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 `1.0 + I`

or `Pi*I`

but not `2 - 3*I`

), then the
computation is done with the precision of the argument.
In the example below, we see that changing the precision to 50 digits does
not matter, because x only had a precision of 19 digits.

? \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 `realprecision`

(in decimal
digits) or `realbitprecision`

(in bits). This precision can be changed
indifferently

***** in decimal digits: use `\p`

or `default(realprecision,...)`

.

***** in bits: use `\pb`

or `default(realbitprecision,...)`

.

After this conversion, the computation proceeds as above for real or complex arguments.

In library mode, the `realprecision`

does not matter; instead the
precision is taken from the `prec`

parameter which every transcendental
function has. As in `gp`

, this `prec`

is not used when the argument
to a function is already inexact. Note that the argument *prec* stands
for the length in words of a real number, including codewords. Hence we must
have *prec* ≥ 3. (Some functions allow a `bitprec`

argument
instead which allow finer granularity.)

Some accuracies attainable on 32-bit machines cannot be attained
on 64-bit machines for parity reasons. For example, an accuracy
of 28 decimal digits on 32-bit machines corresponds to *prec* having
the value 5, for a mantissa of 3 x 32 = 96 bits. But this cannot be
attained on 64-bit machines: we can attain either 64 or 128 bits, but values
in between.

***** 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 `t_POLMOD`

modulus.

***** If the argument is an intmod or a p-adic, at present only a
few functions like `sqrt`

(square root), `sqr`

(square), `log`

,
`exp`

, powering, `teichmuller`

(Teichmüller character) and
`agm`

(arithmetic-geometric mean) are implemented.

Note that in the case of a 2-adic number, `sqr`

(x) may not be
identical to x*x: for example if x = 1+O(2^{5}) and y = 1+O(2^{5}) then
x*y = 1+O(2^{5}) while `sqr`

(x) = 1+O(2^{6}). Here, x * x yields the
same result as `sqr`

(x) since the two operands are known to be
*identical*. The same statement holds true for p-adics raised to the
power n, where v_{p}(n) > 0.

**Remark.** If we wanted to be strictly consistent with
the PARI philosophy, we should have x*y = (4 mod 8) and `sqr`

(x) =
(4 mod 32) when both x and y are congruent to 2 modulo 4.
However, since intmod is an exact object, PARI assumes that the modulus
must not change, and the result is hence (0 mod 4) in both cases. On
the other hand, p-adics are not exact objects, hence are treated
differently.

***** 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 `seriesprecision`

. This precision
(the number of significant terms) can be changed using `\ps`

or
`default(seriesprecision,...)`

. Then the Taylor series expansion of the
function around X = 0 (where X is the main variable) is computed to a
number of terms depending on the number of terms of the argument and the
function being computed.

Under `gp`

this again is transparent to the user. When programming in
library mode, however, it is *strongly* advised to perform an explicit
conversion to a power series first, as in

x = gtoser(x, gvar(x), seriesprec)

where the number of significant terms `seriesprec`

can be specified
explicitly. If you do not do this, a global variable `precdl`

is used
instead, to convert polynomials and rational functions to a power series with
a reasonable number of terms; tampering with the value of this global
variable is *deprecated* and strongly discouraged.

***** 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, are not built-in. For this you can use the
following technique, which is neither very efficient nor numerical stable,
but is often good enough provided we restrict to diagonalizable matrices:

mateval(f, M) = { my([L, H] = mateigen(M, 1)); H * matdiagonal(f(L)) * H^(-1); } ? A = [13,2;10,14]; ? a = mateval(sqrt, A) /* approximates sqrt{A} */ %2 = [3.5522847498307933... 0.27614237491539669...] [1.3807118745769834... 3.69035593728849174...] ? exponent(a^2 - A) %3 = -123 \\ OK ? b = mateval(exp, A); ? exponent(mateval(log, b) - A) %5 = -115 \\ tolerable

The absolute error depends on the condition number of the base
change matrix H and on the largest |f(λ)|, where λ runs
through the eigenvalues. If M is real symmetric, you may use
`qfjacobi`

instead of `mateigen`

.

Of course when the input is not diagonalizable, this function produces junk:

? mateval(sqrt, [0,1;0,0]) %6 = \\ oops ... [0.E-57 0] [ 0 0]

Catalan's constant G = ∑_{n >= 0}((-1)^{n})/((2n+1)^{2})
= 0.91596....
Note that `Catalan`

is one of the few reserved names which cannot be
used for user variables.

The library syntax is `GEN `

.**mpcatalan**(long prec)

Euler's constant γ = 0.57721.... Note that
`Euler`

is one of the few reserved names which cannot be used for
user variables.

The library syntax is `GEN `

.**mpeuler**(long prec)

The complex number sqrt{-1}.

The library syntax is `GEN `

.**gen _{I}**()

The constant π (3.14159...). Note that `Pi`

is one of the few
reserved names which cannot be used for user variables.

The library syntax is `GEN `

.**mppi**(long prec)

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`

and an exact result is returned if possible.

? 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 `GEN `

.**gabs**(GEN x, long prec)

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 `GEN `

.**gacos**(GEN x, long prec)

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 `GEN `

.**gacosh**(GEN x, long prec)

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 `GEN `

.**agm**(GEN x, GEN y, long prec)

Airy [Ai,Bi] functions of argument z.

