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In this section the exponential and logarithmic functions to
the natural base e are described, as well as the conversion
to the logarithm with respect to any base.
The power series expansions are ez= the sum from n=0 to infinity of (zn/n!)
and ln(1 + z)= the sum from n=1 to infinity of ( - 1)n - 1(zn/n).
Further information on the Dilog and Polylog functions can be found in Lewin
[Lew81].
Given a power series f defined over a real or complex field, return
the exponential power series of f.
Exp(r) : FldReElt -> FldReElt
Given an arbitrary real or complex number c, return the exponential
ec of c. Here c is allowed to be free or of fixed precision,
and the result is in the same field as c.
Given a power series f defined over a real or complex field, return
the logarithm of f. The valuation of f must be zero.
Log(c) : FldComElt -> FldComElt
Given a non-zero real or complex number c, return the
logarithm of c (to the natural base e). The principal value with
imaginary part in ( - π, π] is chosen. The result will be
a complex number, unless the argument is real and positive, in which
case a real number is returned.
Given non-negative real numbers b and r, return
the logarithm logb(r) of a to the base b.
Automatic coercion is applied if necessary.
Dilog(s) : FldReElt -> FldReElt
For a given complex s,
this returns the value of
the principal branch of the dilogarithm Li2(s),
which can be defined by Li2(s)=the negative of the integral from 0 to s of (log(1 - s)/s)ds,
and forms the analytic continuation of the power series sum from n=1 to infinity of (sn/n2),
(which is convergent for |s|≤1).
For large values of the argument a functional equation like
Li2((-1/s)) + Li2( - s)=2Li2( - 1) - (1/2)log2(s)
should be used.
For an integer m≥2 and power series f defined over a real
or complex field, return the m-th polylogarithm of the series f. The
valuation of f must be positive for m>1.
For given integer m≥2 and complex s
this returns the value of
the principal branch of the polylogarithm Lim(s),
defined for m≥3 by Lim(s)= the integral from 0 to s given by (Lim - 1(s)/s)ds
(and for m=2 as the dilogarithm Li2).
Then Lim is the analytic continuation of the sum from n=1 to infinity of (sn/nm),
(which is convergent for |s|≤1).
For large values of the argument a functional equation like
( - 1)mLim((-1/s)) + Lim( - s)= - (1/m!)logm(s) + 2∑r=1⌊m/2⌋(logm - 2r(s)/(m - 2r)!)Li2r( - 1)
should be used.
Pari is used here.
PolylogDold(m, s) : RngIntElt, FldComElt -> FldComElt
PolylogP(m, s) : RngIntElt, FldComElt -> FldComElt
Given integer m≥2 and complex s,
this returns the value of
the principal branch of the modified versions tilde Dm, Dm and
Pm of the polylogarithm Lim(s); all of these satisfy
functional equations of the form fm(1/s)=( - 1)mfm(s).
For their definition and main properties, see Zagier
[Zag91].
Pari is used here.
The trigonometric functions may be computed for real and complex arguments
or for power series defined over a real or complex field. The basic power
series expansions are sin(z)=the sum from n=0 to infinity of (( - 1)n + 1z2n + 1/(2n + 1)!),
and cos(z)=the sum from n=0 to infinity of (( - 1)nz2n/(2n)!).
Euler's formulas relate these with the exponential functions via
sin(z)=(ei z - e - i z/2i), cos(z)=(ei z + e - i z/2).
Given a power series f defined over a real or complex field, return
the power series sin(f).
Sin(r) : FldReElt -> FldReElt
Given a real or complex number c, return the value sin(c).
Given a power series f defined over a real or complex field, return
the power series cos(f).
Cos(r) : FldReElt -> FldReElt
Given a real or complex number c, return the value
cos(c).
Given a power series f defined over a real or complex field, return
the two power series sin(f) and cos(f).
Sincos(s) : FldComElt -> FldComElt, FldComElt
Given a real or complex number s, return the two values
sin(s) and cos(s).
