Pari/GP Reference Documentation | Contents - Global index - GP keyboard shortcuts |
Catalan Euler I Pi ^ abs acos acosh agm arg asin asinh atan atanh bernfrac bernpol bernreal bernvec besselh1 besselh2 besseli besselj besseljh besselk besseln cos cosh cotan cotanh dilog eint1 erfc eta exp expm1 gamma gammah gammamellininv gammamellininvasymp gammamellininvinit hyperu incgam incgamc lambertw lngamma log polylog psi sin sinc sinh sqr sqrt sqrtn tan tanh teichmuller theta thetanullk weber zeta zetamult | |
Since the values of transcendental functions cannot be exactly represented, these functions will always return an inexact object: a real number, a complex number, a p-adic number or a power series. All these objects have a certain finite precision. As a general rule, which of course in some cases may have exceptions, transcendental functions operate in the following way:
* If the argument is either a real number or an inexact complex number
(like
? \p 15 realprecision = 19 significant digits (15 digits displayed) ? x = Pi/4 %1 = 0.785398163397448 ? \p 50 realprecision = 57 significant digits (50 digits displayed) ? sin(x) %2 = 0.7071067811865475244 Note that even if the argument is real, the result may be complex (e.g. acos(2.0) or acosh(0.0)). See each individual function help for the definition of the branch cuts and choice of principal value.
* If the argument is either an integer, a rational, an exact complex
number or a quadratic number, it is first converted to a real
or complex number using the current precision, which can be
view and manipulated using the defaults
* in decimal digits: use
* in bits: use After this conversion, the computation proceeds as above for real or complex arguments.
In library mode, the
Some accuracies attainable on 32-bit machines cannot be attained
on 64-bit machines for parity reasons. For example the default
* If the argument is a polmod (representing an algebraic number),
then the function is evaluated for every possible complex embedding of that
algebraic number. A column vector of results is returned, with one component
for each complex embedding. Therefore, the number of components equals
the degree of the
* If the argument is an intmod or a p-adic, at present only a
few functions like
Note that in the case of a 2-adic number,
Remark. If we wanted to be strictly consistent with
the PARI philosophy, we should have x*y = (4 mod 8) and
* If the argument is a polynomial, a power series or a rational function,
it is, if necessary, first converted to a power series using the current
series precision, held in the default
Under * If the argument is a vector or a matrix, the result is the componentwise evaluation of the function. In particular, transcendental functions on square matrices, which are not implemented in the present version 2.9.2, will have a different name if they are implemented some day.
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^ | |
The expression x^n is powering.
* If the exponent n is an integer, then exact operations are performed
using binary (left-shift) powering techniques. If x is a p-adic number, its
precision will increase if v_p(n) > 0. Powering a binary quadratic form
(types
PARI is able to rewrite the multiplication x * x of two identical
objects as x^2, or * If the exponent n is not an integer, powering is treated as the transcendental function exp(nlog x), and in particular acts componentwise on vector or matrices, even square matrices ! (See Section se:trans.) * As an exception, if the exponent is a rational number p/q and x an integer modulo a prime or a p-adic number, return a solution y of y^q = x^p if it exists. Currently, q must not have large prime factors. Beware that
? Mod(7,19)^(1/2) %1 = Mod(11, 19) /* is any square root */ ? sqrt(Mod(7,19)) %2 = Mod(8, 19) /* is the smallest square root */ ? Mod(7,19)^(3/5) %3 = Mod(1, 19) ? %3^(5/3) %4 = Mod(1, 19) /* Mod(7,19) is just another cubic root */
* If the exponent is a negative integer, an inverse must be computed.
For non-invertible
? Mod(4,6)^(-1) *** at top-level: Mod(4,6)^(-1) *** ^----- *** _^_: impossible inverse modulo: Mod(2, 6). (Here, a factor 2 is obtained directly. In general, take the gcd of the representative and the modulus.) This is most useful when performing complicated operations modulo an integer N whose factorization is unknown. Either the computation succeeds and all is well, or a factor d is discovered and the computation may be restarted modulo d or N/d.