? [A,B] = airy(1); ? A %2 = 0.13529241631288141552414742351546630617 ? B %3 = 1.2074235949528712594363788170282869954

The library syntax is `GEN `

.**airy**(GEN z, long prec)

Argument of the complex number x, such that -π < arg(x) ≤ π.

The library syntax is `GEN `

.**garg**(GEN x, long prec)

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 `GEN `

.**gasin**(GEN x, long prec)

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 `GEN `

.**gasinh**(GEN x, long prec)

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 `GEN `

.**gatan**(GEN x, long prec)

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 `GEN `

.**gatanh**(GEN x, long prec)

H^{1}-Bessel function of index *nu* and argument x.

The library syntax is `GEN `

.**hbessel1**(GEN nu, GEN x, long prec)

H^{2}-Bessel function of index *nu* and argument x.

The library syntax is `GEN `

.**hbessel2**(GEN nu, GEN x, long prec)

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 `GEN `

.**ibessel**(GEN nu, GEN x, long prec)

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 `GEN `

.**jbessel**(GEN nu, GEN x, long prec)

J-Bessel function of half integral index.
More precisely, `besseljh`

(n,x) computes J_{n+1/2}(x) where n
must be of type integer, and x is any element of ℂ. In the
present version **2.16.2**, this function is not very accurate when x is small.

The library syntax is `GEN `

.**jbesselh**(GEN n, GEN x, long prec)

k-th zero of the J-Bessel function of index *nu*, close
to π(ν/2 + k - 1/4), usually noted j_{ν,k}.

? besseljzero(0) \\ {first zero of J_{0}} %1 = 2.4048255576957727686216318793264546431 ? besselj(0, %) %2 = 7.1951595399463653939930598011247182898 E-41 ? besseljzero(0, 2) \\ {second zero} %3 = 5.5200781102863106495966041128130274252 ? besseljzero(I) \\ {also works for complex orders, here J_{i}} %4 = 2.5377... + 1.4753...*I

The function uses a Newton iteration due to Temme.
If ν is real and nonnegative, the result is guaranteed and the function really
returns the k-th positive zero of J_{ν}. For general ν, the result
is not well defined, given by the Newton iteration with π(ν/2 + k - 1/4)
as a starting value. (N.B. Using this method for large real ν would give
completely different results than the j_{ν,k} unless k is large enough.)

The library syntax is `GEN `

.**besseljzero**(GEN nu, long k, long bitprec)

K-Bessel function of index *nu* and argument x.

The library syntax is `GEN `

.**kbessel**(GEN nu, GEN x, long prec)

Deprecated alias for `bessely`

.

The library syntax is `GEN `

.**ybessel**(GEN nu, GEN x, long prec)

Y-Bessel function of index *nu* and argument x.

The library syntax is `GEN `

.**ybessel**(GEN nu, GEN x, long prec)

k-th zero of the Y-Bessel function of index *nu*, close
to π(ν/2 + k - 3/4), usually noted y_{ν,k}.

? besselyzero(0) \\ {first zero of Y_{0}} %1 = 0.89357696627916752158488710205833824123 ? bessely(0, %) %2 = 1.8708573650996561952 E-39 ? besselyzero(0, 2) \\ {second zero} %3 = 3.9576784193148578683756771869174012814 ? besselyzero(I) \\ {also works for complex orders, here Y_{i}} %4 = 1.03930... + 1.3266...*I

The function uses a Newton iteration due to Temme.
If ν is real and nonnegative, the result is guaranteed and the function really
returns the k-th positive zero of Y_{ν}. For general ν, the result
is not well defined, given by Newton iteration with π(ν/2 + k - 3/4)
as a starting value. (N.B. Using this method for large real ν would give
completely different results than the y_{ν,k} unless k is large enough.)

The library syntax is `GEN `

.**besselyzero**(GEN nu, long k, long bitprec)

Cosine of x. Note that, for real x, cosine and sine can be obtained simultaneously as

cs(x) = my(z = exp(I*x)); [real(z), imag(z)];

and for general complex x as

cs2(x) = my(z = exp(I*x), u = 1/z); [(z+u)/2, (z-u)/2];

Note that the latter function suffers from catastrophic cancellation
when z^{2} ~ ±1.

The library syntax is `GEN `

.**gcos**(GEN x, long prec)

Hyperbolic cosine of x.

The library syntax is `GEN `

.**gcosh**(GEN x, long prec)

Cotangent of x.

The library syntax is `GEN `

.**gcotan**(GEN x, long prec)

Hyperbolic cotangent of x.

The library syntax is `GEN `

.**gcotanh**(GEN x, long prec)

Principal branch of the dilogarithm of x,
i.e. analytic continuation of the power series
Li_{2}(x) = ∑_{n ≥ 1}x^{n}/n^{2}.

The library syntax is `GEN `

.**dilog**(GEN x, long prec)

Exponential integral ∫_{x}^{ oo } (e^{-t})/(t)dt =
`incgam`

(0, x), where the latter expression extends the function
definition from real x > 0 to all complex x ! = 0.

