %emode is true, %e^(%pi %i x) is simplified as
follows.
%e^(%pi %i x) simplifies to cos (%pi x) + %i sin (%pi x) if
x is a floating point number, an integer, or a multiple of 1/2, 1/3, 1/4,
or 1/6, and then further simplified.
For other numerical x, %e^(%pi %i x) simplifies to
%e^(%pi %i y) where y is x - 2 k for some integer k
such that abs(y) < 1.
When %emode is false, no special simplification of
%e^(%pi %i x) is carried out.
(%i1) %emode; (%o1) true
(%i2) %e^(%pi*%i*1); (%o2) - 1
(%i3) %e^(%pi*%i*216/144); (%o3) - %i
(%i4) %e^(%pi*%i*192/144);
sqrt(3) %i 1
(%o4) (- ----------) - -
2 2
(%i5) %e^(%pi*%i*180/144);
%i 1
(%o5) (- -------) - -------
sqrt(2) sqrt(2)
(%i6) %e^(%pi*%i*120/144);
%i sqrt(3)
(%o6) -- - -------
2 2
(%i7) %e^(%pi*%i*121/144);
121 %i %pi
----------
144
(%o7) %e
Default value: false
When %enumer is true, %e is replaced by its numeric value
2.718… whenever numer is true.
When %enumer is false, this substitution is carried out
only if the exponent in %e^x evaluates to a number.
(%i1) %enumer; (%o1) false
(%i2) numer; (%o2) false
(%i3) 2*%e; (%o3) 2 %e
(%i4) %enumer: not %enumer; (%o4) true
(%i5) 2*%e; (%o5) 2 %e
(%i6) numer: not numer; (%o6) true
(%i7) 2*%e; (%o7) 5.43656365691809
(%i8) 2*%e^1; (%o8) 5.43656365691809
(%i9) 2*%e^x;
x
(%o9) 2 2.718281828459045
Represents the exponential function. Instances of exp (x) in input
are simplified to %e^x; exp does not appear in simplified
expressions.
demoivre if true causes %e^(a + b %i) to simplify to
%e^(a (cos(b) + %i sin(b))) if b is free of %i.
See demoivre.
%emode, when true, causes %e^(%pi %i x) to be simplified.
See %emode.
%enumer, when true causes %e to be replaced by
2.718… whenever numer is true. See %enumer.
(%i1) demoivre; (%o1) false
(%i2) %e^(a + b*%i);
%i b + a
(%o2) %e
(%i3) demoivre: not demoivre; (%o3) true
(%i4) %e^(a + b*%i);
a
(%o4) %e (%i sin(b) + cos(b))
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) 1.248273182099423 - 6.113257028817991 %i
(%i5) li [3] (7);
(%o5) li (7)
3
(%i6) li [3] (7), numer; (%o6) 5.319257992145674 - 5.94792444808033 %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) [0.0, 0.2676526390827326, 0.5822405264650125, 0.978469392930306, 1.644934066848226, 2.190177011441645 - 0.7010261415046585 %i, 2.37439527027248 - 1.2738062049196 %i, 2.448686765338205 - 1.758084848210787 %i, 2.467401100272339 - 2.177586090303601 %i]
(%i9) map (lambda ([x], li [3] (x)), L); (%o9) [0.0, 0.2584613953442624, 0.537213192678042, 0.8444258046482203, 1.2020569, 1.642866878950322 - 0.07821473130035025 %i, 2.060877505514697 - 0.2582419849982037 %i, 2.433418896388322 - 0.4919260182322965 %i, 2.762071904015935 - 0.7546938285978846 %i]
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:
logexpandcauses 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.
logsimpif false then no simplification of %e to a power containing
log’s is done.
lognegintif true implements the rule log(-n) -> log(n)+%i*%pi for
n a positive integer.
%e_to_numlogwhen true, r some rational number, and x some expression,
the 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.
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.
The function logarc(expr) carries out the replacement of
inverse circular and hyperbolic functions with equivalent logarithmic
functions for an expression expr without setting the global
variable logarc.
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.
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 )
The declaration declare(n,integer) causes
logcontract(2*a*n*log(x)) to simplify to 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)).
Default value: true
If true, that is the default value, 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.
When logexpand is set to all or super,
the logarithm of a product expression simplifies to a summation of logarithms.
Examples:
When logexpand is true,
log(a^b) simplifies to b*log(a).
(%i1) log(n^2), logexpand=true; (%o1) 2 log(n)
When logexpand is all,
log(a*b) simplifies to log(a)+log(b).
(%i1) log(10*x), logexpand=all; (%o1) log(x) + log(10)
When logexpand is super,
log(a/b) simplifies to log(a)-log(b)
for rational numbers a/b with a#1.
(%i1) log(a/(n + 1)), logexpand=super; (%o1) log(a) - log(n + 1)
When logexpand is set to all or super,
the logarithm of a product expression simplifies to a summation of logarithms.
(%i1) my_product : product (X(i), i, 1, n);
n
/===\
! !
(%o1) ! ! X(i)
! !
i = 1
(%i2) log(my_product), logexpand=all;
n
====
\
(%o2) > log(X(i))
/
====
i = 1
(%i3) log(my_product), logexpand=super;
n
====
\
(%o3) > log(X(i))
/
====
i = 1
When logexpand is false,
these simplifications are disabled.
(%i1) logexpand : false $
(%i2) log(n^2);
2
(%o2) log(n )
(%i3) log(10*x);
(%o3) log(10 x)
(%i4) log(a/(n + 1));
a
(%o4) log(-----)
n + 1
(%i5) log ('product (X(i), i, 1, n));
n
/===\
! !
(%o5) log( ! ! X(i))
! !
i = 1
Default value: false
If true implements the rule
log(-n) -> log(n)+%i*%pi for n a positive integer.
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 .
The square root of x. It is represented internally by
x^(1/2). See also rootscontract and radexpand.
Next: Trigonometric Functions, Previous: Combinatorial Functions, Up: Mathematical Functions [Contents][Index]