# Constants¶

## Yacas-specific constants¶

%

previous result

% evaluates to the previous result on the command line. % is a global variable that is bound to the previous result from the command line. Using % will evaluate the previous result. (This uses the functionality offered by the {SetGlobalLazyVariable} command).

Typical examples are Simplify(%) and PrettyForm(%) to simplify and show the result in a nice form respectively.

Example

In> Taylor(x,0,5)Sin(x)
Out> x-x^3/6+x^5/120;
In> PrettyForm(%)

3    5
x    x
x - -- + ---
6    120

EndOfFile

end-of-file marker

End of file marker when reading from file. If a file contains the expression {EndOfFile;} the operation will stop reading the file at that point.

## Mathematical constants¶

True
False

boolean constants representing true and false

True and False are typically a result of boolean expressions such as 2 < 3 or True And False.

And(), Or(), Not()

Infinity

constant representing mathematical infinity

Infinity represents infinitely large values. It can be the result of certain calculations.

Note that for most analytic functions yacas understands Infinity as a positive number. Thus Infinity*2 will return Infinity, and a < Infinity will evaluate to True.

Example

In> 2*Infinity
Out> Infinity;
In> 2<Infinity
Out> True;

Pi

mathematical constant, $$\pi$$

The constant represents the number π. It is available symbolically as Pi or numerically through N(Pi).

This is a cached constant which is recalculated only when precision is increased.

Example

In> Sin(3*Pi/2)
Out> -1;
In> Pi+1
Out> Pi+1;
In> N(Pi)
Out> 3.14159265358979323846;

Undefined

constant signifying an undefined result

Undefined is a token that can be returned by a function when it considers its input to be invalid or when no meaningful answer can be given. The result is then undefined.

Most functions also return Undefined when evaluated on it.

Example

In> 2*Infinity
Out> Infinity;
In> 0*Infinity
Out> Undefined;
In> Sin(Infinity);
Out> Undefined;
In> Undefined+2*Exp(Undefined);
Out> Undefined;

GoldenRatio

the golden ratio

The constant represents the golden ratio

$\phi := \frac{1+\sqrt{5}}{2} = 1.6180339887\ldots$

It is available symbolically as GoldenRatio or numerically through N(GoldenRatio).

This is a cached constant which is recalculated only when precision is increased.

Example

In> x:=GoldenRatio - 1
Out> GoldenRatio-1;
In> N(x)
Out> 0.6180339887;
In> N(1/GoldenRatio)
Out> 0.6180339887;
In> V(N(GoldenRatio,20));

CachedConstant: Info: constant GoldenRatio is
being recalculated at precision 20
Out> 1.6180339887498948482;


Catalan

Catalan’s constant

The constant represents the Catalan’s constant

$G := \beta(2) = \sum_{n=0}^\infty\frac{-1^n}{(2n+1)^2}=0.9159655941\ldots$

It is available symbolically as Catalan or numerically through N(Catalan).

This is a cached constant which is recalculated only when precision is increased.

Example

In> N(Catalan)
Out> 0.9159655941;
In> DirichletBeta(2)
Out> Catalan;
In> V(N(Catalan,20))

CachedConstant: Info: constant Catalan is
being recalculated at precision 20
Out> 0.91596559417721901505;


gamma

Euler–Mascheroni constant $$\gamma$$

The constant represents the Euler–Mascheroni constant

$\gamma := \lim_{n\to\infty}\left(-\ln(n)+\sum_{k=1}^n\frac{1}{k}\right)=0.5772156649\ldots$

It is available symbolically as gamma or numerically through N(gamma).

This is a cached constant which is recalculated only when precision is increased.

Note

Euler’s $$\Gamma(x)$$ function is the capitalized Gamma() in yacas.

Example

In> gamma+Pi
Out> gamma+Pi;
In> N(gamma+Pi)
Out> 3.7188083184;
In> V(N(gamma,20))

CachedConstant: Info: constant gamma is being
recalculated at precision 20
GammaConstNum: Info: used 56 iterations at
working precision 24
Out> 0.57721566490153286061;