If n is present, we must have x > 0; the function returns the
n-dimensional vector [`eint1`

(x),...,`eint1`

(nx)]. Contrary to
other transcendental functions, and to the default case (n omitted), the
values are correct up to a bounded *absolute*, rather than relative,
error 10^{-n}, where n is `precision`

(x) if x is a `t_REAL`

and defaults to `realprecision`

otherwise. (In the most important
application, to the computation of L-functions via approximate functional
equations, those values appear as weights in long sums and small individual
relative errors are less useful than controlling the absolute error.) This is
faster than repeatedly calling `eint1(i * x)`

, but less precise.

The library syntax is `GEN `

.
Also available is **veceint1**(GEN x, GEN n = NULL, long prec)`GEN `

.**eint1**(GEN x, long prec)

Complete elliptic integral of the second kind
E(k) = ∫_{0}^{π/2}(1-k^{2}sin(t)^{2})^{1/2}dt for the
complex parameter k using the agm.

In particular, the perimeter of an ellipse of semi-major and semi-minor axes a and b is given by

e = sqrt(1 - (b/a)^2); \\ eccentricity 4 * a * ellE(e) \\ perimeter

The library syntax is `GEN `

.**ellE**(GEN k, long prec)

Complete elliptic integral of the first kind
K(k) = ∫_{0}^{π/2}(1-k^{2}sin(t)^{2})^{-1/2}dt for the
complex parameter k using the agm.

The library syntax is `GEN `

.**ellK**(GEN k, long prec)

Complementary error function, analytic continuation of
(2/sqrtπ)∫_{x}^{ oo } e^{-t^{2}}dt
= {sign(x)}`incgam`

(1/2,x^{2})/sqrtπ for real x ! = 0.
The latter expression extends the function definition from real x to
complex x with positive real part (or zero real part and positive
imaginary part). This is extended to the whole complex plane by
the functional equation `erfc`

(-x) = 2 - `erfc`

(x).

? erfc(0) %1 = 1.0000000000000000000000000000000000000 ? erfc(1) %2 = 0.15729920705028513065877936491739074071 ? erfc(1+I) %3 = -0.31615128169794764488027108024367036903 - 0.19045346923783468628410886196916244244*I

The library syntax is `GEN `

.**gerfc**(GEN x, long prec)

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 `t_PADIC`

, or can be converted to a
*power series* (which must then have positive valuation).

If *flag* is nonzero, 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 `GEN `

.**eta0**(GEN z, long flag, long prec)

Also available is `GEN `

(**trueeta**(GEN x, long prec)*flag* = 1).

Exponential of x. p-adic arguments with positive valuation are accepted.

The library syntax is `GEN `

.
For a **gexp**(GEN x, long prec)`t_PADIC`

x, the function
`GEN `

is also available.**Qp_exp**(GEN x)

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 `expm1`

is recommended instead:

? default(realprecision, 10000); x = 1e-100; ? a = expm1(x); time = 4 ms. ? b = exp(x)-1; time = 4 ms. ? default(realprecision, 10040); x = 1e-100; ? c = expm1(x); \\ reference point ? abs(a-c)/c \\ relative error in expm1(x) %7 = 1.4027986153764843997 E-10019 ? 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,
`expm1`

is more accurate than `exp(x)-1`

.

The library syntax is `GEN `

.**gexpm1**(GEN x, long prec)

For s a complex number, evaluates Euler's gamma
function, which is the analytic continuation of
Γ(s) = ∫_{0}^{ oo } t^{s-1}exp(-t)dt, Re(s) > 0.
Error if s is a nonpositive integer, where Γ has a (simple) pole.

? gamma(5) \\ Γ(n) = (n-1)! for a positive integer n %1 = 24.000000000000000000000000000000000000 ? gamma(0) *** at top-level: gamma(0) *** ^ — — -- *** gamma: domain error in gamma: argument = nonpositive integer ? gamma(x + O(x^3)) %2 = x^-1 - 0.57721566490153286060651209008240243104 + O(x)

For s a `t_PADIC`

, evaluates the Morita gamma function at s, that
is the unique continuous p-adic function on the p-adic integers
extending Γ_{p}(k) = (-1)^{k} ∏_{j < k}'j, where the prime means that
p does not divide j.

? 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 `GEN `

.
For a **ggamma**(GEN s, long prec)`t_PADIC`

x, the function `GEN `

is
also available.**Qp_gamma**(GEN x)

Gamma function evaluated at the argument x+1/2.

The library syntax is `GEN `

.**ggammah**(GEN x, long prec)

Returns the value at t of the inverse Mellin transform
G initialized by `gammamellininvinit`

. If the optional parameter
m is present, return the m-th derivative G^{(m)}(t).

? G = gammamellininvinit([0]); ? gammamellininv(G, 2) - 2*exp(-Pi*2^2) %2 = -4.484155085839414627 E-44

The shortcut

gammamellininv(A,t,m)

for

gammamellininv(gammamellininvinit(A,m), t)

is available.

