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Package sage :: Package functions :: Module transcendental |
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Function Summary | |
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Returns the exponential integral E_1(x). | |
Gamma function at s. | |
Incomplete Gamma function Gamma(s,t). | |
pi(prec)
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Riemann zeta function at s with s a real or complex number. | |
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 | |
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ComplexNumber |
I = 1.0*I
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Function Details |
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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 |
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I
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