For non-invertible
? Mod(x^2, x^3-x)^(-1) *** at top-level: Mod(x^2,x^3-x)^(-1) *** ^----- *** _^_: impossible inverse in RgXQ_inv: Mod(x^2, x^3 - x). Note that the underlying algorihm (subresultant) assumes the base ring is a domain:
? a = Mod(3*y^3+1, 4); b = y^6+y^5+y^4+y^3+y^2+y+1; c = Mod(a,b); ? c^(-1) *** at top-level: Mod(a,b)^(-1) *** ^----- *** _^_: impossible inverse modulo: Mod(2, 4). In fact c is invertible, but ℤ/4ℤ is not a domain and the algorithm fails. It is possible for the algorithm to succeed in such situations and any returned result will be correct, but chances are an error will occur first. In this specific case, one should work with 2-adics. In general, one can also try the following approach
? inversemod(a, b) = { my(m, v = variable(b)); m = polsylvestermatrix(polrecip(a), polrecip(b)); m = matinverseimage(m, matid(#m)[,1]); Polrev(m[1..poldegree(b)], v); } ? inversemod(a,b) %2 = Mod(2,4)*y^5 + Mod(3,4)*y^3 + Mod(1,4)*y^2 + Mod(3,4)*y + Mod(2,4)
This is not guaranteed to work either since
For a
? x = Mat([1;2]) %1 = [1] [2] ? x^(-1) %2 = [1 0]
The library syntax is
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Catalan | |
Catalan's constant G = ∑_{n >= 0}((-1)^n)/((2n+1)^2) = 0.91596....
Note that
The library syntax is
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Euler | |
Euler's constant γ = 0.57721.... Note that
The library syntax is
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I | |
The complex number sqrt{-1}.
The library syntax is
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Pi | |
The constant π (3.14159...). Note that
The library syntax is
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abs | |
Absolute value of x (modulus if x is complex).
Rational functions are not allowed. Contrary to most transcendental
functions, an exact argument is not converted to a real number before
applying
? abs(-1) %1 = 1 ? abs(3/7 + 4/7*I) %2 = 5/7 ? abs(1 + I) %3 = 1.414213562373095048801688724 If x is a polynomial, returns -x if the leading coefficient is real and negative else returns x. For a power series, the constant coefficient is considered instead.
The library syntax is
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acos | |
Principal branch of cos^{-1}(x) = -i log (x + isqrt{1-x^2}). In particular, Re(acos(x)) ∈ [0,π] and if x ∈ ℝ and |x| > 1, then acos(x) is complex. The branch cut is in two pieces: ]- oo ,-1] , continuous with quadrant II, and [1,+ oo [, continuous with quadrant IV. We have acos(x) = π/2 - asin(x) for all x.
The library syntax is
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acosh | |
Principal branch of cosh^{-1}(x) = 2 log(sqrt{(x+1)/2} + sqrt{(x-1)/2}). In particular, Re(acosh(x)) ≥ 0 and Im(acosh(x)) ∈ ]-π,π]; if x ∈ ℝ and x < 1, then acosh(x) is complex.
The library syntax is
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agm | |
Arithmetic-geometric mean of x and y. In the case of complex or negative numbers, the optimal AGM is returned (the largest in absolute value over all choices of the signs of the square roots). p-adic or power series arguments are also allowed. Note that a p-adic agm exists only if x/y is congruent to 1 modulo p (modulo 16 for p = 2). x and y cannot both be vectors or matrices.
The library syntax is
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arg | |
Argument of the complex number x, such that -π < arg(x) ≤ π.
The library syntax is
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asin | |
Principal branch of sin^{-1}(x) = -i log(ix + sqrt{1 - x^2}). In particular, Re(asin(x)) ∈ [-π/2,π/2] and if x ∈ ℝ and |x| > 1 then asin(x) is complex. The branch cut is in two pieces: ]- oo ,-1], continuous with quadrant II, and [1,+ oo [ continuous with quadrant IV. The function satisfies i asin(x) = asinh(ix).
The library syntax is
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asinh | |
Principal branch of sinh^{-1}(x) = log(x + sqrt{1+x^2}). In particular Im(asinh(x)) ∈ [-π/2,π/2]. The branch cut is in two pieces: ]-i oo ,-i], continuous with quadrant III and [+i,+i oo [, continuous with quadrant I.
