[Maxima-commits] CVS: maxima/doc/info/ru Number.texi, NONE, 1.1 maxima.texi, 1.14, 1.15

[Vadim V. Z. <vv...@us...>]

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--- NEW FILE: Number.texi ---
@c Language=Russian
@c Encoding=CP1251
@c File=Number.texi
@c OriginalRevision=1.25
@c TranslatedBy: (c) 2007-09-07 Alexey V. Beshenov <al...@be...>

@menu
* Ôóíêöèè è ïåðåìåííûå äëÿ òåîðèè ÷èñåë::  
@end menu

@node Ôóíêöèè è ïåðåìåííûå äëÿ òåîðèè ÷èñåë,  , Òåîðèÿ ÷èñåë, Òåîðèÿ ÷èñåë
@section Ôóíêöèè è ïåðåìåííûå äëÿ òåîðèè ÷èñåë

@deffn {Ôóíêöèÿ} bern (@var{n})

Âîçâðàùàåò @var{n}-å ÷èñëî Áåðíóëëè äëÿ öåëîãî @var{n}.
×èñëà Áåðíóëëè, ðàâíûå íóëþ, îïóñêàþòñÿ, åñëè @code{zerobern} ðàâíî @code{false}.

Ñì. òàêæå @code{burn}.

@example
(%i1) zerobern: true$
(%i2) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
                  1  1       1      1        1
(%o2)       [1, - -, -, 0, - --, 0, --, 0, - --]
                  2  6       30     42       30
(%i3) zerobern: false$
(%i4) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
            1  1    1   5     691   7    3617  43867
(%o4) [1, - -, -, - --, --, - ----, -, - ----, -----]
            2  6    30  66    2730  6    510    798
@end example

@end deffn

Âîçâðàùàåò çíà÷åíèå ìíîãî÷ëåíà Áåðíóëëè ïîðÿäêà @var{n} â òî÷êå @var{x}.

@end deffn

@deffn {Ôóíêöèÿ} bfzeta (@var{s}, @var{n})

Âîçâðàùàåò äçýòà-ôóíêöèþ Ðèìàíà äëÿ àðãóìåíòà @var{s}.
Âîçâðàùàåìîå çíà÷åíèå - ÷èñëî ñ ïëàâàþùåé òî÷êîé ïîâûøåííîé òî÷íîñòè (bfloat);
@var{n} - êîëè÷åñòâî öèôð â âîçâðàùàåìîì çíà÷åíèè.

Ôóíêöèþ çàãðóæàåò êîìàíäà @code{load ("bffac")}.

@end deffn

@deffn {Ôóíêöèÿ} bfhzeta (@var{s}, @var{h}, @var{n})


Âîçâðàùàåò äçýòà-ôóíêöèþ Ãóðâèöà äëÿ àðãóìåíòîâ @var{s} è @var{h}.
Âîçâðàùàåìîå çíà÷åíèå - ÷èñëî ñ ïëàâàþùåé òî÷êîé ïîâûøåííîé òî÷íîñòè (bfloat);
@var{n} - êîëè÷åñòâî öèôð â âîçâðàùàåìîì çíà÷åíèè.
Äçýòà-ôóíêöèÿ Ãóðâèöà îïðåäåëÿåòñÿ êàê

@example
sum ((k+h)^-s, k, 0, inf)
@end example

Ôóíêöèþ çàãðóæàåò êîìàíäà @code{load ("bffac")}.

@end deffn

@deffn {Ôóíêöèÿ} binomial (@var{x}, @var{y})

Áèíîìèàëüíûé êîýôôèöèåíò @code{@var{x}!/(@var{y}! (@var{x} - @var{y})!)}.

Åñëè @var{x} è @var{y} - öåëûå, ðàññ÷èòûâàåòñÿ ÷èñëåííîå çíà÷åíèå áèíîìèàëüíîãî êîýôôèöèåíòà.
Åñëè @var{y} èëè @var{x - y} - öåëîå, áèíîìèàëüíûé êîýôôèöèåíò âûðàæàåòñÿ ÷åðåç ìíîãî÷ëåí.

