(directly go to documentation on : Gamma, Zeta, Bernoulli, Euler, LambertW.
)
8. Special functions
In this chapter, special and transcendental mathematical functions are described.
Gamma |
Euler's Gamma function |
Zeta |
Riemann's Zeta function |
Bernoulli |
Bernoulli numbers and polynomials |
Euler |
Euler numbers and polynomials |
LambertW |
Lambert's W function |
Gamma -- Euler's Gamma function
Standard library
Calling format:
Parameters:
x -- expression
number -- expression that can be evaluated to a number
Description:
Gamma(x) is an interface to Euler's Gamma function Gamma(x). It returns exact values on integer and half-integer arguments. N(Gamma(x) takes a numeric parameter and always returns a floating-point number in the current precision.
Note that Euler's constant gamma<=>0.57722 is the lowercase gamma in Yacas.
Examples:
In> Gamma(1.3)
Out> Gamma(1.3);
In> N(Gamma(1.3),30)
Out> 0.897470696306277188493754954771;
In> Gamma(1.5)
Out> Sqrt(Pi)/2;
In> N(Gamma(1.5),30);
Out> 0.88622692545275801364908374167;
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See also:
!
,
N
,
gamma
.
Zeta -- Riemann's Zeta function
Standard library
Calling format:
Parameters:
x -- expression
number -- expression that can be evaluated to a number
Description:
Zeta(x) is an interface to Riemann's Zeta function zeta(s). It returns exact values on integer and half-integer arguments. N(Zeta(x) takes a numeric parameter and always returns a floating-point number in the current precision.
Examples:
In> Precision(30)
Out> True;
In> Zeta(1)
Out> Infinity;
In> Zeta(1.3)
Out> Zeta(1.3);
In> N(Zeta(1.3))
Out> 3.93194921180954422697490751058798;
In> Zeta(2)
Out> Pi^2/6;
In> N(Zeta(2));
Out> 1.64493406684822643647241516664602;
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See also:
!
,
N
.
Bernoulli -- Bernoulli numbers and polynomials
Standard library
Calling format:
Bernoulli(index)
Bernoulli(index, x)
|
Parameters:
x -- expression that will be the variable in the polynomial
index -- expression that can be evaluated to an integer
Description:
Bernoulli(n) evaluates the n-th Bernoulli number. Bernoulli(n, x) returns the n-th Bernoulli polynomial in the variable x. The polynomial is returned in the Horner form.
Example:
In> Bernoulli(20);
Out> -174611/330;
In> Bernoulli(4, x);
Out> ((x-2)*x+1)*x^2-1/30;
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See also:
Gamma
,
Zeta
.
Euler -- Euler numbers and polynomials
Standard library
Calling format:
Euler(index)
Euler(index,x)
|
Parameters:
x -- expression that will be the variable in the polynomial
index -- expression that can be evaluated to an integer
Description:
Euler(n) evaluates the n-th Euler number. Euler(n,x) returns the n-th Euler polynomial in the variable x.
Examples:
In> Euler(6)
Out> -61;
In> A:=Euler(5,x)
Out> (x-1/2)^5+(-10*(x-1/2)^3)/4+(25*(x-1/2))/16;
In> Simplify(A)
Out> (2*x^5-5*x^4+5*x^2-1)/2;
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See also:
Bin
.
LambertW -- Lambert's W function
Standard library
Calling format:
Parameters:
x -- expression, argument of the function
Description:
Lambert's W function is (a multiple-valued, complex function) defined for any (complex) z by
W(z)*Exp(W(z))=z.
This function is sometimes useful to represent solutions of transcendental equations. For example, the equation Ln(x)=3*x can be "solved" by writing x= -3*W(-1/3). It is also possible to take a derivative or integrate this function "explicitly".
For real arguments x, W(x) is real if x>= -Exp(-1).
To compute the numeric value of the principal branch of Lambert's W function for real arguments x>= -Exp(-1) to current precision, one can call N(LambertW(x)) (where the function N tries to approximate its argument with a real value).
Examples:
In> LambertW(0)
Out> 0;
In> N(LambertW(-0.24/Sqrt(3*Pi)))
Out> -0.0851224014;
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See also:
Exp
.