The library syntax is `GEN `

.**gammamellininv**(GEN G, GEN t, long m, long bitprec)

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 `A`

is the vector [a_{1},...,a_{d}] and
Γ_{ℝ}(s) = π^{-s/2} Γ(s/2) (Euler's `gamma`

).
The result is a vector
[M[1]...M[n]] with M[1] = 1, such that
K^{(m)}(t) = sqrt{2^{d+1}/d}t^{a+m(2/d-1)}e^{-dπ t^{2/d}}
∑_{n ≥ 0} M[n+1] (π t^{2/d})^{-n}
with a = (1-d+∑_{1 ≤ j ≤ d}a_{j})/d. We also allow A to be the
output of `gammamellininvinit`

.

The library syntax is `GEN `

.**gammamellininvasymp**(GEN A, long precdl, long n)

Initialize data for the computation by `gammamellininv`

of
the m-th derivative of the inverse Mellin transform of the function
f(s) = Γ_{ℝ}(s+a_{1})...Γ_{ℝ}(s+a_{d})
where `A`

is the vector [a_{1},...,a_{d}] and
Γ_{ℝ}(s) = π^{-s/2} Γ(s/2) (Euler's `gamma`

). This is the
special case of Meijer's G functions used to compute L-values via the
approximate functional equation. By extension, A is allowed to be an
`Ldata`

or an `Linit`

, understood as the inverse Mellin transform
of the L-function gamma factor.

**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 `GEN `

.**gammamellininvinit**(GEN A, long m, long bitprec)

General hypergeometric function, where `N`

and `D`

are
the vector of parameters in the numerator and denominator respectively,
evaluated at the argument z, which may be complex, p-adic or a power
series.

This function implements hypergeometric functions
_{p}F_{q}((a_{i})_{1 ≤ i ≤ p},(b_{j})_{1 ≤ j ≤ q};z)
= ∑_{n ≥ 0}(∏_{1 ≤ i ≤ p}(a_{i})_{n})/(∏_{1 ≤ j ≤ q}(b_{j})_{n})
(z^{n})/(n!) ,
where (a)_{n} = a(a+1)...(a+n-1) is the rising Pochhammer symbol. For this
to make sense, none of the b_{j} must be a negative or zero integer. The
corresponding general GP command is

hypergeom([a1,a2,...,ap], [b1,b2,...,bq], z)

Whenever p = 1 or q = 1, a one-element vector can be
replaced by the element it contains. Whenever p = 0 or q = 0, an empty
vector can be omitted. For instance hypergeom(,b,z) computes
_{0}F_{1}(;b;z).

The non-archimedean cases (z a p-adic or power series) are handled trivially. We now discuss the case of a complex z; we distinguish three kinds of such functions according to their radius of convergence R:

***** q ≥ p: R = oo .

***** q = p-1: R = 1. Nonetheless, by integral representations, _{p}F_{q}
can be analytically continued outside the disc of convergence.

***** q ≤ p-2: R = 0. By integral representations, one can make sense of
the function in a suitable domain, by analytic continuation.

The list of implemented functions and their domain of validity in our implementation is as follows:

`F01`

: `hypergeom(,a,z)`

(or `[a]`

).
This is essentially a Bessel function and computed as such. R = oo .

`F10`

: `hypergeom(a,,z)`

This is (1-z)^{-a}.

`F11`

: `hypergeom(a,b,z)`

is the Kummer confluent hypergeometric
function, computed by summing the series. R = oo

`F20`

: `hypergeom([a,b],,z)`

. R = 0, computed as
(1)/(Γ(a))∫_{0}^{ oo } t^{a-1}(1-zt)^{-b}e^{-t}dt .

`F21`

: `hypergeom([a,b],c,z)`

(or `[c]`

).
R = 1, extended by
(Γ(c))/(Γ(b)Γ(c-b))
∫_{0}^{1} t^{b-1}(1-t)^{c-b-1}(1-zt)^{a}dt .
This is Gauss's Hypergeometric function, and almost all of the implementation
work is done for this function.

`F31`

: `hypergeom([a,b,c],d,z)`

(or `[d]`

). R = 0, computed as
(1)/(Γ(a))∫_{0}^{ oo } t^{a-1}e^{-t}
_{2}F_{1}(b,c;d;tz)dt .

`F32`

: `hypergeom([a,b,c],[d,e],z)`

. R = 1, extended by
(Γ(e))/(Γ(c)Γ(e-c))
∫_{0}^{1}t^{c-1}(1-t)^{e-c-1}_{2}F_{1}(a,b;d;tz)dt .

For other inputs: if R = oo or R = 1 and |z| < 1- ϵ is not too close to the circle of convergence, we simply sum the series.

? hypergeom([3,2], 3.4, 0.7) \\ 2F1(3,2; 3.4; 0.7) %1 = 7.9999999999999999999999999999999999999 ? a=5/3; T1=hypergeom([1,1,1],[a,a],1) \\ 3F2(1,1,1; a,a; 1) %2 = 3.1958592952314032651578713968927593818 ? T2=hypergeom([2,1,1],[a+1,a+1],1) %3 = 1.6752931349345765309211012564734179541 ? T3=hypergeom([2*a-1,1,1],[a+1,a+1],1) %4 = 1.9721037126267142061807688820853354440 ? T1 + (a-1)^2/(a^2*(2*a-3)) * (T2-2*(a-1)*T3) \\ - gamma(a)^2/((2*a-3)*gamma(2*a-2)) %5 = -1.880790961315660013 E-37 \\ ~ 0

This identity is due to Bercu.