Ïðèìåðû:

@c ===beg===
@c binomial (11, 7);
@c 11! / 7! / (11 - 7)!;
@c binomial (x, 7);
@c binomial (x + 7, x);
@c binomial (11, y);
@c ===end===
@example
(%i1) binomial (11, 7);
(%o1)                          330
(%i2) 11! / 7! / (11 - 7)!;
(%o2)                          330
(%i3) binomial (x, 7);
        (x - 6) (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) x
(%o3)   -------------------------------------------------
                              5040
(%i4) binomial (x + 7, x);
      (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x + 6) (x + 7)
(%o4) -------------------------------------------------------
                               5040
(%i5) binomial (11, y);
(%o5)                    binomial(11, y)
@end example

@end deffn

@deffn {Ôóíêöèÿ} burn (@var{n})

Âîçâðàùàåò @var{n}-å ÷èñëî Áåðíóëëè äëÿ öåëîãî @var{n}.
@code{burn} ìîæåò áûòü áîëåå ýôôåêòèâíûì, ÷åì @code{bern} äëÿ îòäåëüíûõ áîëüøèõ @var{n}
(âîçìîæíî, åñëè @var{n} áîëüøå 105 èëè â ðàéîíå ýòîãî), òàê êàê @code{bern} ðàññ÷èòûâàåò âñå
÷èñëà Áåðíóëëè äî @var{n}-ãî ïåðåä âûäà÷åé ðåçóëüòàòà.

@code{burn} èñïîëüçóåò âûðàæåíèå ÷èñåë Áåðíóëëè ÷åðåç äçýòà-ôóíêöèþ Ðèìàíà.

Ôóíêöèþ çàãðóæàåò êîìàíäà @code{load ("bffac")}.

@end deffn

@deffn {Ôóíêöèÿ} cf (@var{expr})

Ïðåîáðàçóåò @var{expr} â öåïíóþ äðîáü.
@var{expr} - âûðàæåíèå, ñîñòàâëåííîå èç öåïíûõ äðîáåé è êâàäðàòíûõ êîðíåé èç öåëûõ ÷èñåë.
Îïåðàíäû âûðàæåíèÿ ìîãóò êîìáèíèðîâàòüñÿ àðèôìåòè÷åñêèìè îïåðàòîðàìè. Ïîìèìî öåïíûõ äðîáåé è
êâàäðàòíûõ êîðíåé, ñîìíîæèòåëè âûðàæåíèÿ äîëæíû áûòü öåëûìè èëè ðàöèîíàëüíûìè ÷èñëàìè. Maxima
íå ðàáîòàåò ñ îïåðàöèÿìè íàä öåïíûìè äðîáÿìè âíå @code{cf}.

@code{cf} âû÷èñëÿåò àðãóìåíòû ïîñëå óñòàíîâêè @code{listarith} ðàâíîé @code{false}.
Öåïíàÿ äðîáü @code{a + 1/(b + 1/(c + ...))} ïðåäñòàâëÿåòñÿ â âèäå ñïèñêà @code{[a, b, c, ...]}.

Ýëåìåíòû ñïèñêà @code{a}, @code{b}, @code{c}, ... äîëæíû ðàñêðûâàòüñÿ â öåëûå ÷èñëà.
@var{expr} ìîæåò ñîäåðæàòü @code{sqrt (n)}, ãäå @code{n} - öåëîå.
 ýòîì ñëó÷àå @code{cf} äàñò ÷èñëî ÷ëåíîâ öåïíîé äðîáè, ðàâíîå ïðîèçâåäåíèþ
@code{cflength} íà ïåðèîä. Öåïíàÿ äðîáü ìîæåò ðàñêðûâàòüñÿ â ÷èñëî ÷åðåç àðèôìåòè÷åñêîå
ïðåäñòàâëåíèå, âîçâðàùåííîå @code{cfdisrep}. Ñì. òàêæå @code{cfexpand} äëÿ äðóãîãî ñïîñîáà
âû÷èñëåíèÿ öåïíîé äðîáè.

Ñì. òàêæå @code{cfdisrep}, @code{cfexpand} è @code{cflength}.

Ïðèìåðû:

@itemize @bullet
@item
@var{expr} - âûðàæåíèå, ñîñòàâëåííîå èç öåïíûõ äðîáåé è êâàäðàòíûõ êîðíåé öåëûõ ÷èñåë.

