Package sage :: Package functions :: Module transcendental
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Module sage.functions.transcendental

Computation of transcendental functions.
Function Summary
  exponential_integral_1(x, n)
Returns the exponential integral E_1(x).
  gamma(s)
Gamma function at s.
  gamma_inc(s, t)
Incomplete Gamma function Gamma(s,t).
  pi(prec)
  zeta(s)
Riemann zeta function at s with s a real or complex number.
  zeta_symmetric(s)
Completed function xi(s) that satisfies xi(s) = xi(1-s) and has zeros at the same points as the Riemann zeta function.

Variable Summary
ComplexNumber I = 1.0*I

Function Details

exponential_integral_1(x, n=0)

Returns the exponential integral E_1(x). If the optional argument
n is given, computes list of the first n values of the exponential
integral E_1(x*m).

The exponential integral E_1(x) is

         E_1(x) = int_{x}^{\infty} e^{-t}/t dt

INPUT:
    x -- a positive real number

    n -- (default: 0) a nonnegative integer; if nonzero,
         then return a list of values E_1(x*m) for
         m = 1,2,3,...,n.   This is useful, e.g., when
         computing derivatives of L-functions.

OUTPUT:
    float -- if n is 0 (the default)
  or
    list -- list of floats if n > 0  

EXAMPLES:
    >>> exponential_integral_1(2)
    0.048900510708061118
    >>> w = exponential_integral_1(2,4); w
    [0.048900510708061118, 0.0037793524098489067, 0.00036008245216265867, 3.76656228439249e-05]


IMPLEMENTATION: We use the PARI C-library functions eint1 and
veceint1.  
    
REFERENCE: See page 262, Prop 5.6.12, of Cohen's book "A Course
in Computational Algebraic Number Theory".

REMARKS: When called with the optional argument n, the PARI
C-library is fast for values of n up to some bound, then very
very slow.  For example, if x=5, then the computation takes less
than a second for n=800000, and takes "forever" for n=900000.

gamma(s)

Gamma function at s.

gamma_inc(s, t)

Incomplete Gamma function Gamma(s,t).

WARNING: This function requires having Mathematica installed on your computer. If you are reading this and know a reference for how to compute Gamma(s,t), let me know so I (or you) can implement it in SAGE. (Pari has the incomplete Gamma function, but only for s and t real.)

zeta(s)

Riemann zeta function at s with s a real or complex number.
EXAMPLES:
    >> zeta(2)
    1.6449340668482264364724151667

zeta_symmetric(s)

Completed function xi(s) that satisfies xi(s) = xi(1-s) and
has zeros at the same points as the Riemann zeta function.

More precisely, xi(s) = gamma(s/2 + 1) * (s-1) * pi^(-s/2) * zeta(s)
EXAMPLES:
    >> zeta_symmetric(0.7)
    0.49758041465112690357779107524
    >> zeta_symmetric(1-0.7)
    0.49758041465112690357779107525

REFERENCE:
  I copied the definition of xi from
    http://www.math.ubc.ca/~pugh/RiemannZeta/RiemannZetaLong.html

Variable Details

I

Type:
ComplexNumber
Value:
1.0*I                                                                  

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