The library syntax is `GEN `

.**hypergeom**(GEN N = NULL, GEN D = NULL, GEN z, long prec)

U-confluent hypergeometric function with complex
parameters a, b, z. Note that _{2}F_{0}(a,b,z)
= (-z)^{-a}U(a, a+1-b, -1/z),

? hyperu(1, 3/2, I) %1 = 0.23219... - 0.80952...*I ? -I * hypergeom([1, 1+1-3/2], [], -1/I) %2 = 0.23219... - 0.80952...*I

The library syntax is `GEN `

.**hyperu**(GEN a, GEN b, GEN z, long prec)

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 `GEN `

.
Also available is **incgam0**(GEN s, GEN x, GEN g = NULL, long prec)`GEN `

.**incgam**(GEN s, GEN x, long prec)

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}^{×}
e^{-t}t^{s-1}dt.

The library syntax is `GEN `

.**incgamc**(GEN s, GEN x, long prec)

Lambert W function, solution of the implicit equation xe^{×} = y.

***** For real inputs y:
If `branch = 0`

, principal branch W_{0} defined for y ≥ -exp(-1).
If `branch = -1`

, branch W_{-1} defined for -exp(-1) ≤ y < 0.

***** For p-adic inputs, p odd: give a solution of xexp(x) = y if y has
positive valuation, of log(x)+x = log(y) otherwise.

***** For 2-adic inputs: give a solution of xexp(x) = y if y has
valuation > 1, of log(x)+x = log(y) otherwise.

**Caveat.**
Complex values of y are also supported but experimental. The other
branches W_{k} for k not equal to 0 or -1 (set `branch`

to k)
are also experimental.

For k ≥ 1, W_{-1-k}(x) = W_{k}(x), and Im(W_{k}(x)) is
close to (π/2)(4k-sign(x)).

The library syntax is `GEN `

.**glambertW**(GEN y, long branch, long prec)

Lerch transcendent Φ(z,s,a) = ∑_{n ≥ 0}z^{n}(n+a)^{-s} and
analytically continued, for reasonable values of the arguments.

The library syntax is `GEN `

.**lerchphi**(GEN z, GEN s, GEN a, long prec)

Lerch zeta function
L(s,a,λ) = ∑_{n ≥ 0}e^{2π iλ n}(n+a)^{-s}
and analytically continued, for reasonable values of the arguments.

The library syntax is `GEN `

.**lerchzeta**(GEN s, GEN a, GEN lam, long prec)

Principal branch of the logarithm of the gamma function of x. This
function is analytic on the complex plane with nonpositive integers
removed, and can have much larger arguments than `gamma`

itself.

For x a power series such that x(0) is not a pole of `gamma`

,
compute the Taylor expansion. (PARI only knows about regular power series
and can't include logarithmic terms.)

? 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

For x a `t_PADIC`

, return the p-adic logΓ_{p} function, which
is the p-adic logarithm of Morita's gamma function for x ∈ ℤ_{p},
and Diamond's function if |x| > 1.

? lngamma(5+O(5^7)) %2 = 4*5^2 + 4*5^3 + 5^4 + 2*5^5 + O(5^6) ? log(gamma(5+O(5^7))) %3 = 4*5^2 + 4*5^3 + 5^4 + 2*5^5 + O(5^6) ? lngamma(1/5+O(5^4)) %4 = 4*5^-1 + 4 + 2*5 + 5^2 + 5^3 + O(5^4) ? gamma(1/5+O(5^4)) *** at top-level: gamma(1/5+O(5^4)) *** ^ — — — — — -- *** gamma: domain error in gamma: v_{p}(x) < 0

The library syntax is `GEN `

.**glngamma**(GEN x, long prec)

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 `GEN `

.
For a **glog**(GEN x, long prec)`t_PADIC`

x, the function
`GEN `

is also available.**Qp_log**(GEN x)

Return log(1+x), computed in a way that is also accurate
when the real part of x is near 0. This is the reciprocal function
of `expm1`

(x) = exp(x)-1.

? default(realprecision, 10000); x = Pi*1e-100; ? (expm1(log1p(x)) - x) / x %2 = -7.668242895059371866 E-10019 ? (log1p(expm1(x)) - x) / x %3 = -7.668242895059371866 E-10019

When x is small, this function is both faster and more accurate than log(1+x):

? \p38 ? x = 1e-20; ? localprec(100); c = log1p(x); \\ reference point ? a = log1p(x); abs((a - c)/c) %6 = 0.E-38 ? b = log(1+x); abs((b - c)/c) \\ slightly less accurate %7 = 1.5930919111324522770 E-38 ? for (i=1,10^5,log1p(x)) time = 81 ms. ? for (i=1,10^5,log(1+x)) time = 100 ms. \\ slower, too

The library syntax is `GEN `

.**glog1p**(GEN x, long prec)

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^{k}B_{k})/(k!)
(log|x|)^{k}Li_{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 `GEN `

.
Also available is
**polylog0**(long m, GEN x, long flag, long prec)`GEN `

(**gpolylog**(long m, GEN x, long prec)*flag* = 0).