@example
(%i1) cf ([5, 3, 1]*[11, 9, 7] + [3, 7]/[4, 3, 2]);
(%o1)               [59, 17, 2, 1, 1, 1, 27]
(%i2) cf ((3/17)*[1, -2, 5]/sqrt(11) + (8/13));
(%o2)        [0, 1, 1, 1, 3, 2, 1, 4, 1, 9, 1, 9, 2]
@end example

@item
@code{cflength} îïðåäåëÿåò ÷èñëî ïåðèîäîâ öåïíîé äðîáè, ðàññ÷èòûâàåìûõ äëÿ àëãåáðàè÷åñêèõ
èððàöèîíàëüíûõ ÷èñåë.

@example
(%i1) cflength: 1$
(%i2) cf ((1 + sqrt(5))/2);
(%o2)                    [1, 1, 1, 1, 2]
(%i3) cflength: 2$
(%i4) cf ((1 + sqrt(5))/2);
(%o4)               [1, 1, 1, 1, 1, 1, 1, 2]
(%i5) cflength: 3$
(%i6) cf ((1 + sqrt(5))/2);
(%o6)           [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]
@end example

@item
Öåïíàÿ äðîáü ìîæåò áûòü ðàññ÷èòàíà ÷åðåç àðèôìåòè÷åñêîå ïðåäñòàâëåíèå, âîçâðàùàåìîå
@code{cfdisrep}.

@example
(%i1) cflength: 3$
(%i2) cfdisrep (cf (sqrt (3)))$
(%i3) ev (%, numer);
(%o3)                   1.731707317073171
@end example

@item
Maxima íå ðàáîòàåò ñ îïåðàöèÿìè íàä öåïíûìè äðîáÿìè âíå @code{cf}.

@example
(%i1) cf ([1,1,1,1,1,2] * 3);
(%o1)                     [4, 1, 5, 2]
(%i2) cf ([1,1,1,1,1,2]) * 3;
(%o2)                  [3, 3, 3, 3, 3, 6]
@end example

@end itemize
@end deffn

@deffn {Ôóíêöèÿ} cfdisrep (@var{list})
Âîçâðàùàåò ïðîñòîå âûðàæåíèå âèäà @code{a + 1/(b + 1/(c + ...))} äëÿ ñïèñî÷íîãî
ïðåäñòàâëåíèÿ öåïíîé äðîáè @code{[a, b, c, ...]}.

@example
(%i1) cf ([1, 2, -3] + [1, -2, 1]);
(%o1)                     [1, 1, 1, 2]
(%i2) cfdisrep (%);
                                  1
(%o2)                     1 + ---------
                                    1
                              1 + -----
                                      1
                                  1 + -
                                      2
@end example

@end deffn

@deffn {Ôóíêöèÿ} cfexpand (@var{x})
Âîçâðàùàåò ìàòðèöó ÷èñëèòåëåé è çíàìåíàòåëåé ïîñëåäíåé (ïåðâûé ñòîëáåö) è ïðåäïîñëåäíåé
(âòîðîé ñòîëáåö) ïîäõîäÿùåé äðîáè äëÿ öåïíîé äðîáè @var{x}.

@example
(%i1) cf (rat (ev (%pi, numer)));

`rat' replaced 3.141592653589793 by 103993/33102 =3.141592653011902
(%o1)                  [3, 7, 15, 1, 292]
(%i2) cfexpand (%); 
                         [ 103993  355 ]
(%o2)                    [             ]
                         [ 33102   113 ]
(%i3) %[1,1]/%[2,1], numer;
(%o3)                   3.141592653011902
@end example

@end deffn

@defvr {Óïðàâëÿþùàÿ ïåðåìåííàÿ} cflength
Çíà÷åíèå ïî óìîë÷àíèþ: 1

Ôóíêöèÿ @code{cf} âîçâðàùàåò ÷èñëî ÷ëåíîâ öåïíîé äðîáè, ðàâíîå ïðîèçâåäåíèþ
@code{cflength} íà ïåðèîä. Òàêèì îáðàçîì, ïî óìîë÷àíèþ âîçâðàùàåòñÿ îäèí ïåðèîä.