Given a power series f defined over the real or complex field, return
the power series tan(f).
Tan(r) : FldReElt -> FldReElt
Given a real or complex number c, return the value
tan(c)=(sin(c)/cos(c)). Note that c should not
be too close to one of the zeroes (π/2 + n .π) of cos(z).
Given a power series f defined over a real or complex field having
valuation zero, return the power series cot(f).
Cot(r) : FldReElt -> FldReElt
Given a real or complex number c, return the value
cot(c)=(cos(c)/sin(c)). Note that c should not
be too close to one of the zeroes n .π of sin(z).
Given a power series f defined over a real or complex field,
return the power series sec(f).
Sec(r) : FldReElt -> FldReElt
Given a real or complex number c, return the value
sec(c)=1/cos(c). Note that c should not
be too close to one of the zeroes (π/2 + n .π) of cos(z).
Given a power series f defined over a real or complex field having
valuation zero, return the power series cosec(f).
Cosec(r) : FldReElt -> FldReElt
Given a real or complex number c, return the value
cosec(c)=1/sin(c). Note that c should not
be too close to one of the zeroes n .π of sin(z).
The inverse trigonometric functions are all available for arbitrary
real or complex arguments.
The principal values are chosen as
indicated.
We mention the power series expansions for the inverse of the sine and
tangent functions (for |z|≤1):
eqalign(
arcsin(z)&=∑n=0^∞(∏k=12n k^(( - 1)k - 1))(z2n + 1/2n + 1),
arctan(z)&=∑n=1^∞( - 1)n(z2n + 1/2n + 1).
)
The important relations with the logarithmic function include
eqalign(
arcsin(z)&=(1/i)log(i z + Sqrt(1 - z2)),
arccos(z)&=(1/i)log(z + Sqrt(z2 - 1)),
arctan(z)&=(1/2i)log((1 + i z/1 - i z)).
)
Given a power series f defined over a real or complex field. return the
inverse sine of the power series f.
Arcsin(r) : FldComElt -> FldComElt
Given a real or complex number s, return a value
t such that sin(t)=s. The principal value with real part in
[ - π/2, π/2] is chosen. The return value is a complex number,
unless s is real and -1≤s≤1, in which case a free
real number is returned.
Given a power series f defined over a real or complex field. return the
inverse cosine of the power series f.
Arccos(r) : FldComElt -> FldComElt
Given a real or complex number s, return a value
t such that cos(t)=s. The principal value with real part in
[0, π] is chosen. The return value is a complex number,
unless s is real and -1≤s≤1, in which case a free
real number is returned.
Given a power series f defined over the real or complex field, return the
inverse tangent of the power series f.
Arctan(r) : FldComElt -> FldComElt
Given a real or complex number s, return a value
t such that tan(t)=s. The principal value with real part in
( - π/2, π/2) is chosen. The return value is a complex number,
unless s is real, in which case a free
real number is returned.
Arctan2(x, y) : FldReElt, FldReElt -> FldReElt
Given the real numbers x and y, return the value v of arctan(y/x)
determined by the choice of signs for x and y.
That is, the value v is chosen in ( - π, π) in such a way
that the signs of y and sin(v) coincide, as well as the signs
of x and cos(v).
An error occurs if x and y are both zero;
if y is zero and x non-zero, the value returned is
sign(x) .π/2.
The arguments are allowed to be in any real field (automatic coercion is
used whenever necessary). Note that the ordering of x and y differs
from the C math library function.
Arccot(r) : FldComElt -> FldComElt
Given a real or complex number s, return a value
t such that cot(t)=s. The principal value with real part in
( - π/2, π/2) is chosen. The return value is a complex number,
unless s is real, in which case a real number is returned.
Arcsec(r) : FldComElt -> FldComElt
Given a real or complex number s, return a value
t such that sec(t)=s. The principal value with real part in
[0, π/2)∪(π/2, π] is chosen. The return value is a complex
number, unless s is real, in which case a
real number is returned.