The library syntax is
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atan | |
Principal branch of tan^{-1}(x) = log ((1+ix)/(1-ix)) / 2i. In particular the real part of atan(x) belongs to ]-π/2,π/2[. The branch cut is in two pieces: ]-i oo ,-i[, continuous with quadrant IV, and ]i,+i oo [ continuous with quadrant II. The function satisfies atan(x) = -iatanh(ix) for all x != ± i.
The library syntax is
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atanh | |
Principal branch of tanh^{-1}(x) = log ((1+x)/(1-x)) / 2. In particular the imaginary part of atanh(x) belongs to [-π/2,π/2]; if x ∈ ℝ and |x| > 1 then atanh(x) is complex.
The library syntax is
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bernfrac | |
Bernoulli number B_x, where B_0 = 1, B_1 = -1/2, B_2 = 1/6,..., expressed as a rational number. The argument x should be of type integer.
The library syntax is
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bernpol | |
Bernoulli polynomial B_n in variable v.
? bernpol(1) %1 = x - 1/2 ? bernpol(3) %2 = x^3 - 3/2*x^2 + 1/2*x
The library syntax is
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bernreal | |
Bernoulli number
B_x, as
The library syntax is
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bernvec | |
This routine is obsolete, kept for backward compatibility only.
The library syntax is
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besselh1 | |
H^1-Bessel function of index nu and argument x.
The library syntax is
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besselh2 | |
H^2-Bessel function of index nu and argument x.
The library syntax is
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besseli | |
I-Bessel function of index nu and argument x. If x converts to a power series, the initial factor (x/2)^ν/Γ(ν+1) is omitted (since it cannot be represented in PARI when ν is not integral).
The library syntax is
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besselj | |
J-Bessel function of index nu and argument x. If x converts to a power series, the initial factor (x/2)^ν/Γ(ν+1) is omitted (since it cannot be represented in PARI when ν is not integral).
The library syntax is
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besseljh | |
J-Bessel function of half integral index.
More precisely,
The library syntax is
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besselk | |
K-Bessel function of index nu and argument x.
The library syntax is
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besseln | |
N-Bessel function of index nu and argument x.
The library syntax is
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cos | |
Cosine of x.
The library syntax is
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cosh | |
Hyperbolic cosine of x.
The library syntax is
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cotan | |
Cotangent of x.
The library syntax is
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cotanh | |
Hyperbolic cotangent of x.
The library syntax is
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dilog | |
Principal branch of the dilogarithm of x, i.e. analytic continuation of the power series log_2(x) = ∑_{n ≥ 1}x^n/n^2.
The library syntax is
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eint1 | |
Exponential integral ∫_x^ oo (e^{-t})/(t)dt =
If n is present, we must have x > 0; the function returns the
n-dimensional vector [
The library syntax is
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erfc | |
Complementary error function, analytic continuation of
(2/sqrtπ)∫_x^ oo e^{-t^2}dt =
The library syntax is
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eta | |
Variants of Dedekind's η function. If flag = 0, return ∏_{n = 1}^ oo (1-q^n), where q depends on x in the following way: * q = e^{2iπ x} if x is a complex number (which must then have positive imaginary part); notice that the factor q^{1/24} is missing!
* q = x if x is a If flag is non-zero, x is converted to a complex number and we return the true η function, q^{1/24}∏_{n = 1}^ oo (1-q^n), where q = e^{2iπ x}.
The library syntax is
Also available is
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exp | |
Exponential of x. p-adic arguments with positive valuation are accepted.
The library syntax is
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expm1 | |
Return exp(x)-1, computed in a way that is also accurate
when the real part of x is near 0.
A naive direct computation would suffer from catastrophic cancellation;
PARI's direct computation of exp(x) alleviates this well known problem at
the expense of computing exp(x) to a higher accuracy when x is small.
Using
? default(realprecision, 10000); x = 1e-100; ? a = expm1(x); time = 4 ms. ? b = exp(x)-1; time = 28 ms. ? default(realprecision, 10040); x = 1e-100; ? c = expm1(x); \\ reference point ? abs(a-c)/c \\ relative error in expm1(x) %7 = 0.E-10017 ? abs(b-c)/c \\ relative error in exp(x)-1 %8 = 1.7907031188259675794 E-9919
As the example above shows, when x is near 0,
The library syntax is
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gamma | |
For s a complex number, evaluates Euler's gamma function Γ(s) = ∫_0^ oo t^{s-1}exp(-t)dt. Error if s is a non-positive integer, where Γ has a pole.