@example
(%i1) cflength: 1$
(%i2) cf ((1 + sqrt(5))/2);
(%o2)                    [1, 1, 1, 1, 2]
(%i3) cflength: 2$
(%i4) cf ((1 + sqrt(5))/2);
(%o4)               [1, 1, 1, 1, 1, 1, 1, 2]
(%i5) cflength: 3$
(%i6) cf ((1 + sqrt(5))/2);
(%o6)           [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]
@end example

@end defvr

@deffn {Ôóíêöèÿ} divsum (@var{n}, @var{k})
@deffnx {Ôóíêöèÿ} divsum (@var{n})

@code{divsum (@var{n}, @var{k})} âîçâðàùàåò ñóììó äåëèòåëåé @var{n}, âîçâåäåííûõ
â ñòåïåíü @var{k}.

@code{divsum (@var{n})} âîçâðàùàåò ñóììó äåëèòåëåé @var{n}.

@example
(%i1) divsum (12);
(%o1)                          28
(%i2) 1 + 2 + 3 + 4 + 6 + 12;
(%o2)                          28
(%i3) divsum (12, 2);
(%o3)                          210
(%i4) 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2;
(%o4)                          210
@end example

@end deffn

@deffn {Ôóíêöèÿ} euler (@var{n})
Âîçâðàùàåò @var{n}-å ÷èñëî Ýéëåðà äëÿ íåîòðèöàòåëüíîãî öåëîãî @var{n}.
Äëÿ ïîñòîÿííîé Ýéëåðà-Ìàñêåðîíè ñì. @code{%gamma}.

@example
(%i1) map (euler, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
(%o1)    [1, 0, - 1, 0, 5, 0, - 61, 0, 1385, 0, - 50521]
@end example

@end deffn

@defvr {Êîíñòàíòà} %gamma
@ifinfo
@vrindex Euler-Mascheroni constant
@end ifinfo
Ïîñòîÿííàÿ Ýéëåðà-Ìàñêåðîíè, 0.5772156649015329 ....

@end defvr

@deffn {Ôóíêöèÿ} factorial (@var{x})
Ïðåäñòàâëÿåò ôàêòîðèàë @var{x}. Maxima ðàáîòàåò ñ @code{factorial (@var{x})}
àíàëîãè÷íî @code{@var{x}!}. Ñì. @code{!}.

@end deffn

@deffn {Ôóíêöèÿ} fib (@var{n})

Âîçâðàùàåò @var{n}-å ÷èñëî Ôèáîíà÷÷è.
@code{fib(0)} ðàâíî 0, @code{fib(1)} ðàâíî 1, @code{fib (-@var{n})} ðàâíî
@code{(-1)^(@var{n} + 1) * fib(@var{n})}.

Ïîñëå âûçîâà @code{fib} @code{prevfib} ðàâíî @code{fib (@var{x} - 1)},
÷èñëó Ôèáîíà÷÷è, ïðåäøåñòâóþùåìó ïîñëåäíåìó ðàññ÷èòàííîìó.

@example
(%i1) map (fib, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
(%o1)         [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
@end example

@end deffn

@deffn {Ôóíêöèÿ} fibtophi (@var{expr})

Âûðàæàåò ÷èñëà Ôèáîíà÷÷è â @var{expr} ÷åðåç ïîñòîÿííóþ @code{%phi},
ðàâíóþ @code{(1 + sqrt(5))/2}, ïðèáëèçèòåëüíî 1.61803399.

Ïðèìåðû:

@c ===beg===
@c fibtophi (fib (n));
@c fib (n-1) + fib (n) - fib (n+1);
@c fibtophi (%);
@c ratsimp (%);
@c ===end===
@example
(%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
@end example

@end deffn

@deffn {Ôóíêöèÿ} ifactors (@var{n})

Äëÿ öåëîãî ïîëîæèòåëüíîãî @var{n} âîçâðàùàåò ôàêòîðèçàöèþ @var{n}.
Åñëè @code{n=p1^e1..pk^nk} åñòü ðàçëîæåíèå @var{n} íà ïðîñòûå ìíîæèòåëè, @code{ifactors}
âîçâðàùàåò @code{[[p1, e1], ... , [pk, ek]]}.

Èñïîëüçóåìûå ìåòîäû ôàêòîðèçàöèè - îáû÷íîå äåëåíèå íà ïðîñòûå ÷èñëà (äî 9973),
ðî-àëãîðèòì Ïîëëàðäà è ìåòîä ýëëèïòè÷åñêèõ êðèâûõ.

