Maxima - руководство на русском языке

Constant: %phi

%phi represents the so-called golden-золотое mean-сечение, (1+sqrt(5))/2. The

numeric value of %phi is the double-precision floating-point value 1.618033988749895d0.

fibtophi expresses Fibonacci numbers fib(n) in terms of %phi.

By default, Maxima does not know the algebraic properties of %phi. After evaluating

tellrat(%phi^2-%phi-1) and algebraic: true, ratsimp can simplify some expressions

containing %phi.

Примеры:

fibtophi expresses Fibonacci numbers fib(n) in terms of %phi:

(%i1) fibtophi (fib (n));

n n

%phi - (1 - %phi)

(%o1) -------------------

2 %phi - 1

(%i2) fib (n-1) + fib (n) - fib (n+1);

(%o2) - fib(n + 1) + fib(n) + fib(n - 1)

(%i3) fibtophi (%);

n + 1 n + 1 n n

%phi - (1 - %phi) %phi - (1 - %phi)

(%o3) - --------------------------- + -------------------

2 %phi - 1 2 %phi - 1

n - 1 n - 1

%phi - (1 - %phi)

+ ---------------------------

2 %phi - 1

(%i4) ratsimp (%);

(%o4) 0

By default-по умолчанию, Maxima does not know the algebraic properties of %phi. After

evaluating tellrat(%phi^2-%phi-1) and algebraic: true, ratsimp can simplify some

expressions containing %phi.

Пример:

(%i1) e : expand ((%phi^2 - %phi - 1) * (A + 1));

2 2

(%o1) %phi A - %phi A - A + %phi - %phi - 1

(%i2) ratsimp (e);

2 2

(%o2) (%phi - %phi - 1) A + %phi - %phi - 1

(%i3) tellrat (%phi^2 - %phi - 1);

2

(%o3) [%phi - %phi - 1]

(%i4) algebraic : true;

(%o4) true

(%i5) ratsimp (e);

(%o5) 0

Constant: %pi

%pi represents the ratio-дробь of the perimeter of a circle-круга to its diameter. The

numeric value of %pi is the double-precision floating-point value 3.141592653589793d0.

131

Constant: true

true represents the Boolean constant of the same name. Maxima implements true by

the value T in Lisp.

15. Logarithms — Логарифмы

15.1 Functions and Variables for Logarithms

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.

Функция: li [s] (z)

Represents-представляет the polylogarithm function of order-порядка s and argument z,

defined-определённую by the infinite-бесконечным series-рядом

inf

==== k

\ z

Li (z) = > --

s / s

==== k

k = 1

Функция li[1] is -log(1-z). Функция li [2] and li [3] are the dilogarithm and

trilogarithm functions, respectively-соответственно.

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.

Примеры:

(%i1) assume (x > 0); /* пусть */

(%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) 1.248273182099424 - 6.113257028817992 %i

(%i5) li [3] (7);

(%o5) li (7)

3

132

(%i6) li [2] (7), numer;

(%o6) 1.248273182099424 - 6.113257028817992 %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, 0.26765263908273, 0.58224052646501, 0.97846939293031,

1.644934066848226, 2.190177011441645 - 0.70102614150466 %i,

2.37439527027248 - 1.2738062049196 %i,

2.448686765338203 - 1.758084848210787 %i,

2.46740110027234 - 2.177586090303602 %i]

(%i9) map (lambda ([x], li [3] (x)), L);

(%o9) [0.0, 0.25846139534426, 0.53721319267804, 0.84442580464822,

1.2020569,

1.642866878950322 - 0.07821473130035 %i,

2.060877505514697 - 0.2582419849982 %i,

2.433418896388322 - 0.4919260182323 %i,

2.762071904015935 - 0.75469382859788 %i]

Функция: log (x)

Represents-представляет the natural-натуральный (base-по_основанию e) logarithm-

логарифм of x.