For s a vector of positive integers and z a vector of complex
numbers of the same length, returns the multiple polylogarithm value (MPV)
ζ(s_{1},..., s_{r}; z_{1},...,z_{r})
= ∑_{n_{1} > ... > n_{r} > 0}
∏_{1 ≤ i ≤ r}z_{i}^{ni}/n_{i}^{si}.
If z is omitted, assume z = [1,...,1], i.e., Multiple Zeta Value.
More generally, return Yamamoto's interpolation between ordinary multiple
polylogarithms (t = 0) and star polylogarithms (t = 1, using the
condition n_{1} ≥ ... ≥ n_{r} > 0), evaluated at t.

We must have |z_{1}...z_{i}| ≤ 1 for all i, and if s_{1} = 1 we
must have z_{1} ! = 1.

? 8*polylogmult([2,1],[-1,1]) - zeta(3) %1 = 0.E-38

**Warning.** The algorithm used converges when the z_{i} are
± 1. It may not converge as some z_{i} ! = 1 becomes too close to 1,
even at roots of 1 of moderate order:

? polylogmult([2,1], (99+20*I)/101 * [1,1]) *** polylogmult: sorry, polylogmult in this range is not yet implemented. ? polylogmult([2,1], exp(I*Pi/20)* [1,1]) *** polylogmult: sorry, polylogmult in this range is not yet implemented.

More precisely, if y_{i} := 1 / (z_{1}...z_{i}) and
v := min_{i < j; y_{i} ! = 1} |(1 - y_{i}) y_{j}| > 1/4
then the algorithm computes the value up to a 2^{-b} absolute error
in O(k^{2}N) operations on floating point numbers of O(N) bits,
where k = ∑_{i} s_{i} is the weight and N = b / log_{2} (4v).

The library syntax is `GEN `

.
Also available is
**polylogmult_interpolate**(GEN s, GEN z = NULL, GEN t = NULL, long prec)`GEN `

(t is **polylogmult**(GEN s, GEN z, long prec)`NULL`

).

The ψ-function of x, i.e. the logarithmic derivative Γ'(x)/Γ(x).

The library syntax is `GEN `

.**gpsi**(GEN x, long prec)

Return the column vector v of all complex N-th roots of 1, where N is a positive integer. In other words, v[k] = exp(2I(k-1)π/N) for k = 1,..., N. Rational components (e.g., the roots ±1 and ± I) are given exactly, not as floating point numbers:

? rootsof1(4) %1 = [1, I, -1, -I]~ ? rootsof1(3) %2 = [1, -1/2 + 0.866025...*I, -1/2 - 0.866025...*I]~

The library syntax is `GEN `

.**grootsof1**(long N, long prec)

Sine of x. Note that, for real x, cosine and sine can be obtained simultaneously as

cs(x) = my(z = exp(I*x)); [real(z), imag(z)];

and for general complex x as

cs2(x) = my(z = exp(I*x), u = 1/z); [(z+u)/2, (z-u)/2];

Note that the latter function suffers from catastrophic cancellation
when z^{2} ~ ±1.

The library syntax is `GEN `

.**gsin**(GEN x, long prec)

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} = `sinc`

(x/2)^{2} / 2
accurately near x = 0.

The library syntax is `GEN `

.**gsinc**(GEN x, long prec)

Hyperbolic sine of x.

The library syntax is `GEN `

.**gsinh**(GEN x, long prec)

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 `%2`

and `%3`

above).

The library syntax is `GEN `

.**gsqr**(GEN x)

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, `t_PADIC`

and `t_FFELT`

are allowed as arguments. In
the first 2 cases (`t_INTMOD`

, `t_PADIC`

), the square root (if it
exists) which is returned is the one whose first p-adic digit is in the
interval [0,p/2]. For other arguments, the result is undefined.

The library syntax is `GEN `

.
For a **gsqrt**(GEN x, long prec)`t_PADIC`

x, the function
`GEN `

is also available.**Qp_sqrt**(GEN x)

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 `GEN `

.
If x is a **gsqrtn**(GEN x, GEN n, GEN *z = NULL, long prec)`t_PADIC`

, the function
`GEN `

is also available.**Qp_sqrtn**(GEN x, GEN n, GEN *z)

Tangent of x.

The library syntax is `GEN `

.**gtan**(GEN x, long prec)

Hyperbolic tangent of x.

The library syntax is `GEN `

.**gtanh**(GEN x, long prec)

Teichmüller character of the p-adic number x, i.e. the unique
(p-1)-th root of unity congruent to x / p^{vp(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 `teichmuller`

, as the
optional argument `tab`

, to speed up later computations.

? 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) + `hammingweight`

(p))
values of `teichmuller`

are to be computed, then it is worthwile to
initialize:

? 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 precomputation induces a speedup by a factor 1293/ 60 ~ 21.5.

**Caveat.**
If the accuracy of `tab`

(the argument n above) is lower than the
precision of x, the *former* is used, i.e. the cached value is not
refined to higher accuracy. It the accuracy of `tab`

is larger, then
the precision of x is used:

? 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 `GEN `

.**teichmuller**(GEN x, GEN tab = NULL)

Also available are the functions `GEN `

(**teich**(GEN x)`tab`

is
`NULL`

) as well as
`GEN `

.**teichmullerinit**(long p, long n)

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 `GEN `

.**theta**(GEN q, GEN z, long prec)

k-th derivative at z = 0 of `theta`

(q,z).