For s a
? gamma(1/4 + O(5^10)) %1= 1 + 4*5 + 3*5^4 + 5^6 + 5^7 + 4*5^9 + O(5^10) ? algdep(%,4) %2 = x^4 + 4*x^2 + 5
The library syntax is
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gammah | |
Gamma function evaluated at the argument x+1/2.
The library syntax is
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gammamellininv | |
Returns the value at t of the inverse Mellin transform
G initialized by
? G = gammamellininvinit([0]); ? gammamellininv(G, 2) - 2*exp(-Pi*2^2) %2 = -4.484155085839414627 E-44 The alternative shortcut
gammamellininv(A,t,m) for
gammamellininv(gammamellininvinit(A,m), t) is available.
The library syntax is
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gammamellininvasymp | |
Return the first n terms of the asymptotic expansion at infinity
of the m-th derivative K^{(m)}(t) of the inverse Mellin transform of the
function
f(s) = Γ_ℝ(s+a_1)...Γ_ℝ(s+a_d) ,
where
The library syntax is
| |
gammamellininvinit | |
Initialize data for the computation by Caveat. Contrary to the PARI convention, this function guarantees an absolute (rather than relative) error bound. For instance, the inverse Mellin transform of Γ_ℝ(s) is 2exp(-π z^2):
? G = gammamellininvinit([0]); ? gammamellininv(G, 2) - 2*exp(-Pi*2^2) %2 = -4.484155085839414627 E-44 The inverse Mellin transform of Γ_ℝ(s+1) is 2 zexp(-π z^2), and its second derivative is 4π z exp(-π z^2)(2π z^2 - 3):
? G = gammamellininvinit([1], 2); ? a(z) = 4*Pi*z*exp(-Pi*z^2)*(2*Pi*z^2-3); ? b(z) = gammamellininv(G,z); ? t(z) = b(z) - a(z); ? t(3/2) %3 = -1.4693679385278593850 E-39
The library syntax is
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hyperu | |
U-confluent hypergeometric function with parameters a and b. The parameters a and b can be complex but the present implementation requires x to be positive.
The library syntax is
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incgam | |
Incomplete gamma function ∫_x^ oo e^{-t}t^{s-1}dt, extended by analytic continuation to all complex x, s not both 0. The relative error is bounded in terms of the precision of s (the accuracy of x is ignored when determining the output precision). When g is given, assume that g = Γ(s). For small |x|, this will speed up the computation.
The library syntax is
| |
incgamc | |
Complementary incomplete gamma function. The arguments x and s are complex numbers such that s is not a pole of Γ and |x|/(|s|+1) is not much larger than 1 (otherwise the convergence is very slow). The result returned is ∫_0^x e^{-t}t^{s-1}dt.
The library syntax is
| |
lambertw | |
Lambert W function, solution of the implicit equation xe^x = y, for y > 0.
The library syntax is
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lngamma | |
Principal branch of the logarithm of the gamma function of x. This
function is analytic on the complex plane with non-positive integers
removed, and can have much larger arguments than
For x a power series such that x(0) is not a pole of
? lngamma(1+x+O(x^2)) %1 = -0.57721566490153286060651209008240243104*x + O(x^2) ? lngamma(x+O(x^2)) *** at top-level: lngamma(x+O(x^2)) *** ^----------------- *** lngamma: domain error in lngamma: valuation != 0 ? lngamma(-1+x+O(x^2)) *** lngamma: Warning: normalizing a series with 0 leading term. *** at top-level: lngamma(-1+x+O(x^2)) *** ^-------------------- *** lngamma: domain error in intformal: residue(series, pole) != 0
The library syntax is
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log | |
Principal branch of the natural logarithm of x ∈ ℂ^*, i.e. such that Im(log(x)) ∈ ]-π,π]. The branch cut lies along the negative real axis, continuous with quadrant 2, i.e. such that lim_{b → 0^+} log (a+bi) = log a for a ∈ ℝ^*. The result is complex (with imaginary part equal to π) if x ∈ ℝ and x < 0. In general, the algorithm uses the formula log(x) ~ (π)/(2agm(1, 4/s)) - m log 2, if s = x 2^m is large enough. (The result is exact to B bits provided s > 2^{B/2}.) At low accuracies, the series expansion near 1 is used. p-adic arguments are also accepted for x, with the convention that log(p) = 0. Hence in particular exp(log(x))/x is not in general equal to 1 but to a (p-1)-th root of unity (or ±1 if p = 2) times a power of p.
