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10.4 Root, Exponential and Logarithmic Functions

Option variable: %e_to_numlog

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.

Option variable: %emode

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
Option variable: %enumer

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.

See also ev and numer.

(%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
Function: exp (x)

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))
Function: li [s] (z)

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]
Function: log (x)

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.

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, 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.

Option variable: logabs

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.

Function: logarc (expr)

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.

Option 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.

Option variable: logconcoeffp

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)).

Function: logcontract (expr)

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)).

Option variable: logexpand

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
Option variable: lognegint

Default value: false

If true implements the rule log(-n) -> log(n)+%i*%pi for n a positive integer.

Option variable: logsimp

Default value: true

If false then no simplification of %e to a power containing log’s is done.

Function: plog (x)

Represents the principal branch of the complex-valued natural logarithm with -%pi < carg(x) <= +%pi .

Function: sqrt (x)

The square root of x. It is represented internally by x^(1/2). See also rootscontract and radexpand.

Categories: Mathematical functions ·

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