@example
(%i1) ifactors(51575319651600);
(%o1)     [[2, 4], [3, 2], [5, 2], [1583, 1], [9050207, 1]]
(%i2) apply("*", map(lambda([u], u[1]^u[2]), %));
(%o2)                        51575319651600
@end example

@end deffn

@deffn {Ôóíêöèÿ} inrt (@var{x}, @var{n})

Âîçâðàùàåò öåëûé @var{n}-é êîðåíü àáñîëþòíîãî çíà÷åíèÿ @var{x}.

@example
(%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
(%i2) map (lambda ([a], inrt (10^a, 3)), l);
(%o2) [2, 4, 10, 21, 46, 100, 215, 464, 1000, 2154, 4641, 10000]
@end example

@end deffn

@deffn {Ôóíêöèÿ} inv_mod (@var{n}, @var{m})
Ðàññ÷èòûâàåò ÷èñëî, îáðàòíîå @var{n} ïî ìîäóëþ @var{m}.
@code{inv_mod (n,m)} âîçâðàùàåò @code{false}, åñëè @var{n} åñòü äåëèòåëü íóëÿ
ïî ìîäóëþ  @var{m}.

@example
(%i1) inv_mod(3, 41);
(%o1)                           14
(%i2) ratsimp(3^-1), modulus=41;
(%o2)                           14
(%i3) inv_mod(3, 42);
(%o3)                          false
@end example

@end deffn

@deffn {Ôóíêöèÿ} jacobi (@var{p}, @var{q})
Ñèìâîë ßêîáè äëÿ @var{p} è @var{q}.

@example
(%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
(%i2) map (lambda ([a], jacobi (a, 9)), l);
(%o2)         [1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0]
@end example

@end deffn

@deffn {Ôóíêöèÿ} lcm (@var{expr_1}, ..., @var{expr_n})
Âîçâðàùàåò íàèáîëüøèé îáùèé äåëèòåëü àðãóìåíòîâ.
Àðãóìåíòû ìîãóò áûòü êàê öåëûìè ÷èñëàìè, òàê è îáùèìè âûðàæåíèÿìè.

Ôóíêöèþ çàãðóæàåò @code{load ("functs")}.

@end deffn

@deffn {Ôóíêöèÿ} minfactorial (@var{expr})
Ïðîâåðÿåò @var{expr} íà íàëè÷èå äâóõ ôàêòîðèàëîâ, ðàçëè÷àþùèõñÿ íà öåëîå ÷èñëî.
Ïîñëå ýòîãî @code{minfactorial} çàìåíÿåò âûðàæåíèå ïðîèçâåäåíèåì ìíîãî÷ëåíîâ.

@example
(%i1) n!/(n+2)!;
                               n!
(%o1)                       --------
                            (n + 2)!
(%i2) minfactorial (%);
                                1
(%o2)                    ---------------
                         (n + 1) (n + 2)
@end example

@end deffn

@deffn {Ôóíêöèÿ} next_prime (@var{n})
Âîçâðàùàåò íàèìåíüøåå ïðîñòîå ÷èñëî, áîëüøåå @var{n}.

@example
(%i1) next_prime(27);
(%o1)                       29
@end example

@end deffn

@deffn {Ôóíêöèÿ} partfrac (@var{expr}, @var{var})
Ðàçëàãàåò âûðàæåíèå @var{expr} íà ïðîñòûå äðîáè îòíîñèòåëüíî ãëàâíîé ïåðåìåííîé @var{var}.
@code{partfrac} äåëàåò ïîëíîå ðàçëîæåíèå íà ïðîñòûå äðîáè. Èñïîëüçóåìûé àëãîðèòì
îñíîâàí íà òîì, ÷òî çíàìåíàòåëè â ðàçëîæåíèè íà ïðîñòûå äðîáè (ñîìíîæèòåëè
èñõîäíîãî çíàìåíàòåëÿ) âçàèìíî ïðîñòû. ×èñëèòåëè ìîãóò áûòü çàïèñàíû êàê
ëèíåéíûå êîìáèíàöèè çíàìåíàòåëåé, îòêóäà âûòåêàåò ðàçëîæåíèå.