The library syntax is
| |
polylog | |
One of the different polylogarithms, depending on flag: If flag = 0 or is omitted: m-th polylogarithm of x, i.e. analytic continuation of the power series Li_m(x) = ∑_{n ≥ 1}x^n/n^m (x < 1). Uses the functional equation linking the values at x and 1/x to restrict to the case |x| ≤ 1, then the power series when |x|^2 ≤ 1/2, and the power series expansion in log(x) otherwise. Using flag, computes a modified m-th polylogarithm of x. We use Zagier's notations; let Re_m denote Re or Im depending on whether m is odd or even: If flag = 1: compute ~ D_m(x), defined for |x| ≤ 1 by Re_m(∑_{k = 0}^{m-1} ((-log|x|)^k)/(k!)Li_{m-k}(x) +((-log|x|)^{m-1})/(m!)log|1-x|). If flag = 2: compute D_m(x), defined for |x| ≤ 1 by Re_m(∑_{k = 0}^{m-1}((-log|x|)^k)/(k!)Li_{m-k}(x) -(1)/(2)((-log|x|)^m)/(m!)). If flag = 3: compute P_m(x), defined for |x| ≤ 1 by Re_m(∑_{k = 0}^{m-1}(2^kB_k)/(k!)(log|x|)^kLi_{m-k}(x) -(2^{m-1}B_m)/(m!)(log|x|)^m). These three functions satisfy the functional equation f_m(1/x) = (-1)^{m-1}f_m(x).
The library syntax is
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psi | |
The ψ-function of x, i.e. the logarithmic derivative Γ'(x)/Γ(x).
The library syntax is
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sin | |
Sine of x.
The library syntax is
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sinc | |
Cardinal sine of x, i.e. sin(x)/x if x != 0, 1 otherwise.
Note that this function also allows to compute
(1-cos(x)) / x^2 =
The library syntax is
| |
sinh | |
Hyperbolic sine of x.
The library syntax is
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sqr | |
Square of x. This operation is not completely straightforward, i.e. identical to x * x, since it can usually be computed more efficiently (roughly one-half of the elementary multiplications can be saved). Also, squaring a 2-adic number increases its precision. For example,
? (1 + O(2^4))^2 %1 = 1 + O(2^5) ? (1 + O(2^4)) * (1 + O(2^4)) %2 = 1 + O(2^4) Note that this function is also called whenever one multiplies two objects which are known to be identical, e.g. they are the value of the same variable, or we are computing a power.
? x = (1 + O(2^4)); x * x %3 = 1 + O(2^5) ? (1 + O(2^4))^4 %4 = 1 + O(2^6)
(note the difference between
The library syntax is
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sqrt | |
Principal branch of the square root of x, defined as sqrt{x} = exp(log x / 2). In particular, we have Arg(sqrt(x)) ∈ ]-π/2, π/2], and if x ∈ ℝ and x < 0, then the result is complex with positive imaginary part.
Intmod a prime p,
The library syntax is
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sqrtn | |
Principal branch of the nth root of x, i.e. such that Arg(sqrtn(x)) ∈ ]-π/n, π/n]. Intmod a prime and p-adics are allowed as arguments. If z is present, it is set to a suitable root of unity allowing to recover all the other roots. If it was not possible, z is set to zero. In the case this argument is present and no nth root exist, 0 is returned instead of raising an error.