The library syntax is `GEN `

.**thetanullk**(GEN q, long k, long prec)

`GEN `

returns the vector
of all (d**vecthetanullk**(GEN q, long k, long prec)^{i}θ)/(dz^{i})(q,0) for all odd i = 1, 3,..., 2k-1.
`GEN `

returns
**vecthetanullk_tau**(GEN tau, long k, long prec)`vecthetanullk_tau`

at q = exp(2iπ `tau`

).

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 `ellj`

).
If *flag* = 1, returns
f_{1}(x) = η(x/2)/η(x) {such that}
j = (f_{1}^{24}+16)^{3}/f_{1}^{24}.
Finally, if *flag* = 2, returns
f_{2}(x) = sqrt{2}η(2x)/η(x) {such that}
j = (f_{2}^{24}+16)^{3}/f_{2}^{24}.
Note the identities f^{8} = f_{1}^{8}+f_{2}^{8} and ff_{1}f_{2} = sqrt2.

The library syntax is `GEN `

.
Also available are **weber0**(GEN x, long flag, long prec)`GEN `

,
**weberf**(GEN x, long prec)`GEN `

and **weberf1**(GEN x, long prec)`GEN `

.**weberf2**(GEN x, long prec)

For s ! = 1 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. Power series
are also allowed:

? zeta(2) - Pi^2/6 %1 = 0.E-38 ? zeta(1+x+O(x^3)) %2 = 1.0000000000000000000000000000000000000*x^-1 + \ 0.57721566490153286060651209008240243104 + O(x)

For s ! = 1 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). Power series are not allowed in this case.

? zeta(-3+O(5^10)) %1 = 4*5^-1 + 4 + 3*5 + 4*5^3 + 4*5^5 + 4*5^7 + O(5^9))))) ? (1-5^3) * zeta(-3) %2 = -1.0333333333333333333333333333333333333 ? bestappr(%) %3 = -31/30 ? zeta(-3+O(5^10)) - (-31/30) %4 = O(5^9)

The library syntax is `GEN `

.**gzeta**(GEN s, long prec)

Hurwitz zeta function ζ(s,x) = ∑_{n ≥ 0}(n+x)^{-s} and
analytically continued, with s ! = 1 and x not a negative or zero
integer. Note that ζ(s,1) = ζ(s). s can also be a polynomial,
rational function, or power series. If `der`

is positive, compute the
`der`

'th derivative with respect to s. Note that the derivative
with respect to x is simply -sζ(s+1,x).

? zetahurwitz(Pi,Pi) %1 = 0.056155444497585099925180502385781494484 ? zetahurwitz(2,1) - zeta(2) %2 = -2.350988701644575016 E-38 ? zetahurwitz(Pi,3) - (zeta(Pi)-1-1/2^Pi) %3 = -2.2040519077917890774 E-39 ? zetahurwitz(-7/2,1) - zeta(-7/2) %4 = -2.295887403949780289 E-41 ? zetahurwitz(-2.3,Pi+I*log(2)) %5 = -5.1928369229555125820137832704455696057\ - 6.1349660138824147237884128986232049582*I ? zetahurwitz(-1+x^2+O(x^3),1) %6 = -0.083333333333333333333333333333333333333\ - 0.16542114370045092921391966024278064276*x^2 + O(x^3) ? zetahurwitz(1+x+O(x^4),2) %7 = 1.0000000000000000000000000000000000000*x^-1\ - 0.42278433509846713939348790991759756896\ + 0.072815845483676724860586375874901319138*x + O(x^2) ? zetahurwitz(2,1,2) \\ zeta''(2) %8 = 1.9892802342989010234208586874215163815

The derivative can be used to compute Barnes' multiple gamma functions. For instance:

? mygamma(z)=exp(zetahurwitz(0,z,1)-zeta'(0)); /* Alternate way to compute the gamma function */ ? BarnesG(z)=exp(-zetahurwitz(-1,z,1)+(z-1)*lngamma(z)+zeta'(-1)); /* Barnes G function, satisfying G(z+1)=gamma(z)*G(z): */ ? BarnesG(6)/BarnesG(5) % = 24.000000000000000000000000000000000002

The library syntax is `GEN `

.**zetahurwitz**(GEN s, GEN x, long der, long bitprec)

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}^{-s1}...n_{k}^{-sk}
of length k and weight ∑_{i} s_{i}.
More generally, return Yamamoto's t-MZV interpolation evaluated at t:
for t = 0, this is the ordinary MZV; for t = 1, we obtain the MZSV
star value, with ≥ instead of strict inequalities;
and of course, for t = `'x`

we obtain Yamamoto's one-variable polynomial.

? zetamult([2,1]) - zeta(3) \\ Euler's identity %1 = 0.E-38 ? zetamult([2,1], 1) \\ star value %2 = 2.4041138063191885707994763230228999815 ? zetamult([2,1], 'x) %3 = 1.20205[...]*x + 1.20205[...]

If the bit precision is B, this function runs in time
Õ(k(B+k)^{2}) if t = 0, and Õ(kB^{3}) otherwise.

