5.3.1 Functions and Variables for Constants

Constant: %e ¶

%e represents the base of the natural logarithm, also known as Euler’s number. The numeric value of %e is the double-precision floating-point value 2.718281828459045d0. (See A&S eqn 4.1.16, A&S 4.1.17.)

Categories: Constants ·

Constant: %i ¶

%i represents the imaginary unit, \(\sqrt{-1}\) .

Categories: Constants ·

Constant: false ¶

false represents the Boolean constant of the same name. Maxima implements false by the value NIL in Lisp.

Categories: Constants ·

Constant: %gamma ¶

The Euler-Mascheroni constant, 0.5772156649015329.... It is defined by (A&S eqn 6.1.3 and DLMF 5.2.ii)

\[\gamma = \lim_{n \rightarrow \infty} \left(\sum_{k=1}^n {1\over k} - \log n\right) \]

Categories: Constants ·

Constant: ind ¶

ind represents a bounded, indefinite result.

See also limit.

Example:

(%i1) limit (sin(1/x), x, 0);
(%o1)                          ind

Categories: Constants ·

Constant: inf ¶

inf represents real positive infinity.

Categories: Constants ·

Constant: infinity ¶

infinity represents complex infinity.

Categories: Constants ·

Constant: minf ¶

minf represents real minus (i.e., negative) infinity.

Categories: Constants ·

Constant: %phi ¶

%phi represents the so-called golden mean, \((1+\sqrt{5})/2\) . 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.

Examples:

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, 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.

(%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

Categories: Constants ·

Constant: %pi ¶

%pi represents the ratio of the perimeter of a circle to its diameter. The numeric value of %pi is the double-precision floating-point value 3.141592653589793d0.

Categories: Constants ·

Constant: true ¶

true represents the Boolean constant of the same name. Maxima implements true by the value T in Lisp.

Categories: Constants ·

Constant: und ¶

und represents an undefined result.

See also limit.

Example:

(%i1) limit (x*sin(x), x, inf);
(%o1)                          und

Categories: Constants ·

Constant: zeroa ¶

zeroa represents an infinitesimal above zero. zeroa can be used in expressions. limit simplifies expressions which contain infinitesimals.

See also zerob and limit.

Example:

limit simplifies expressions which contain infinitesimals:

(%i1) limit(zeroa);
(%o1)                           0
(%i2) limit(x+zeroa);
(%o2)                           x

Categories: Constants ·

Constant: zerob ¶

zerob represents an infinitesimal below zero. zerob can be used in expressions. limit simplifies expressions which contain infinitesimals.

See also zeroa and limit.

Categories: Constants ·