? sqrtn(Mod(2,7), 2) %1 = Mod(3, 7) ? sqrtn(Mod(2,7), 2, &z); z %2 = Mod(6, 7) ? sqrtn(Mod(2,7), 3) *** at top-level: sqrtn(Mod(2,7),3) *** ^----------------- *** sqrtn: nth-root does not exist in gsqrtn. ? sqrtn(Mod(2,7), 3, &z) %2 = 0 ? z %3 = 0 The following script computes all roots in all possible cases:
sqrtnall(x,n)= { my(V,r,z,r2); r = sqrtn(x,n, &z); if (!z, error("Impossible case in sqrtn")); if (type(x) == "t_INTMOD" || type(x)=="t_PADIC", r2 = r*z; n = 1; while (r2!=r, r2*=z;n++)); V = vector(n); V[1] = r; for(i=2, n, V[i] = V[i-1]*z); V } addhelp(sqrtnall,"sqrtnall(x,n):compute the vector of nth-roots of x");
The library syntax is
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tan | |
Tangent of x.
The library syntax is
| |
tanh | |
Hyperbolic tangent of x.
The library syntax is
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teichmuller | |
Teichmüller character of the p-adic number x, i.e. the unique
(p-1)-th root of unity congruent to x / p^{v_p(x)} modulo p.
If x is of the form [p,n], for a prime p and integer n,
return the lifts to ℤ of the images of i + O(p^n) for
i = 1,..., p-1, i.e. all roots of 1 ordered by residue class modulo
p. Such a vector can be fed back to
? z = teichmuller(2 + O(101^5)) %1 = 2 + 83*101 + 18*101^2 + 69*101^3 + 62*101^4 + O(101^5) ? z^100 %2 = 1 + O(101^5) ? T = teichmuller([101, 5]); ? teichmuller(2 + O(101^5), T) %4 = 2 + 83*101 + 18*101^2 + 69*101^3 + 62*101^4 + O(101^5)
As a rule of thumb, if more than
p / 2(log_2(p) +
? p = 101; n = 100; T = teichmuller([p,n]); \\ instantaneous ? for(i=1,10^3, vector(p-1, i, teichmuller(i+O(p^n), T))) time = 60 ms. ? for(i=1,10^3, vector(p-1, i, teichmuller(i+O(p^n)))) time = 1,293 ms. ? 1 + 2*(log(p)/log(2) + hammingweight(p)) %8 = 22.316[...] Here the precompuation induces a speedup by a factor 1293/ 60 ~ 21.5.
Caveat.
If the accuracy of
? Tlow = teichmuller([101, 2]); \\ lower accuracy ! ? teichmuller(2 + O(101^5), Tlow) %10 = 2 + 83*101 + O(101^5) \\ no longer a root of 1 ? Thigh = teichmuller([101, 10]); \\ higher accuracy ? teichmuller(2 + O(101^5), Thigh) %12 = 2 + 83*101 + 18*101^2 + 69*101^3 + 62*101^4 + O(101^5)
The library syntax is
Also available are the functions
| |
theta | |
Jacobi sine theta-function θ_1(z, q) = 2q^{1/4} ∑_{n ≥ 0} (-1)^n q^{n(n+1)} sin((2n+1)z).
The library syntax is
| |
thetanullk | |
k-th derivative at z = 0 of
The library syntax is
| |
weber | |
One of Weber's three f functions.
If flag = 0, returns
f(x) = exp(-iπ/24).η((x+1)/2)/η(x) {such that}
j = (f^{24}-16)^3/f^{24},
where j is the elliptic j-invariant (see the function
The library syntax is
| |
zeta | |
For s a complex number, Riemann's zeta function ζ(s) = ∑_{n ≥ 1}n^{-s}, computed using the Euler-Maclaurin summation formula, except when s is of type integer, in which case it is computed using Bernoulli numbers for s ≤ 0 or s > 0 and even, and using modular forms for s > 0 and odd. For s a p-adic number, Kubota-Leopoldt zeta function at s, that is the unique continuous p-adic function on the p-adic integers that interpolates the values of (1 - p^{-k}) ζ(k) at negative integers k such that k = 1 (mod p-1) (resp. k is odd) if p is odd (resp. p = 2).
The library syntax is
| |
zetamult | |
For s a vector of positive integers such that s[1] ≥ 2, returns the multiple zeta value (MZV) ζ(s_1,..., s_k) = ∑_{n_1 > ... > n_k > 0} n_1^{-s_1}...n_k^{-s_k}.
? zetamult([2,1]) - zeta(3) \\ Euler's identity %1 = 0.E-38
The library syntax is
| |