Module: sage.functions.transcendental
Module-level Functions
x) |
x) |
x, [n=0]) |
Returns the exponential integral
. If the optional argument
is given, computes list of the first
values of the exponential
integral
.
The exponential integral
is
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
sage: exponential_integral_1(2) 0.048900510708061118 sage: 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.
s) |
Gamma function at s.
s, t) |
Incomplete Gamma function Gamma(s,t).
s, t) |
Incomplete Gamma function Gamma(s,t).
s) |
Riemann zeta function at s with s a real or complex number.
INPUT: s -- real or complex number
If s is a real number the computation is done using the MPFR library. When the input is not real, the computation is done using the PARI C library.
sage: zeta(2) 1.6449340668482264 sage: RR = RealField(200) sage: zeta(RR(2)) 1.6449340668482264364724151666460251892189499012067984377355578
s) |
Completed function
that satisfies
and
has zeros at the same points as the Riemann zeta function.
INPUT: s -- real or complex number
If s is a real number the computation is done using the MPFR library. When the input is not real, the computation is done using the PARI C library.
More precisely,
sage: zeta_symmetric(0.7) 0.49758041465112701 sage: zeta_symmetric(1-0.7) 0.49758041465112696 sage: RR = RealField(200) sage: zeta_symmetric(RR('0.7')) 0.49758041465112690357779107525638385212657443284080589766061548 sage: I = CC.0 sage: zeta_symmetric(RR('0.5') + I*RR('14.0')) 0.00020129444423525752 + 0.000000000000000000060986372202309624*I sage: zeta_symmetric(RR('0.5') + I*RR('14.1')) 0.000048989348325568226 + 0.0000000000000000000084703294725430034*I sage: zeta_symmetric(RR('0.5') + I*RR('14.2')) -0.000086893128262009693 - 0.000000000000000000035575383784680614*I
REFERENCE: I copied the definition of xi from http://www.math.ubc.ca/~pugh/RiemannZeta/RiemannZetaLong.html
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