In addition to the above format (`avec`

), the function
also accepts a binary word format `evec`

(each s_{i} is replaced
by s_{i} bits, all of them 0 but the last one) giving the MZV
representation as an iterated integral, and an `index`

format
(if e is the positive integer attached the `evec`

vector of
bits, the index is the integer e + 2^{k-2}). The function
`zetamultconvert`

allows to pass from one format to the other; the
function `zetamultall`

computes simultaneously all MZVs of weight
∑_{i ≤ k} s_{i} up to n.

The library syntax is `GEN `

.
Also available is **zetamult_interpolate**(GEN s, GEN t = NULL, long prec)`GEN `

for t = 0.**zetamult**(GEN s, long prec)

List of all multiple zeta values (MZVs) for weight s_{1} +...+ s_{r}
up to k. Binary digits of *flag* mean : 0 = star values if set;
1 = values up to to duality if set (see `zetamultdual`

, ignored if
star values); 2 = values of weight k if set (else all values up to weight
k); 3 = return the 2-component vector `[Z, M]`

, where M is the vector
of the corresponding indices m, i.e., such that
`zetamult(M[i])`

= `Z[i]`

. Note that it is necessary to use
`zetamultconvert`

to have the corresponding `avec`

(s_{1},..., s_{r}).

With the default value *flag* = 0, the function returns a vector with 2^{k-1}-1
components whose i-th entry is the MZV of `index`

i (see
`zetamult`

). If the bit precision is B, this function runs in time
O(2^{k} k B^{2}) for an output of size O(2^{k} B).

? Z = zetamultall(5); #Z \\ 2^4 - 1 MZVs of weight <= 5 %1 = 15 ? Z[10] %2 = 0.22881039760335375976874614894168879193 ? zetamultconvert(10) %3 = Vecsmall([3, 2]) \\ {index 10 corresponds to ζ(3,2)} ? zetamult(%) \\ double check %4 = 0.22881039760335375976874614894168879193 ? zetamult(10) \\ we can use the index directly %5 = 0.22881039760335375976874614894168879193

If we use flag bits 1 and 2, we avoid unnecessary computations and copying, saving a potential factor 4: half the values are in lower weight and computing up to duality save another rough factor 2. Unfortunately, the indexing now no longer corresponds to the new shorter vector of MZVs:

? Z = zetamultall(5, 2); #Z \\ up to duality %6 = 9 ? Z = zetamultall(5, 2); #Z \\ only weight 5 %7 = 8 ? Z = zetamultall(5, 2 + 4); #Z \\ both %8 = 4

So how to recover the value attached to index 10 ? Flag bit 3 returns the actual indices used:

? [Z, M] = zetamultall(5, 2 + 8); M \\ other indices were not included %9 = Vecsmall([1, 2, 4, 5, 6, 8, 9, 10, 12]) ? Z[8] \\ index m = 10 is now in M[8] %10 = 0.22881039760335375976874614894168879193 ? [Z, M] = zetamultall(5, 2 + 4 + 8); M %11 = Vecsmall([8, 9, 10, 12]) ? Z[3] \\ index m = 10 is now in M[3] %12 = 0.22881039760335375976874614894168879193

The following construction automates the above programmatically, looking up the MZVs of index 10 ( = ζ(3,2)) in all cases, without inspecting the various index sets M visually:

? Z[vecsearch(M, 10)] \\ works in all the above settings %13 = 0.22881039760335375976874614894168879193

The library syntax is `GEN `

.**zetamultall**(long k, long flag, long prec)

`a`

being either an `evec`

, `avec`

, or index `m`

,
converts into `evec`

(*flag* = 0), `avec`

(*flag* = 1), or
index `m`

(*flag* = 2).

? zetamultconvert(10) %1 = Vecsmall([3, 2]) ? zetamultconvert(13) %2 = Vecsmall([2, 2, 1]) ? zetamultconvert(10, 0) %3 = Vecsmall([0, 0, 1, 0, 1]) ? zetamultconvert(13, 0) %4 = Vecsmall([0, 1, 0, 1, 1])

The last two lines imply that [3,2] and [2,2,1]
are dual (reverse order of bits and swap 0 and 1 in `evec`

form).
Hence they have the same zeta value:

? zetamult([3,2]) %5 = 0.22881039760335375976874614894168879193 ? zetamult([2,2,1]) %6 = 0.22881039760335375976874614894168879193

The library syntax is `GEN `

.**zetamultconvert**(GEN a, long flag)

s being either an `evec`

, `avec`

, or index `m`

,
return the dual sequence in `avec`

format.
The dual of a sequence of length r and weight k has length k-r and
weight k. Duality is an involution and zeta values attached to
dual sequences are the same:

? zetamultdual([4]) %1 = Vecsmall([2, 1, 1]) ? zetamultdual(%) %2 = Vecsmall([4]) ? zetamult(%1) - zetamult(%2) %3 = 0.E-38

In `evec`

form, duality simply reverses the order of bits and swaps 0
and 1:

? zetamultconvert([4], 0) %4 = Vecsmall([0, 0, 0, 1]) ? zetamultconvert([2,1,1], 0) %5 = Vecsmall([0, 1, 1, 1])

The library syntax is `GEN `

.**zetamultdual**(GEN s)