@example
(%i1) 1/(1+x)^2 - 2/(1+x) + 2/(2+x);
                      2       2        1
(%o1)               ----- - ----- + --------
                    x + 2   x + 1          2
                                    (x + 1)
(%i2) ratsimp (%);
                                 x
(%o2)                 - -------------------
                         3      2
                        x  + 4 x  + 5 x + 2
(%i3) partfrac (%, x);
                      2       2        1
(%o3)               ----- - ----- + --------
                    x + 2   x + 1          2
                                    (x + 1)
@end example
@end deffn

@deffn {Ôóíêöèÿ} power_mod (@var{a}, @var{n}, @var{m})
Èñïîëüçóåò ìîäóëÿðíûé àëãîðèòì âû÷èñëåíèÿ @code{a^n mod m}, ãäå @var{a} è @var{n} - öåëûå
è @var{m} - ïîëîæèòåëüíîå öåëîå. Åñëè @var{n} îòðèöàòåëüíî,
äëÿ ïîèñêà îáðàòíîãî ïî ìîäóëþ @var{m} ÷èñëà èñïîëüçóåòñÿ @code{inv_mod}.

@example
(%i1) power_mod(3, 15, 5);
(%o1)                          2
(%i2) mod(3^15,5);
(%o2)                          2
(%i3) power_mod(2, -1, 5);
(%o3)                          3
(%i4) inv_mod(2,5);
(%o4)                          3
@end example

@end deffn

@deffn {Ôóíêöèÿ} primep (@var{n})
Ïðîâåðêà íà ïðîñòîòó. Åñëè @code{primep (@var{n})} âîçâðàùàåò @code{false}, òî @var{n}
ÿâëÿåòñÿ ñîñòàâíûì ÷èñëîì; åñëè âîçâðàùàåò @code{true}, òî @var{n} ñ áîëüøîé
âåðîÿòíîñòüþ ÿâëÿåòñÿ ïðîñòûì ÷èñëîì.

Äëÿ @var{n} ìåíüøå 341550071728321 èñïîëüçóåòñÿ äåòåðìèíèðîâàííàÿ âåðñèÿ òåñòà
Ìèëëåðà-Ðàáèíà. Åñëè @code{primep (@var{n})} âîçâðàùàåò @code{true}, òî @var{n} åñòü ïðîñòîå ÷èñëî.

Äëÿ @var{n} áîëüøå 34155071728321 @code{primep} èñïîëüçóåò @code{primep_number_of_tests}
òåñòîâ Ìèëëåðà-Ðàáèíà íà ïñåâäîïðîñòîòó è îäèí òåñò Ëþêàñà íà ïñåâäîïðîñòîòó.
Âåðîÿòíîñòü òîãî, ÷òî @var{n} ïðîéäåò îäèí òåñò Ìèëëåðà-Ðàáèíà, ìåíåå 1/4. Äëÿ çíà÷åíèÿ ïî
óìîë÷àíèþ 25 ïåðåìåííîé @code{primep_number_of_tests} âåðîÿòíîñòü òîãî, ÷òî @var{n}
áóäåò ñîñòàâíûì, ìíîãî ìåíüøå 10^-15.

@end deffn

@defvr {Óïðàâëÿþùàÿ ïåðåìåííàÿ} primep_number_of_tests
Çíà÷åíèå ïî óìîë÷àíèþ: 25

×èñëî òåñòîâ Ìèëëåðà-Ðàáèíà, èñïîëüçóåìûõ â @code{primep}.
@end defvr

@deffn {Ôóíêöèÿ} prev_prime (@var{n})
Âîçâðàùàåò íàèáîëüøåå ïðîñòîå ÷èñëî, ìåíüøåå @var{n}.

@example
(%i1) prev_prime(27);
(%o1)                       23
@end example
@end deffn

@deffn {Ôóíêöèÿ} qunit (@var{n})
Âîçâðàùàåò ýëåìåíò ïîëÿ @code{sqrt (@var{n})} ñ åäèíè÷íîé íîðìîé, ÷òî ðàâíîñèëüíî
ðåøåíèþ óðàâíåíèÿ Ïåëëÿ @code{a^2 - @var{n} b^2 = 1}.

@example
(%i1) qunit (17);
(%o1)                     sqrt(17) + 4
(%i2) expand (% * (sqrt(17) - 4));
(%o2)                           1
@end example

@end deffn

@deffn {Ôóíêöèÿ} totient (@var{n})
Âîçâðàùàåò ÷èñëî öåëûõ ÷èñåë, ìåíüøèõ èëè ðàâíûõ @var{n}, êîòîðûå âçàèìíî ïðîñòû ñ @var{n}.