Arccosec(r) : FldComElt -> FldComElt
Given a real or complex number s, return a value
t such that cosec(t)=s. The principal value with real part in
[ - π/2, 0)∪(0, π/2] is chosen. The return value is a complex number,
unless s is real, in which case a
real number is returned.
The hyperbolic functions are available for real and complex
arguments, as specified below.
The hyperbolic functions are defined using
eqalign(
sinh(z)&=(ez - e - z/2),
cosh(z)&=(ez + e - z/2).
)
Given a power series f defined over a real or complex field, return the
hyperbolic sine of the power series f.
Sinh(r) : FldReElt -> FldReElt
Given a real or complex number s, return sinh(s).
The result will be a real or complex value, in accordance with the
argument.
Given a power series f defined over a real or complex field, return the
hyperbolic cosine of the power series f.
Cosh(r) : FldComElt -> FldComElt
Given a real or complex number s, return cosh(s).
The result will be a real or complex value, in accordance with the
argument.
Given a power series f defined over a real or complex field, return the
hyperbolic tangent of the power series f.
Tanh(r) : FldComElt -> FldComElt
Given a real or complex number s, return tanh(s)=(sinh(s)
/cosh(s)).
The result will be a real or complex value, in accordance with the
argument.
Coth(r) : FldComElt -> FldComElt
Given a real or complex number s, return coth(s)=(cosh(s)
/sinh(s)).
The result will be a real or complex value, in accordance with the
argument.
Sech(r) : FldComElt -> FldComElt
Given a real or complex number s, return sech(s)=1/cosh(s).
The result will be a real or complex value, in accordance with the
argument.
Cosech(r) : FldComElt -> FldComElt
Given a real or complex number s, return cosech(s)=1/sinh(s).
The result will be a real or complex value, in accordance with the
argument.
The inverse hyperbolic functions are available for
real or complex arguments.
The principal values are chosen as
indicated.
Given a power series f defined over a real or complex field, return the
inverse hyperbolic sine of the power series f.
Argsinh(r) : FldComElt -> FldComElt
Given a real or complex number s, return t such that
sinh(t)=s; the principal value with imaginary part in
[ - π/2, π/2] is chosen. The return value is a complex number, unless
the argument is real, in which case a real number is returned.
Given a power series f defined over a real or complex field, return the
inverse hyperbolic cosine of the power series f.
Argcosh(r) : FldComElt -> FldComElt
Given a real or complex number s, return t such that
cosh(t))=s; the principal value with imaginary part in
[0, π] is chosen. The return value is a complex number, unless
the argument is real and s≥1, in which case a real number is returned.
Given a power series f defined over a real or complex field, return the
inverse hyperbolic tangent of the power series f.
Argtanh(r) : FldComElt -> FldComElt
Given a real or complex number s, return t such that tanh(t)=s; the
principal value with imaginary part in [ - π/2, π/2] is chosen. The return
value is a complex number, unless the argument is real and -1<s<1, in which
case a real number is returned.
Argsech(r) : FldComElt -> FldComElt
Given a real or complex number s, return t such that
sech(t))=s; the principal value with imaginary part in
[0, π] is chosen. The return value is a complex number, unless
the argument is real and |s|≥1, in which case a real number is
returned.
Argcosech(r) : FldComElt -> FldComElt
Given a real or complex number s, return t such that
cosech(t))=s; the principal value with imaginary part in
[ - π/2, π/2] is chosen. The return value is a complex number, unless
the argument is real, in which case a real number is returned.
Argcoth(r) : FldComElt -> FldComElt
Given a real or complex number s, return t such that
coth(t))=s; the principal value with imaginary part in
[ - π/2, π/2] is chosen. The return value is a complex number, unless
the argument is real and 0<s≤1, in which case free real number is returned.
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