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#### 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, \(\sqrt{-1}\).

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Constant: false

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

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Constant: %gamma

  The Euler-Mascheroni constant, 0.5772156649015329.... It is defined by (A&S eqn 6.1.3 and DLMF 5.2.ii) $$ \gamma = \lim_{n \rightarrow \infty} \left(\sum_{k=1}^n {1\over k} - \log n\right) $$

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Constant: ind

  `ind` represents a bounded, indefinite result.

  See also `limit`.

  Example:

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

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Constant: inf

  `inf` represents real positive infinity.

  Categories:Constants ·

Constant: infinity

  `infinity` represents complex infinity.

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Constant: least_negative_float

  The least negative floating-point number in Maxima. That is, the negative floating-point number closest to 0. It is approximately -4.94065e-324, when denormal numbers are supported. Otherwise it is the same as `least_negative_normalized_float`.

  Categories:Constants ·

Constant: least_negative_normalized_float

  The least negative normalized floating-point number in Maxima. That is, the negative normalized floating-point number closest to 0. It is approximately -2.22507e-308.

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Constant: least_positive_float

  The least positive floating-point number in Maxima. That is, the positive floating-point number closest to 0. It is approximately 4.94065e-324, when denormal numbers are supported. Otherwise it is the same as `least_positive_normalized_float`.

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Constant: least_positive_normalized_float

  The least positive normalized floating-point number in Maxima. That is, the positive normalized floating-point number closest to 0. It is approximately 2.22507e-308.

  Categories:Constants ·

Constant: minf

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

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Constant: most_negative_float

  The most negative floating-point number in Maxima. It is approximately -1.79769e+308.

  Categories:Constants ·

Constant: most_positive_float

  The most positive floating-point number in Maxima. It is approximately 1.797693e+308.

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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.

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Constant: true

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

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Constant: und

  `und` represents an undefined result.

  See also `limit`.

  Example:

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

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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 ·

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