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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.
‘Category: Exponential and logarithm functions’ ‘Category: Simplification flags and variables’
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
‘Category: Exponential and logarithm functions’ ‘Category: Simplification flags and variables’
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
‘Category: Exponential and logarithm functions’ ‘Category: Evaluation flags’
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))
‘Category: Exponential and logarithm functions’
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]
‘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 an 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.
‘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.
‘Category: Exponential and logarithm functions’ ‘Category: Integral calculus’ ‘Category: Global flags’
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.
‘Category: Exponential and logarithm functions’ ‘Category: Simplification flags and variables’ ‘Category: Simplification functions’
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.
‘Category: Exponential and logarithm functions’ ‘Category: Simplification flags and variables’ ‘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)).
‘Category: Exponential and logarithm functions’ ‘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 )
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)).
‘Category: Exponential and logarithm functions’
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
‘Category: Exponential and logarithm functions’ ‘Category: Simplification flags and variables’
Default value: false
If true implements the rule
log(-n) -> log(n)+%i*%pi for n a positive integer.
‘Category: Exponential and logarithm functions’ ‘Category: Simplification flags and variables’
Default value: true
If false then no simplification of %e to a
power containing log’s is done.
‘Category: Exponential and logarithm functions’ ‘Category: Simplification flags and variables’
Represents the principal branch of the complex-valued natural
logarithm with -%pi < carg(x) <= +%pi .
‘Category: Exponential and logarithm functions’ ‘Category: Complex variables’
The square root of x. It is represented internally by
x^(1/2). See also rootscontract and radexpand.
‘Category: Mathematical functions’
Next: Trigonometric Functions, Previous: Combinatorial Functions, Up: Mathematical Functions [Contents][Index]