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.
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.
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.
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.
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).
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).
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).
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).
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).
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).
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.