Maxima does not have a built-in function for the base 10 logarithm (десятичного) or other

bases. Можно воспользоваться формулой log10(x):=log(x)/log(10) is a useful

definition.

Simplification-упрощение and evaluation-вычисление of logarithms is governed-

управляется by several-несколькими global-глобальными flags-флагами:

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. Функция (log(1/b), for b integer, always

simplifies.) If it is set to false, all of these simplifications-упрощения will be turned off-

отключены.

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-правило log(-n)->log(n)+%i*%pi for n a

positive-положительное integer-целое.

%e_to_numlog - when-когда true, то 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: 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.

133

Option variable: logarc

Функция: logarc (expr)

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.

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)).

Функция: logcontract (expr)

Recursively-рекурсивно scans-сканирует the expression expr, transforming-

трансформируя subexpressions-подвыражения of the form a1*log(b1)+a2*log(b2)+c

into log(ratsimp(b1^a1* b2^a2))+c

Пример:

(%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)).

Option variable: logexpand

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.

Option variable: lognegint

Default value: false

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

134

Option variable: lognumer

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.

Option variable: logsimp

Default value: true

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

Функция: plog (x)

Represents the principal-направление branch-ветвления of the complex-valued natural

logarithm with -%pi < carg(x) <= +%pi .

16. Trigonometric — Тригонометрия

16.1 Introduction to Trigonometric

Maxima has many trigonometric functions defined. Not all trigonometric identities are

programmed, but it is possible for the user to add many of them using the pattern matching

capabilities of the system. The trigonometric functions defined in Maxima are: acos, acosh,

acot, acoth, acsc, acsch, asec, asech, asin, asinh, atan, atanh, cos, cosh, cot, coth, csc, csch,

sec, sech, sin, sinh, tan, and tanh. There are a number of commands especially for handling

trigonometric functions, see trigexpand, trigreduce, and the switch trigsign. Two share packages

extend the simplification rules built into Maxima, ntrig and atrig1. Do describe(command) for

details.

16.2 Functions and Variables for Trigonometric

Option variable: %piargs

Default value: true

When %piargs is true, trigonometric functions are simplified to algebraic constants when

the argument is an integer multiple of %pi, %pi/2, %pi/3, %pi/4, or %pi/6

Maxima knows some identities which can be applied when %pi, etc., are multiplied by an

integer variable (that is, a symbol declared to be integer).

Примеры:

(%i1) %piargs : false;

(%o1) false

(%i2) [sin (%pi), sin (%pi/2), sin (%pi/3)];

%pi %pi

(%o2) [sin(%pi), sin(---), sin(---)]

2 3

(%i3) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];

%pi %pi %pi

(%o3) [sin(---), sin(---), sin(---)]

4 5 6

(%i4) %piargs : true;

135

(%o4) true

(%i5) [sin (%pi), sin (%pi/2), sin (%pi/3)];

sqrt(3)

(%o5) [0, 1, -------]

2

(%i6) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];

1 %pi 1

(%o6) [-------, sin(---), -]

sqrt(2) 5 2

(%i7) [cos (%pi/3), cos (10*%pi/3), tan (10*%pi/3), cos (sqrt(2)*%pi/3)];

1 1 sqrt(2) %pi

(%o7) [-, - -, sqrt(3), cos(-----------)]

2 2 3

Some identities are applied when %pi and %pi/2 are multiplied by an integer variable:

(%i1) declare (n, integer, m, even);

(%o1) done

(%i2) [sin (%pi * n), cos (%pi * m), sin (%pi/2 * m), cos (%pi/2 * m)];

m/2

(%o2) [0, 1, 0, (- 1) ]

19. Plotting — Графики

19.1 Functions and Variables for Plotting

Функция: contour_plot (expr, x_range, y_range, options, ...)

Plots the contours (curves-кривые of equal-равных value-значений) of expr over the

region x_range by y_range. Any additional arguments are treated the same as in plot3d.

contour_plot only works when the plot format is gnuplot or gnuplot_pipes.

Maxima - руководство на русском языке

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