@end deffn

@defvr {Óïðàâëÿþùàÿ ïåðåìåííàÿ} zerobern
Çíà÷åíèå ïî óìîë÷àíèþ: @code{true}

Åñëè @code{zerobern} ðàâíî @code{false}, @code{bern} èñêëþ÷àåò ÷èñëà Áåðíóëëè, ðàâíûå íóëþ.
Ñì. @code{bern}.

@end defvr

@deffn {Ôóíêöèÿ} zeta (@var{n})
Âîçâðàùàåò äçýòà-ôóíêöèþ Ðèìàíà, åñëè @var{n} - îòðèöàòåëüíîå öåëîå, 0, 1,
èëè ïîëîæèòåëüíîå ÷åòíîå ÷èñëî, è âîçâðàùàåò íåâû÷èñëÿåìóþ ôîðìó @code{zeta (@var{n})}
äëÿ âñåõ äðóãèõ àðãóìåíòîâ, âêëþ÷àÿ íå öåëûå ðàöèîíàëüíûå, ÷èñëà ñ ïëàâàþùåé òî÷êîé è
êîìïëåêñíûå.

Ñì. òàêæå @code{bfzeta} è @code{zeta%pi}.

@example
(%i1) map (zeta, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5]);
                                     2              4
           1        1     1       %pi            %pi
(%o1) [0, ---, 0, - --, - -, inf, ----, zeta(3), ----, zeta(5)]
          120       12    2        6              90
@end example

@end deffn

@defvr {Óïðàâëÿþùàÿ ïåðåìåííàÿ} zeta%pi
Çíà÷åíèå ïî óìîë÷àíèþ: @code{true}


Çíà÷åíèå ïî óìîë÷àíèþ: @code{true}

Åñëè @code{zeta%pi} ðàâíî @code{true}, @code{zeta} âîçâðàùàåò âûðàæåíèå,
ïðîïîðöèîíàëüíîå @code{%pi^n} äëÿ öåëîãî ÷åòíîãî @code{n}.  ïðîòèâíîì ñëó÷àå äëÿ
öåëîãî ÷åòíîãî @code{n} âîçâðàùàåòñÿ íåâû÷èñëÿåìàÿ ôîðìà @code{zeta (n)}.

@example
(%i1) zeta%pi: true$
(%i2) zeta (4);
                                 4
                              %pi
(%o2)                         ----
                               90
(%i3) zeta%pi: false$
(%i4) zeta (4);
(%o4)                        zeta(4)
@end example

@end defvr

Index: maxima.texi
===================================================================
RCS file: /cvsroot/maxima/maxima/doc/info/ru/maxima.texi,v
retrieving revision 1.14
retrieving revision 1.15
diff -u -d -r1.14 -r1.15
--- maxima.texi	7 Oct 2007 13:08:05 -0000	1.14
+++ maxima.texi	7 Oct 2007 15:01:38 -0000	1.15
@@ -128,7 +128,7 @@
 * Ïàêåò ctensor::                 Êîìïîíåíòíûå òåíçîðíûå âû÷èñëåíèÿ.
 * Ïàêåò atensor::                 Âû÷èñëåíèÿ ñ òåíçîðíûìè àëãåáðàìè.
 * Series::                        Taylor and power series.
-* Number Theory::                 Number theory.
+* Òåîðèÿ ÷èñåë::                  Òåîðèÿ ÷èñåë.
 * Symmetries::                  
 * Groups::                        Abstract algebra.
 
@@ -261,7 +261,7 @@
 Òðèãîíîìåòðèÿ
 
 * Òðèãîíîìåòðèÿ â Maxima::  
-* Ôóíêöèè è ïåðåìåííûå èç òðèãîíîìåòðèè::  
+* Ôóíêöèè è ïåðåìåííûå äëÿ òðèãîíîìåòðèè::  
 
 Special Functions
 
@@ -339,9 +339,9 @@
 * Introduction to Series::      
 * Definitions for Series::      
 
-Number Theory
+Òåîðèÿ ÷èñåë
 
-* Definitions for Number Theory::  
+* Ôóíêöèè è ïåðåìåííûå äëÿ òåîðèè ÷èñåë::  
 
 Symmetries